Pre-Calculus Lessons

Course Lesson List

Below are the lessons for Pre-Calculus. View All Lessons » to see all lessons from all courses.

Pre-Calculus

Advanced Function Analysis - Properties of Functions

A Review of Function Basics

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Properties of Functions

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Tags

  • Review
  • Functions

Priority: High

Calculus is almost entirely about working with functions, so to begin a Calculus readiness course like Pre-Calculus, we'll start by making sure we're up to speed on what we are already supposed to know.

  • Review what functions are and how they work
  • Recap the knowledge we gained in Algebra Two about function notation, and domain and range
Function Composition

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Properties of Functions

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Tags

  • Functions
  • Function Analysis

Priority: VIP Knowledge

Function composition is the "Inception" of functions - a function within a function. At first glance, function composition confuses many students, just as the movie equivalent might. But once you get familiar with what it means, you'll be able to work with these the same way you work with standalone functions.

  • Understand conceptually how function composition works
  • Navigate function notation for function composition, using what you already know about notation
  • Work with variable expression inputs and/or work backwards to an unknown input from a known output
  • Be able to break apart a complex function into a series of function compositions of simpler functions
Even and Odd Functions

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Properties of Functions

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Tags

  • Functions
  • Function Analysis

Priority: Normal

Whether or not your teacher is going to test you on this property of functions outright, in Calculus it is occasionally necessary to know whether a function is odd, even, or neither. This lesson will help you understand exactly what that means, as well as the best tips and shortcuts for knowing whether or not a function is indeed odd, even, or neither.

  • Know what is means to say a function is even or odd, algebraically
  • Understand the implications that being odd or even has on a function's graph
  • Learn shortcuts to determine whether a polynomial function is even, odd, or neither
  • See and remember which other common functions we know are even or odd
One-to-one Functions

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Properties of Functions

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Tags

  • Functions
  • Function Analysis

Priority: Normal

One-to-one functions are slightly easier to study than many-to-one functions, and having this quality (or not having this quality) makes the difference between whether we can or cannot work with it in the next lesson. So while this isn't the most earth-shattering function property, it is worth knowing and understanding - especially in the context of studying function inverses in the next lesson.

  • Define conceptually what it means for a function to be "one-to-one"
  • Be able to identify one-to-one functions from a function's graph using visual clues
  • Understand what will be true about a function definition algebraically if it is one-to-one
Inverse Functions

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Properties of Functions

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Tags

  • Functions
  • Function Analysis

Priority: High

Two functions are inverses of each other if they "undo" one another. This lesson will define what this means using function language, and help us identify what types of functions do and do not have inverses.

  • See how we write function inverses using function notation
  • Be able to tell whether or not a function has an inverse based on its graph
  • Find function inverses algebraically by swapping $x$ and $y$
  • Make a connection between the domain of a function and the range of that function's inverse
  • Know the relationship between the graph of a function and the graph of its inverse
Misc Function Characteristics

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Properties of Functions

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Tags

  • Functions
  • Function Analysis

Priority: Optional

This lesson covers the function characteristics of continuous, smooth, and monotone slope, and recaps what you recently learned about one-to-one and even/odd functions by looking at all of these properties simultaneously.

  • Learn what the terms "monotone increasing" and "monotone decreasing" refer to for functions
  • Review the function property of "continuous" and add on a stronger property called "smooth"
  • Think about functions of y instead of our typical functions of x
Periodic Functions

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Properties of Functions

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Tags

  • Functions
  • Function Analysis

Priority: Optional

While many functions we study extend left and right forever, periodic ones are special in that they repeat over and over, and do not approach very large or small values for very large or small $x$. This lesson will show you all you should know about studying periodic functions (at least, until we get to trigonometry).

  • Define what periodic functions are
  • Understand both graphical and algebraic techniques for finding the "wavelength" or "period"
  • Characterize periodic functions as even, odd, or neither

Advanced Function Analysis - Working with Functions

Arithmetic of Functions

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Working with Functions

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Tags

  • Functions
  • Function Analysis
  • Algebra Manipulation

Priority: Normal

Just like variables, functions can be added, subtracted, multiplied, or divided with one another. This lesson shows you how to do that, and looks at what we might expect for results.

  • Learn to add, subtract, multiply, and divide two functions similar to how you would work with two variables
  • Understand why these operations create new functions (not relations) and how the resulting functions are similar and different to their comprising pieces
  • See shortcuts for evaluating arithmetic expressions involving entire functions
Set-Builder and Interval Notation

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Working with Functions

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Tags

  • Functions
  • Number Properties
  • Real Numbers

Priority: Normal

The classic way to describe a range of numbers is to use an inequality. However, there are other ways to express a range of numbers, and not only do we have to learn them because our teachers will use it, but they are also sometimes easier and less work to use.

  • Review what we know up to this point about describing ranges and sets of numbers
  • Learn interval notation to describe a range of real numbers
  • Learn set-builder notation to describe a set of numbers
  • Understand the motivation for these notations and the advantages each has
Function Translations

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Working with Functions

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Tags

  • Functions
  • Graphing
  • Function Analysis

Priority: High

If a function is translated, its original shape remains intact but its graph shifts up, down, left, or right. This lesson shows us how to draw relationships between two functions that are otherwise the same, but translated, or shifted over from one another. We will continue using translations throughout the entirety of Pre-Calculus, including trigonometry.

  • Identify translations based on graphs only
  • Use function notation to understand how to translate a function up, down, left, or right
  • Algebraically find the new translated function from the original function, given the desired translation
Advanced Function Transformations

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Working with Functions

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Tags

  • Functions
  • Function Analysis
  • Algebra Manipulation

Priority: Normal

In Algebra Two we learned how to algebraically shift functions by moving them left, right, up, and down. This lesson will introduce ways to manipulate functions by stretching, compressing, or reflecting. We'll also see how to manage several simultaneous transformations.

  • Review translations (aka sliding) type transformations that we learned in Algebra Two
  • Understand how stretch-type transformations visually change the graph of a function
  • Know how to apply stretch and compression transformations to a function algebraically
  • Apply translation and stretch / compression transformations simultaneously
  • Learn how to reflect a function over the $x$ or $y$ axis
  • Learn how to reflect a function over any horizontal or vertical line
Finding Domain and Range

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Working with Functions

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Tags

  • Functions
  • Function Analysis
  • Algebra Manipulation

Priority: High

Up to this point, the only ways we have been able to find domain or range have been by inspection or by examining graphs. This lesson will go beyond what we already know about domain and range, giving us a better understanding and a definite approach to finding them.

  • Review what Domain and Range each is and what we already know about finding them from inspection
  • Compile the authoritative list of what to consider when seeking the domain of a function
  • Define what to do algebraically to determine the domain of a function
  • Find the range of a function using function inverse techniques
  • Determine the domain of composite and piecewise functions
  • Identify general patterns of domain and range based on function family
Multivariable Functions

Pre-Calculus  $\rightarrow$  Advanced Function Analysis  $\rightarrow$  Working with Functions

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Tags

  • Functions

Priority: Optional

It is possible (and common in the world of computer programming) for a function to require more than one input. This lesson gives a quick yet useful overview at how multivariable functions operate conceptually.

  • Understand how functions work when they require more than one input
  • Use common multivariable functions

Exponential Relationships - Understanding Variable Exponents

Real Number Exponents

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Variable Exponents

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Tags

  • Exponents
  • Real Numbers
  • Review

Priority: Normal

Building up to the next several lessons on exponential functions and relationships, we need to understand that we can let any real number be an exponent, from integer and fraction exponents that we have already seen, to irrational numbers like $\sqrt{2}$ and $\pi$.

  • Quickly recap all previously learned exponent rules
  • Make connections between rational exponents, which we've seen, and terminating decimal exponents, which we haven't yet seen
  • Understand how irrational exponents work and what they mean, even though they are not as intuitive
Relationships With Variable Exponents

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Variable Exponents

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Tags

  • Exponents
  • Variable Exponents
  • Word Problems

Priority: High

For the first time, we will begin to understand relationships that involve constants raised to variable exponents instead of the other way around (e.g. $3^x$ instead of $x^3$). We will examine the behavior of these relationships using integer values for $x$ before seeing what happens when we allow exponent variable $x$ to be non-integer values. Finally, we will look at exponential growth versus exponential decay, and begin to work with basic growth and decay models.

  • Introduce the concept of having the unknown variable in an equation be the exponent
  • See and understand the general patterns associated with relationships that involve a variable as the exponent, using x-y tables
  • Further justify evaluating expressions when the exponent is an irrational number
  • Define the concepts of exponential growth and exponential decay, and know how to tell which one describes a given relationship
  • Investigate basic growth and decay models that use rate of growth or decay $r$ and initial quantity $a$
  • Be able to set up and interpret a simple growth or decay model in the context of a word problem
Basic Exponential Equations

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Variable Exponents

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Tags

  • Exponents
  • Variable Exponents
  • Solving Equations

Priority: High

Exponential equations are equations in which the unknown variable is an exponent. There are two ways to solve these: either using integer and fraction knowledge or using logarithms. While we will study logarithms very soon, this lesson shows us how to solve exponential equations using only what we know about integers and fractions. We'll also understand when we can use this technique, and when we instead need to rely on logarithms.

  • Solve for variable exponents in an equation using only what we know about integers and fractions
  • Learn the "same base" trick for solving certain equations that involve variable exponents, and know when the trick can and can't be used
  • Understand that when the "same base" trick cannot be used, that there is necessarily no rational solution (solution requires forthcoming logarithm techniques)
  • Solve special kinds of exponential equations that are quadratic equations in disguise
Exponential Functions

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Variable Exponents

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Tags

  • Exponents
  • Functions
  • Variable Exponents
  • Graphing

Priority: VIP Knowledge

This lesson will put together several ideas that we saw in the prior three lessons, defining exponential relationships in the context of looking at a function. Not only will be employ techniques for solving for a variable like we saw in the prior lesson, but we will also define general properties of an exponential function based on the coefficients in the typical exponential form we see.

  • Apply recently acquired knowledge about exponential relationships to the input/output behavior we are familiar with for functions
  • Know and understand the form of a basic exponential function, $f(x) = ab^x$
  • Describe common properties of exponential functions, and continue to understand the difference between growth and decay
  • Interpret the coefficients of an exponential function, and what happens when the coefficient is positive vs negative
  • Using growth or decay and a positive or negative coefficient, learn and understand the four major categories that an exponential function can fall into: positive growth, positive decay, negative growth, and negative decay
  • See the general shape of exponential graphs, and how that shape changes in each of the four exponential categories
Graphing Exponential Functions

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Variable Exponents

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Tags

  • Exponents
  • Functions
  • Variable Exponents
  • Graphing

Priority: High

Using what we now know about how exponential functions work, we can understand the general shape of their graphs, and the common properties that the graphs of all exponential functions share.

  • Learn how to graph specific exponential functions by plotting a few points and eyeballing it
  • Know general properties of the graph of an exponential function, as well as what points are commonly on the graph
  • Be able to graph an exponential function quickly by applying transformations to its parent function, visually
Analyzing Exponential Transformations

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Variable Exponents

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Tags

  • Exponents
  • Functions
  • Variable Exponents
  • Graphing

Priority: Normal

In the last lesson, we learned how to graph exponential functions, including some basic transformations such as shifting and scaling. Here, we will practice working with transformations algebraically, and learn specific implications for exponentials that certain transformations have.

  • Apply what we know about algebraic function transformations to exponential functions
  • Be able to look at an exponential written function definition and know quickly what its parent function is and the transformations that it has
  • Know how to start with a parent function in written form and obtain a new transformed function in simplest form
  • Understand how reflection transformations are connected to the exponential properties of growth vs decay, and positive vs negative
Working With the Number e

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Variable Exponents

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Tags

  • Exponents
  • Variable Exponents
  • Word Problems

Priority: Normal

This lesson defines the number $e$ - one of the most important numbers in math. It is what we call the "natural exponential base". Like $\pi$, it is an irrational number that comes from a special definition.

  • See the limit definition of the number $e$, but more importantly understand what it represents conceptually
  • Set up basic continuous growth models with given information
  • Given an exponential model using $e$, find the equivalent model that uses a base of $1+r$ and vice versa

Exponential Relationships - Understanding Logarithms

Logarithms

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Logarithms

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Tags

  • Logarithms
  • Exponents
  • Variable Exponents

Priority: VIP Knowledge

This important lesson introduces logarithms: what they are, and their relationship to what you already know about exponentials.

  • Know he definition of a logarithm forward and backward (literally)
  • Learn which two log bases are most common to use and why
  • Evaluate simple logarithm expressions without a calculator using integer relationships
  • Turn simple equations with logarithms into exponential equations that we know how to solve
  • Use rounding and estimation techniques to approximate the numerical value of a logarithm
Logarithmic Functions

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Logarithms

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Tags

  • Logarithms
  • Variable Exponents
  • Exponents
  • Functions

Priority: High

Now that we know what logarithms are, we will think about them in the context of functions to further understand their nature and their relationship to exponential functions.

  • Examine functions of the form $f(x) = log_b (x)$ and understand their properties
  • Use the definition of logarithms to show that logarithmic functions are inverses of exponential functions and vice versa
  • Understand and be able to find the domain and range of any logarithmic functions
Graphs of Log Functions

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Logarithms

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Tags

  • Logarithms
  • Functions
  • Graphing

Priority: Normal

As we often do after learning about a new function type, let's see what graphs of logarithmic relationships look like, and how they relate to other graphs that we've already seen.

  • See what shape and pattern to expect from the graph of a logarithmic function
  • Compare graphs of logarithmic and exponential functions, keeping in mind their inverse relationship to one another
  • Further validate the domain and range patterns of logarithmic functions with graphs
  • Apply typical function transformations to log functions, and connect the transformation's visual effect to the changes in the function definition, and vice versa
Logarithm Rules

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Logarithms

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Tags

  • Logarithms
  • Exponents
  • Algebra Manipulation
  • Algebra Rules

Priority: High

Because logarithms really represent exponents, the rules that apply to them look oddly similar to things we know about working with exponents. The three rules in this lesson may look unusual at first for the same reason, but we use them often, because they allow us to manipulate logarithms to make it easier to solve certain problems.

  • Learn the three important logarithm manipulation rules
  • Practice using them backward and forward (literally), since we need to use the rules backward just as often as we do forward
Expanding and Condensing Logarithmic Expressions

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Logarithms

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Tags

  • Logarithms
  • Algebra Manipulation

Priority: High

Using the logarithmic rules we recently learned, we can now operate on logarithms by either taking a single logarithm of a complex object and breaking it up into simpler logarithms, or conversely, taking several simple logarithms and combining them into a single logarithm. These opposite processes each have their place in your math future, each making certain complex tasks easier.

  • Apply multiple logarithm manipulation rules at the same time
  • Begin with a single logarithm of a complicated term and end with sums and differences of several much simpler logarithms (expanding)
  • Begin with a string of sums and differences of logarithms and end with a single logarithm by using logarithm manipulation formulas (condensing)
Log Change of Base Formula

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Understanding Logarithms

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Tags

  • Logarithms
  • Algebra Manipulation

Priority: Normal

It is possible, but not intuitive, to turn a logarithm of base b into a logarithm of some other base. This formula is particularly important if you are working with only variable, or if you need a decimal approximation and only have an older calculator that does not allow you to determine the base (many calculators only have input functions for base 10 and base $e$). This short lesson demonstrates how to change a log expression into an equivalent expression with a different base.

  • Learn the logarithm change of base formula
  • Understand the motivation for such a formula, and why most calculators only have natural and common logarithm functionality
  • Practice using the formula both with hand written problems and with calculator based problems

Exponential Relationships - Exponential and Logarithmic Equations

Solving Exponential Equations Using Logarithms

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Exponential and Logarithmic Equations

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Tags

  • Logarithms
  • Variable Exponents
  • Solving Equations

Priority: VIP Knowledge

Exponential equations can be definitely solved using logarithm manipulation, regardless of whether or not the equation has a "nice" integer solution. This lesson shows you how to solve equations with unknown variable exponents, using logarithms.

  • Learn how to solve equations that contain unknown variable exponents using logarithms
  • If asked for a decimal approximation, get an answer from your calculator after solving for the variable using logarithms
  • Understand the difference between exponential equations we solve in this lesson that require logarithms and same-base exponential equations we solved recently without logarithms
  • Identify the types of exponential equations that can and cannot be solved with algebra (even with logarithms)
  • Know how to use a graphing calculator to find the solutions to ones that cannot be solved any other way
Solving Logarithmic Equations

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Exponential and Logarithmic Equations

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Tags

  • Logarithms
  • Exponents
  • Solving Equations
  • Algebra Manipulation

Priority: High

This brief lesson shows you how to solve equations that have logarithms on one or both sides of the equal sign, as well as other equations with logarithms of unknown bases, by using the exponent definition of logarithms.

  • Know how to solve equations that have logarithms on one side of the equation
  • Know how to solve equations that have logarithms on both sides of the equation
  • Learn how to solve equations that have logarithms with unknown base $x$
  • Understand why to check for extraneous solutions, and know how to check
  • Know which types of logarithmic equations cannot be solved explicitly, and therefore must be solved with a computer
Using Common Exponential Models

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Exponential and Logarithmic Equations

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Tags

  • Logarithms
  • Variable Exponents
  • Word Problems
  • Applied Math

Priority: Normal

Recall from much earlier lesson on variable exponent relationships that exponentials come in two flavors - growth and decay. In this lesson, we will examine specific scenarios of both growth and decay models and how to solve for unknown quantities with these models. We will also see some examples of exponential relationships in the real world.

  • Learn how to set up commonly used exponential growth and decay models (but not money accumulation with interest - that is the topic of the next lesson!)
  • Learn how to solve for the missing quantity in all kinds of exponential models, including initial amount, growth rate, and time
Simple, Compound, and Continuous Interest

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Exponential and Logarithmic Equations

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Tags

  • Logarithms
  • Variable Exponents
  • Word Problems
  • Financial Math
  • Applied Math

Priority: Normal

Now that we can manipulate equations with logarithms, we can look at all three common ways that interest on money may accrue. This lesson will show us how to set up interest models, and solve for the unknown missing quantity, which may be initial amount, interest rate, or time.

  • Learn how each interest accrual method works (simple, compound, or continuous interest)
  • Practice setting up the correct equation to model out a given money growth situation
  • Be able to solve for the missing quantity whether that is the initial amount, the interest rate, or the amount of time, regardless of which of the there interest methods is being used
Advanced Modeling With Exponential Functions

Pre-Calculus  $\rightarrow$  Exponential Relationships  $\rightarrow$  Exponential and Logarithmic Equations

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Tags

  • Logarithms
  • Functions
  • Variable Exponents
  • Word Problems

Priority: Optional

The complexity of exponential modeling can go far beyond the typical growth and decay situations we have already seen, such as half-life or multiplying bacteria. Here we will look at more complex real life situations that use exponentials to model advanced behavior.

  • Get acquainted with situations that use exponentials in a non-traditional way to model situations, such as Newton's Law of Cooling or Carrying Capacity Population Growth
  • Interpret the coefficients and horizontal asymptotes in advanced exponential models
  • Solve for any missing unknown in advanced exponential models

Trigonometry - Right Triangles

Right Triangle Trigonometry

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Right Triangles

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Tags

  • Trigonometry
  • Right Triangles

Priority: VIP Knowledge

The three major trigonometry ratios exist in any right triangle - the sine, cosine, and tangent. While we will look at these big three a lot throughout the course of trigonometry, this first lesson helps us understand what the ratios mean and where they come from.

  • Define the sine, cosine, and tangent ratios for a given non-right angle in a right triangle (aka SOH-CAH-TOA)
  • Using the Pythagorean Theorem and the definition of sine, cosine, and tangent either to answer questions about unknown triangle side lengths or to find trig ratio values
Solving a Right Triangle

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Right Triangles

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Tags

  • Trigonometry
  • Right Triangles

Priority: VIP Knowledge

With very little initial information, you can find all three sides and all three angles of a right triangle, particularly with the help of trig ratios that we just learned about. In this lesson we'll see the common ways in which we'll be asked to do this.

  • Know what it means to "solve a triangle"
  • Use the Pythagorean Theorem and basic properties of triangles, in conjunction with trig ratios, to find all missing values of a right triangle
  • Learn how to use trig ratios with your calculator to solve for missing sides in a right triangle
  • Learn how to use your calculator's "inverse" trig ratio functionality to solve for missing angles in a right triangle
Special Right Triangles

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Right Triangles

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Tags

  • Trigonometry
  • Right Triangles
  • SAT / ACT Math

Priority: High

For right triangles with special angles, we can skip the trig ratio magic and use some standard ratios. Not only is this a faster approach, but it is also commonly prevalent on tests like the SAT.

  • Learn the two special right triangles, and the side ratios that accompany each
  • Given any one of the three sides of these special triangles, be able to find the other two
Applied Right Triangle Problems

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Right Triangles

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Tags

  • Trigonometry
  • Right Triangles
  • Word Problems
  • Applied Math

Priority: Normal

Using the very same skills we recently learned to solve a right triangle, we can set up and solve word problems that describe right triangles with unknown sides or angles. We'll look at problems involving angles of elevation and depression, including complicated diagrams.

  • Independently set up geometry word problems for situations that measure angles of elevation and depression
  • Set up other types of geometry word problems that are described by a right triangle with unknown sides or angles
  • Solve complex applied right triangle word problems using a combination of Pythagorean Theorem, trig ratios, and algebra

Trigonometry - Angles and Angle Measurement

Measuring Angles in Degrees

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Angles and Angle Measurement

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Tags

  • Angle Measurement

Priority: Normal

At this point in your math career, you've probably measured angles using degrees in the past. This first lesson on angle measurement will reinforce what we already know about degrees, and also discuss a couple of common ways to work with partial degrees.

  • Specifically define / recall what exactly degrees are
  • Define partial degrees as fractions of degrees in a system called DMS (degrees minutes seconds)
  • Learn how to convert partial degrees between our natural decimal system and DMS measure
Measuring Angles in the Coordinate Plane

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Angles and Angle Measurement

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Tags

  • Angle Measurement
  • Coordinate Plane

Priority: High

Having re-remembered all the important facts from the last lesson about measuring angles with degrees, we take our first look at a very common way of working with angles - by putting them in the coordinate plane. Then we can open up angles at any measure we want, and look at the properties that result, such as which quadrant the angle ends up in.

  • Look at the standard way in which we measure angles in the coordinate plane
  • Understand why and how negative angles are possible, and interpret how they are different from positive angles
  • Understand why and how angles can be larger than 360 degrees (one full circle rotation)
  • Know what co-terminal angles are and how to tell if two angles are co-terminal
Radian Angle Measure

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Angles and Angle Measurement

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Tags

  • Angle Measurement
  • Coordinate Plane

Priority: High

For much of the ride ahead through Trigonometry (and very much so in Calculus), we aren't able to use degrees as a way to measure angles. This lesson introduces the very important alternative, dimension-less way to measure angles, measured in units we call radians.

  • Learn the definition of radians and understand why it is a measure without dimension
  • Translate between common degree measure angles and their equivalent radian measure
  • Revisit measuring angles in the coordinate plane but this time using radians
  • Become an expert on measuring angles with radians instead of degrees - you'll need it for the future!
Angular and Linear Speed of Rotation

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Angles and Angle Measurement

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Tags

  • Angle Measurement
  • Word Problems
  • Applied Math

Priority: Normal

This lesson, which is both popular and potentially confusing, looks at the major difference between how fast a wheel is rotating and how fast a wheel is actually moving (which depends entirely on the size of the wheel). We'll see lots of examples and practice to understand how teachers test you on this.

  • Understand the difference between angular speed and linear speed
  • Using basic circle properties, be able to convert angular speed to linear speed and vice versa
  • Determine the speed relationships between two differently sized spinning connected wheels, such as on a bicycle
Arc Length and Sector Area

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Angles and Angle Measurement

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Tags

  • Angle Measurement
  • Circles

Priority: Normal

When we measure angles in radians, we can quickly calculate the arc length and area of partial circles, commonly referred to as sectors. This lesson shows you how to measure each in radians, and then how to do it similarly with degrees.

  • Understand exactly what a sector is and how it is defined
  • Know the difference between sector arc length and sector perimeter, and how to find either when using radian measure
  • Learn how to find the area of a sector measured in radians
  • See the analogous formulas to use when working with degree measure

Trigonometry - Trigonometry Functions

The Sine and Cosine Functions

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Trigonometry Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Angle Measurement
  • Coordinate Plane
  • Functions

Priority: VIP Knowledge

Trigonometric ratios are introduced to us in a way that has a nice visual interpretation with right triangles, but the reality is that we can take the sine or cosine of any angle - not just acute angles. This lesson explores this idea by looking at a definition of sine and cosine in the coordinate plane.

  • Look at sine and cosine with triangles in the coordinate plane
  • Define sine and cosine values based on the actual coordinates of points in the plane
  • With the coordinate plane definitions of sine and cosine, find the sine and cosine of any angle in any quadrant, not just acute angles
  • Create functions of the sine and cosine ratios with the angle you are measuring at as the input
  • Understand basic function properties that $y=\sin(x)$ and $y=\cos(x)$ each have
The Unit Circle

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Trigonometry Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Angle Measurement
  • Coordinate Plane

Priority: VIP Knowledge

Because of the way we define sine and cosine with coordinates in the plane, it turns out that looking at the coordinates on a unit length circle will give us a great reference table for values that we will need to memorize, or at least know how to quickly look up without a calculator.

  • See and understand what the unit circle is in the coordinate plane
  • Learn how to use the unit circle to obtain the sine and cosine values for the most common angle measures
The Tangent Function

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Trigonometry Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Angle Measurement
  • Coordinate Plane
  • Functions

Priority: High

Much of what we just learned about the sine and cosine functions apply to working with the tangent trig ratio as well - particularly creating a coordinate based definition in the coordinate plane. This lesson works through the content we just saw for sine and cosine, but for the tangent trig ratio.

  • Define the tangent trig ratio using coordinates in the plane
  • Build the important relationship between sine, cosine, and tangent, that tan = sin / cos
  • Define the tangent function similar to the sine and cosine Functions, where the input is the angle and the output is the trig ratio
  • Be able to find the tan of any angle in any quadrant, not just acute angles
  • Calculate the tangent of an unknown angle given information about the point on the plane that the angle passes through
  • Calculate the tangent of an unknown angle given which quadrant the angle lies in, and the value of the sine or cosine of that angle
Reference Angles

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Trigonometry Functions

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Tags

  • Angle Measurement
  • Coordinate Plane
  • Trigonometry

Priority: Normal

Understanding and being able to identify reference angles is a major step in the general process of evaluating trig functions. This lesson focuses on getting you to understand what reference angles are, and to become masterful at finding them.

  • For angles oriented in the coordinate plane, understand what reference angles are
  • Practice finding the reference angle of angles in any quadrant
  • Learn a big shortcut for finding reference angles when measuring angles with radians
Evaluating Trig Functions

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Trigonometry Functions

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Tags

  • Trigonometry
  • Angle Measurement
  • Coordinate Plane
  • Trig Functions

Priority: VIP Knowledge

One of the most important and common tasks we are required to perform in trigonometry (and again in Calculus) is to evaluate a trig function at a specific value. For common angles we are often expected to do this without a calculator. This lesson teaches you how to do just that, and provides a three-step approach to make sure you handle it the right way every time.

  • Review recent lesson material all together to understand which trig functions in which quadrants are positive or negative
  • Leveraging recent lesson material, evaluate sine, cosine, and tangent at common angles in all four quadrants without a calculator
  • Learn the Mister Math three-step approach to correctly evaluate sine, cosine, and tangent for common angles
  • Determine the sine, cosine, and tangent of an unknown angle knowing only which point on the unit circle it passes through
  • Better your understanding of what you will and will not be expected to do without the aide of a calculator
All Six Trig Functions

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Trigonometry Functions

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Tags

  • Trigonometry
  • Coordinate Plane
  • Trig Functions

Priority: High

So far we have looked at sine, cosine, and tangent. Additionally, the reciprocal of each of these creates a new trig function, and because of their use and commonality, we give them their own names and study their properties. We will learn these new three trig function names and coordinate plane definitions of each.

  • Learn how to obtain three new trig functions (cotangent, secant, cosecant) by reciprocating each of the three original trig functions, and which new function comes from which old function
  • Define the cotangent, secant, and cosecant functions using points in the plane ($x$, $y$, and $r$), like we did before for sine, cosine, and tangent
  • Learn basic identities that show how all 6 trig functions can be easily defined using only sine and cosine
  • Learn how to evaluate the three new trig functions at common angles, analogous to how we learned to do it for sine, cosine, and tangent
  • Learn how to evaluate the three new trig functions at non-common angles with a calculator, since most calculators do not have buttons for them
  • Learn how to find the values of all six trig functions of an unknown angle by drawing a right triangle, given information about one of the function values and which quadrant the angle falls in
Inverse Trig Functions

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Trigonometry Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Functions
  • Function Analysis

Priority: High

Like many typical algebra functions, we can find an inverse function for trig functions such that we input a trig ratio value into the inverse trig function and it gives the original angle as output (as opposed to normal trig functions that take an angle as the input and give a ratio value as the output). This lesson will look at the definitions and properties of these inverse trig functions, before showcasing their applications and common types of exam questions.

  • Understand what reverse trig functions are in relation to trig functions
  • Understand the inherent difficulty of defining a perfect inverse function due to the periodic nature of normal trig functions, and learn how we mitigate that issue
  • Know whether a question is asking you for a specific answer versus a list of all possible answers
  • Based on knowledge of common angle trig values, know when we can and cannot answer a questions without a calculator
  • Use the domain of inverse trig functions to answer questions that involve both inverse and normal trig functions together
  • Draw triangles in the coordinate plane to find either exact values or generic relationships between inverse trig functions and normal trig functions

Trigonometry - Using and Graphing Trig Functions

Pythagorean Trigonometric Identities

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Using and Graphing Trig Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Simplifying
  • Algebra Manipulation

Priority: VIP Knowledge

One of the most common identities of trigonometry is derived from applying the Pythagorean Theorem to our coordinate plane triangle scheme. This lesson covers the derivation of Pythagorean trig identities, and showcases ways to use them to answer questions

  • Learn the most fundamental Pythagorean trigonometric identity, which shows up a lot in advanced mathematics
  • Derive two other equivalent Pythagorean identities from the first one
  • Use these identities to answer questions about unknown quantities
Trig Function Relationships

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Using and Graphing Trig Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Algebra Manipulation

Priority: Normal

This lesson demonstrates relationships that exist among trig functions, first by reviewing a few fundamental inverse relationships that we've already seen, and then introducing ones based on negatives and angle shifts.

  • Review relationships we already know about trig functions, such as the reciprocal relationship between sine and cosecant, for example
  • Explore other relationships such as negative argument (e.g. $sin(-x) = -sin(x)$) and shifted argument relationships (e.g. $sin(x) = cos(\pi/2-x)$).
Basic Trig Identity Proofs

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Using and Graphing Trig Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Algebra Manipulation
  • Simplifying

Priority: High

This lesson introduces us to the infamous task of proving trig identities. This puzzle-like skill requires us to use our algebra and trig function relationship knowledge to manipulate two expressions to show that they are the same.

  • Learn what it means when you're asked to prove a trig identity
  • Practice proving simpler trig identities (complex trig identity proofs are covered in a future lesson)
Solving Basic Trigonometric Equations

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Using and Graphing Trig Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Solving Equations

Priority: Normal

Now that we know more about how to work with inverse trig functions, we can solve fairly basic equations for unknown x when x is the argument of a trigonometric function, such as solving $2\sin(x) - 1 = 0$. Note that an upcoming lesson is dedicated to solving similar types of equations, but focusing on equations with more complex expressions present.

  • Similar to solving for a variable, learn to isolate the trig function in the equation
  • Properly apply the inverse trig function operation to both sides of an equation, once one side of the equation contains an isolated trig function
  • Pay attention to the directions - they are often the only indicator as to whether your answer should be a finite or infinite list
  • As with recent lessons, know the difference between questions that can and cannot be solved without a calculator
Trigonometric Graphs of Sine and Cosine

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Using and Graphing Trig Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Graphing
  • Function Analysis
  • Word Problems

Priority: High

Functions that are transformed versions of $y = \sin(x)$ and $y = \cos(x)$ are wave-shaped patterns that we refer to as sinusoidal. This lesson, which is very commonly covered on exams, shows us how to graph functions of this form, and conversely write down functions of this form based on a graph. We'll also see some application word problems.

  • Learn the general properties of the graphs of sine and cosine functions, both of which are referred to as sinusoidal
  • Leverage our knowledge of function transformations to see how we might account for vertical and horizontal stretching, and vertical and horizontal shifting, relative to the base functions $y = \sin(x)$ and $y = \cos(x)$
  • Practice writing down functions based on graphs by examining which transformations appear on the graph, relative to the base function
  • Use the Mister Math four-step approach to graphing written sinusoidal functions by accounting for any transformations in the order that makes it easiest for you
  • Work practice problems and word problems that require working with and manipulating sinusoidal functions
Graphs of Other Trigonometric Functions

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Using and Graphing Trig Functions

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Tags

  • Trigonometry
  • Trig Functions
  • Graphing
  • Function Analysis

Priority: Optional

When it comes to graphing trig functions, the general focus is definitely on the sine and cosine forms we saw in the last lesson. However, many teachers will quickly cover the graphs of the other four trig functions, and so, this lesson will show you their forms and properties, leveraging some of the work we accomplished in the last lesson when possible.

  • Learn how to graph basic forms of $y = \tan(x)$ and $y = \cot(x)$, without transformations
  • Discuss how transformations can be implemented in tan and cot graphs, although this is not commonly studied
  • Graph basic and transformed versions of functions of the form $y = \sec(x)$ and $y = \csc(x)$ by first graphing their reciprocal functions using what we recently learned about graphing sinusoids

Trigonometry - Advanced Trigonometry

Common Angle Combination Identities

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Advanced Trigonometry

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Tags

  • Trigonometry
  • Trig Functions
  • Simplifying
  • Algebra Manipulation

Priority: Normal

Special identities exist that relate trig functions of the sum or difference of two angles to trig functions of each angle. This lesson will show you the formulas, which we will practice using with carful plug and play techniques.

  • Learn and (maybe) memorize the angle combination trig identities for sums and differences of angles
  • Learn and (maybe) memorize the angle combination trig identities for double angles, and understand how they can be quickly derived from the angle combination trig identities
  • Use these formulas to prove trig identities
Other Angle Combination Identities

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Advanced Trigonometry

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Tags

  • Trigonometry
  • Trig Functions
  • Simplifying
  • Algebra Manipulation

Priority: Optional

This lesson covers some of the less common trigonometric angle combination identities that were not covered in the prior lesson. These are less common and not always covered by a standard Pre-Calculus curriculum, but are sometimes utilized in advanced courses.

  • See and (hopefully not) memorize the half angle identity formulas
  • Learn the fairly obscure and uncommon identities that relate the sum of trig functions to various products of trig functions
  • Gain familiarity with the types of questions that teachers ask about this uncommon topic (but not trig identity proofs - we will practice those in the next lesson!)
Advanced Trig Identity Proofs

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Advanced Trigonometry

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Tags

  • Trigonometry
  • Trig Functions
  • Simplifying
  • Algebra Manipulation

Priority: Normal

Though there are no direct applications of this topic, teachers love having students prove trig identities. We've seen the basic idea before, so in this lesson we start with a reminder of the concept of proving identities, and then use all the trig knowledge we now have.

  • Review the types of trig identity proofs we have already been exposed to
  • Work on trig identity proofs with more complex expressions, and learn tips and tricks about common things to look for
Solving Advanced Trigonometric Equations

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Advanced Trigonometry

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Tags

  • Trigonometry
  • Solving Equations
  • Algebra Manipulation

Priority: Normal

In a prior lesson, we saw how to solve basic trigonometric equations where the unknown variable to solve for was the angle measured in radians - but only how to solve equations with a single trig function in them. Now we'll turn our attention to more complicated trig equations that require either trig identities or advanced ideas to solve.

  • Learn to solve any equation that involves trig functions, regardless of complexity, using tricks and knowledge of trig identities
  • Categorize equations by what type of expressions are present, and become familiar with common categories and their solution processes
  • Give appropriate answers based on the problem set instructions (finite vs infinite lists of solutions)
  • Know when to check for extraneous solutions

Trigonometry - Solving Scalene Triangles

The Law of Sines

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Solving Scalene Triangles

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Tags

  • Trigonometry
  • Word Problems
  • Triangles

Priority: Normal

We've learned a lot about solving right triangles, but some of the trig knowledge we acquired will also help us solve scalene (non-right) triangles. In this lesson we'll see how the Law of Sines works on any triangle, and when we can and can't use it.

  • Learn what the Law of Sines states, and understand when and how to use it
  • Learn which situations have one solution vs multiple solutions, because most teachers mark it wrong if you only give one
  • Set up word problems from scratch and apply the Law of Sines to answer questions
The Law of Cosines

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Solving Scalene Triangles

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Tags

  • Trigonometry
  • Word Problems
  • Triangles

Priority: Normal

Similar to the Law of Sines, the Law of Cosines is a tool for working with scalene triangles. It is slightly more complicated but works well for situations in which the Law of Sines fails to work. This lesson will teach us when this is the right approach, and how it works.

  • Learn the Law of Cosines conceptually, and how it may be rewritten three ways - one for each side of the triangle
  • Know the right situations to use Law of Cosines versus the preferred Law of Sines method, since Law of Sines is faster and less complicated
  • Recognize when there is potentially more than one solution, in the same ambiguous case we saw with Law of Sines (neither LoS or LoC removes the ambiguity)
  • Set up triangle word problems and solve them using either Law of Sines or Law of Cosines - whichever is most efficient
The Area of a Triangle Using Sin

Pre-Calculus  $\rightarrow$  Trigonometry  $\rightarrow$  Solving Scalene Triangles

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Tags

  • Trigonometry
  • Triangles

Priority: Optional

In geometry we had to do a lot of work to find the area of a scalene triangle, because the formula $A = 1/2 \, bh$ required that the base and height be perpendicular. With a quick formula involving the sine function, we can find the area of any triangle.

  • Learn and practice the formula for the area of any triangle using two sides and the angle between them

Conic Sections and Analytic Geometry - Conic Sections, Circles, and Ellipses

What a Conic Section Is

Pre-Calculus  $\rightarrow$  Conic Sections and Analytic Geometry  $\rightarrow$  Conic Sections, Circles, and Ellipses

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Tags

  • Quadratics
  • Graphing
  • Coordinate Plane

Priority: Normal

We've learned a lot about parabolas through our study of quadratics, but equations of parabolas, circles, ellipses, and hyperbolas are all related in a special way. This lesson is our first look at what conic sections are.

  • Learn the four main categories of conic sections (circles, ellipses, parabolas, and hyperbolas), and why they are all related and referred to as "conic"
  • See the generic algebraic equation form of a conic section, and understand that the equation can represent any of the four categories depending solely on the coefficients
  • Get a cursory look at the coefficient patterns that produce each category, before we look at each category with its own upcoming lesson
The Equation of a Circle

Pre-Calculus  $\rightarrow$  Conic Sections and Analytic Geometry  $\rightarrow$  Conic Sections, Circles, and Ellipses

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Tags

  • Quadratics
  • Graphing
  • Coordinate Plane
  • Algebra Manipulation
  • Circles

Priority: VIP Knowledge

One of the most common and useful quadratic forms is the circle. This lesson will examine conic section equation circles, and what we need to know about both the quadratic equation and the accompanying graph in the coordinate plane.

  • Understand the quadratic form conic equation of a circle, and its derivation via the Distance Formula
  • Use the Completing the Square technique to rewrite a generic quadratic form equation into the standard circle form
  • Learn how to graph circles based on the equation, and conversely, how to obtain an equation based on a graph
Equations and Properties of Ellipses

Pre-Calculus  $\rightarrow$  Conic Sections and Analytic Geometry  $\rightarrow$  Conic Sections, Circles, and Ellipses

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Tags

  • Quadratics
  • Graphing
  • Coordinate Plane
  • Algebra Manipulation

Priority: Normal

Though ellipses are visually similar to circles, their oblique shape without a fixed radius is not as easy to define as a circle. This lesson explores the definitions, equations, and properties of ellipses, and what we need to know about working with them.

  • Learn the quadratic equation forms that represent ellipses, and how they are similar to and different from circles
  • Define vertices and axes for ellipses
  • Define focus points and focal distance in an ellipse
  • Define the directrix line associated with an ellipse
  • Interpret an ellipse's measure of eccentricity
  • Practice graphing ellipses based on the quadratic equations, and conversely finding a equation that matches the graph

Conic Sections and Analytic Geometry - Parabolas and Hyperbolas

The Conic Equation of a Parabola

Pre-Calculus  $\rightarrow$  Conic Sections and Analytic Geometry  $\rightarrow$  Parabolas and Hyperbolas

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Tags

  • Quadratics
  • Graphing
  • Coordinate Plane

Priority: Normal

Parabolas that we study in the context of conic sections look and behave similarly to the ones we studied during out work on quadratic functions in algebra. However, the form and attributes of conic section parabolas that we will look at is really like taking on a new point of view about something we are already familiar with. Here we will work with a specific equation form of parabolas, and quantify attributes such as its axis, focus, vertex, and directrix line.

  • Learn the standard form for conic section parabolas, for both vertically and horizontally oriented parabolas
  • For a parabola with vertex $(h,k)$, know what its focal point and directrix line are, formulaically
  • Understand a geometric interpretation of what the focus point is of a parabola
The Equations of Hyperbolas

Pre-Calculus  $\rightarrow$  Conic Sections and Analytic Geometry  $\rightarrow$  Parabolas and Hyperbolas

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Tags

  • Quadratics
  • Graphing
  • Coordinate Plane

Priority: Normal

Hyperbolas are perhaps the least familiar conic section, because we work with them the least. This lesson will describe exactly what they are, and show you what you are supposed to know about them.

  • Learn about the general shape and orientation of horizontal and vertical hyperbolas
  • Find the equation of the hyperbola and the equations of its asymptotes from basic given information (such as its center, its focus, etc.)
  • Graph hyperbolas quickly by putting them in standard conic section equation form and utilizing the vertex and asymptotes

Conic Sections and Analytic Geometry - Analytic Geometry

Classifying Conic Sections

Pre-Calculus  $\rightarrow$  Conic Sections and Analytic Geometry  $\rightarrow$  Analytic Geometry

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Tags

  • Quadratics
  • Graphing
  • Coordinate Plane

Priority: Normal

At this point, we've seen all the types of conic sections. This short lesson focuses on how to look at equations in conic form and figure out which type of conic section you're dealing with, based on the coefficients.

  • Identify conic sections using a determinant method similar to the quadratic formula, for standard form conic equations
  • Recall and be able to categorize conic sections by their literal definition (e.g. which one is the set of points such that the sum of distances from any point to the two focus points is constant)
Nonlinear Systems of Equations

Pre-Calculus  $\rightarrow$  Conic Sections and Analytic Geometry  $\rightarrow$  Analytic Geometry

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Tags

  • Solving Equations
  • Systems of Equations

Priority: Normal

The solutions to nonlinear systems can be a topic so complicated that it wouldn't belong in this course. However, if we restrict our discussion to polynomials of degree 2 or less, we'll see some manageable processes, and familiar conic section graphs.

  • See common techniques for solving systems of equations that involve quadratics
  • Understand the inherent differences from solving linear systems
  • Know why we graphically interpret solutions as intersections, just like we did for linear systems
  • Solve these systems with a graphing calculator and know the intricacies of doing that
Conic Rotations

Pre-Calculus  $\rightarrow$  Conic Sections and Analytic Geometry  $\rightarrow$  Analytic Geometry

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Tags

  • Quadratics
  • Trig Functions
  • Coordinate Plane

Priority: Optional

Usually, conic sections are naturally oriented along either the $x$ or $y$ axis. If we want to create ones that are rotated at an angle, can use trig functions to achieve this. This lesson will show you how to transform conic sections via rotation.

  • Understand which conic sections can be rotated and which ones cannot
  • Learn the formulaic approach for applying a rotational transformation to a conic section

Miscellaneous and Calculus Prep Topics - Polar Coordinates

The Polar Axis System

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Polar Coordinates

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Tags

  • Angle Measurement
  • Coordinate Plane
  • Polar Coordinates

Priority: High

For some topics, it will be more convenient (and often required) to measure the location of points in the plane using a different approach than the $(x,y)$ system we're used to using. This lesson introduces the Polar Axis coordinate measurement system, as well as its relationship to the usual rectangular system.

  • Describe the location of points in the plane using the polar axis system $(r,\theta)$ instead of rectangular $(x,y)$ system we usually use
  • Define relationships among $x$, $y$, $r$, and $\theta$.
  • Be able to swiftly convert between rectangular and polar coordinates
Equations and Functions in Polar Coordinates

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Polar Coordinates

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Tags

  • Functions
  • Algebra Manipulation
  • Coordinate Plane
  • Polar Coordinates

Priority: High

In general functions are operations that require an input and compute an output. This lesson introduces functions of the form $r(\theta)$ instead of the usual $f(x)$ that we are accustomed to. Here we input some angle $\theta$ into the function and it gives distance from the origin as output.

  • Define polar functions conceptually and understand what the input / output means
  • See what kind of unique graphs are possible using polar equations and functions
  • Graph simple polar functions by hand using common angles
  • Convert equations from polar to rectangular coordinates, if possible
  • Learn how to graph polar functions with your graphing calculator
Polar Form of Complex Numbers

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Polar Coordinates

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Tags

  • Imaginary Numbers
  • Trig Functions
  • Multiplying
  • Dividing
  • Coordinate Plane
  • Polar Coordinates

Priority: Normal

In Algebra Two we learned about imaginary and complex numbers, which had a few convenient uses in advanced algebra concepts such as solving polynomial equations. We saw how we could represent complex numbers in the real-imaginary coordinate plane with $x$ and $y$ style coordinates, but they have even more use if we look at that same plane using polar coordinates.

  • Apply polar coordinate methods to the Argand Real-Imaginary Plane to plot complex numbers
  • Be able to translate complex numbers between rectangular coordinate form and polar coordinate form
  • Be able to consistently define a complex number in polar form using the cis notation
  • Learn how to quickly multiply or divide two complex numbers in polar form, and compare the process to complex number arithmetic using $a+bi$ form (which is more difficult)
CIS and De Moivre's Theorem

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Polar Coordinates

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Tags

  • Imaginary Numbers
  • Coordinate Plane
  • n-th Roots
  • Square Roots
  • Polar Coordinates

Priority: Normal

When we studied radicals and polynomials, the Fundamental Theorem of Algebra dictated that there are $n$ $n$-th roots for any number, thought they do not all have to be real. Using polar complex "cis" form that we learned in the last lesson, here we will learn how to find all the roots of any real or complex number.

  • Continue building familiarity with the cis polar form which was introduced in the last lesson
  • Learn De Moivre's Theorem, which has to do with raising a complex number to an exponent
  • Use De Moivre's Theorem to find all $n$ of the $n$th-roots of a complex number
  • Since real numbers are a subset of complex numbers, see how to use De Moivre's Theorem to find all $n$ of the $n$th-roots of a real number
Classic Polar Functions

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Polar Coordinates

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Tags

  • Polar Coordinates
  • Coordinate Plane
  • Graphing
  • Functions

Priority: Normal

Continuing on with our study of polar functions, this lesson examines some common polar function forms that create graphs of spirals, roses, and cardioids, among other things. Beside the function form that each has, we will also strive to understand some common properties they share as well.

  • Become familiar with functions that graph as spirals, circles, cardioids, lemniscates, and roses
  • For each form, understand how the coefficients being positive or negative affect the graph
  • For each form that involves a trig function, know the graphical differences between using sine vs cosine in otherwise identical polar functions

Miscellaneous and Calculus Prep Topics - Vectors

Vectors as Arrows

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Vectors

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Tags

  • Vectors
  • Coordinate Plane
  • Adding
  • Subtracting

Priority: High

The concept of vectors is an incredibly useful one for applied science. In this introductory lesson on them, we will look at what they are visually and how to combine two vectors.

  • Define vectors as objects, similar to numbers but also having a direction property
  • Visually represent a vector as an arrow, the length of which dictates the magnitude
  • Learn the end-to-end method of adding and subtracting vectors
  • Know the outcome of multiplying a vector by a real number, which we call a scalar
Vector Properties and Operations

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Vectors

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Tags

  • Vectors
  • Matrices
  • Adding
  • Subtracting
  • Number Properties

Priority: High

Vectors may be given in the form of magnitude and direction or in the form of $x$ and $y$ components. This lesson focuses on the common things we do when working with vectors in component form.

  • Examine vectors in component form and understand the difference between this and examining vectors with magnitude and direction
  • Learn how to add, subtract, and scale vectors when using component form
  • Calculate the norm and / or direction of a vector based on its components
  • Define unit length vectors and learn how to find them
  • Calculate the angle between two component form vectors using the law of cosines
The Vector Dot Product

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Vectors

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Tags

  • Vectors
  • Multiplying
  • Algebra Rules
  • Number Properties

Priority: Normal

Similar to matrices, vectors are not simple numbers and thus using them in multiplication is not as plainly defined as it is for numbers. This lesson discusses one approach to the product of two vectors - the dot product - and what this operation implies conceptually and graphically

  • Examine and understand the dot product process for vectors
  • Learn various properties and axioms for the vector dot product
  • Know what is true about the dot product of perpendicular or parallel vectors
  • Learn how to find a perpendicular vector to a given vector using the dot product
Vector Application Problems

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Vectors

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Tags

  • Vectors
  • Word Problems
  • Applied Math

Priority: Normal

Using the concepts of magnitude and direction together, we'll see some common motion and force word problems, and develop an approach to vector based word problems in general.

  • Use a combination of algebra and graphing to set up vector based word problems
  • Represent moving objects and / or forces on a diagram using vectors in the form of arrows with both a magnitude and direction
  • Combine vectors acting on the same object by adding their components and finding the resultant vector in word problem situations
The Vector Cross Product

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Vectors

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Tags

  • Vectors
  • Multiplying
  • Algebra Rules
  • Number Properties

Priority: Optional

Another vector product operation is referred to as the cross product. This approach to vector multiplication has a different conceptual and graphic interpretation from the dot product. We will examine the cross product and its differences from the dot product in this lesson.

  • Define and understand the process of taking the cross product of two vectors
  • Be able to describe the graphic interpretation of this operation, and the difference between it and the interpretation of the dot product
  • Understand why and when we would want to take the cross product of two vectors
  • Practice the cross product process for vectors of dimension 2 and 3

Miscellaneous and Calculus Prep Topics - Intro to Sequences and Series

General Recursive Formulas

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Intro to Sequences and Series

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Tags

  • Number Properties
  • Sequences
  • Number Patterns

Priority: Normal

To kick off this section on the study of formula defined patterns, we will examine recursive relationships, where each number in the pattern is defined in terms of the number that precedes it.

  • Understand conceptually what recursive formulas do, and why we need a given first term
  • Practice generating the $n$-th term in a pattern using a recursive formula
  • Use what we know about functions, inverses, and solving equations to move backward and discern a previous term using recursive formulas
Understanding Sequences

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Intro to Sequences and Series

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Tags

  • Number Properties
  • Sequences
  • Number Patterns

Priority: Normal

Sequences are similar to recursive formulas, except in a sequence, instead of using the prior term to define the next term, we will use $n$ to define the $n$-th term (where $n$ is literally representative of which term number you're looking at).

  • Understand exactly what sequences are and how they are both similar to and different from recursive relationships
  • Label sequences as finite or infinite based on whether or not we want the list to terminate
  • Identify common patterns and be able to write a term-generating rule using $n$, as to be able to find any term in a sequence
Arithmetic and Geometric Sequences

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Intro to Sequences and Series

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Tags

  • Sequences
  • Adding
  • Multiplying
  • Number Patterns
  • SAT / ACT Math

Priority: High

When a sequence uses only addition or only multiplication to get from one term to the next, the sequence has special properties (and hence a special name). We will examine common formulas and calculations associated with these types of sequences - both of which are popular topics for in-class exams and college entrance exams.

  • Define an arithmetic sequence and know its formula for quickly generating the $n$-th term
  • Define a geometric sequence and know its formula for quickly generating the $n$-th term
  • Given a list of terms in an arithmetic sequence, be able to find several missing successive middle terms based on how many terms are missing
  • Be able to quickly determine whether or not a sequence is arithmetic, geometric, or neither
  • Be able to write a pattern specific formula for any arithmetic or geometric sequence, given a list of successive sequence terms.
Finite Differences

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Intro to Sequences and Series

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Tags

  • Number Properties
  • Number Patterns
  • Sequences
  • Polynomials

Priority: Optional

If a list of numbers can be obtained by a polynomial sequence-generating term, there will be a special relationship between the differences between the terms and the degree of that polynomial. This lesson looks at the topic of finite differences, which not only yields a way to generate a polynomial for sequence generating, but also similarly can be used to fit a polynomial of degree $n-1$ to $n$ data points, and find out what that polynomials is.

  • Understand what finite differences are and how to calculate them
  • Know the relationship between the number of non-constant finite differences and the degree of the polynomial that will generate the list
  • Leverage this polynomial relationship among $n$ terms to be able to find a polynomial to fit through $n$ ordered pairs in the coordinate plane
Series and Sigma Notation

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Intro to Sequences and Series

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Tags

  • Sequences
  • Number Patterns
  • Series
  • Adding

Priority: Normal

When you add up the terms of a sequence, we call it a series. This lesson introduces us to the notation used in series, and the common things we do with them.

  • Define series in terms of sequences and understand the conceptual difference between the two
  • Learn the notation and vocal commonly associated with studying series
  • Apply linear operator principles to series, such as factoring out a constant from the series, or that the series of a sum is the sum of each series
  • Develop an understanding of the difference between a finite and infinite series
  • Learn (but hopefully not memorize) formulas for special types of finite series
Arithmetic and Geometric Series

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Intro to Sequences and Series

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Tags

  • Sequences
  • Adding
  • Multiplying
  • Series
  • Word Problems

Priority: Normal

Just like arithmetic and geometric sequences had special properties that set them apart from any old sequence, so too do arithmetic and geometric series. This lesson will teach us how to work with important formulas to sum up finite arithmetic or geometric series, and flex our practice muscle with word problems.

  • Define finite arithmetic and geometric series based on how we defined finite arithmetic and geometric sequences
  • With the skills learned during study of sequences, continue to practice taking a list or sum of numbers and not only determining whether it is arithmetic, geometric, or neither, but also what its generating term is
  • Use properties of the arithmetic mean to derive the formula for the finite sum of an arithmetic series, in terms of the first and last term in the series
  • Derive a formula for a finite geometric series based on the number of terms and the multiplicative ratio
  • In word problems, distinguish between the need to use an arithmetic or geometric series to represent the described situation, and practice successfully setting up and solving series based word problems
Exploring Infinite Series

Pre-Calculus  $\rightarrow$  Miscellaneous and Calculus Prep Topics  $\rightarrow$  Intro to Sequences and Series

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Tags

  • Series
  • Adding
  • Number Patterns

Priority: Normal

Series, like sequences, can be finite or infinite, even though series means to sum terms together. If the terms have certain properties, it turns out that the sum of an the infinite list of terms is exactly equal to a specific number. Here we will understand when an infinite series does and does not add up to a finite result.

  • Learn which types of series will and will not have finite sums when there are an infinite number of terms
  • Understand what is meant when we say that a particular infinite series is divergent