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Manipulating Functions

Manipulating Functions

When we think of manipulating a function, what comes to mind? All those years of math we've taken up until now? Maybe some cobwebs in our brains start to get dusted off and some gears start to turn as we try to remember all the rules, identities, and properties of all these different types of functions, expressions, and equations (and their graphs)? And maybe we get a headache trying to recall all this information?

Well sweat not! We will review all of those things, and more, as we delve into manipulating functions for AP Calculus.

In this article, we will review all the algebra and trigonometry basics needed to manipulate these functions, equations, and expressions. Then, we will introduce how to use all these tools to manipulate algebraic equations, exponential functions, logarithmic functions, and trigonometric functions. Let's get started!

  • Manipulating functions: algebra - a quick review
    • Manipulating fractions
    • Absolute values
    • Manipulating powers
    • Manipulating roots
    • Manipulating logarithms
    • Factoring
    • Solving quadratic equations
  • Manipulating functions: a refresher on trig functions
    • SohCahToa
    • Special right triangles
    • The unit circle
    • Graphing trig functions
    • Trig identities and formulas
  • Manipulating functions: an introduction

Manipulating Functions: Algebra - A Quick Review

While functions are at the heart of calculus, algebra, and therefore algebraic manipulation, is the language of calculus. We can’t do calculus without algebra! So, if your memory is the least bit foggy on the rules for fractions, absolute values, powers, roots, logs, factoring, and solving quadratic equations, this is where we review those.

Manipulating Fractions

Fractions are everywhere in calculus. No matter how much we want to, we can’t escape them. Here are some rules for, properties of, and techniques to use with fractions that we need to remember:

  • Rules:

    • Never divide by zero! If the denominator of a fraction is zero, then it is undefined.

    • BUT, the numerator of a fraction can be zero.

      • Any fraction with a zero in the numerator is equal to zero, as long as zero is not in the denominator.

    • When adding or subtracting fractions, the denominators must be equal.

      • To add or subtract fractions, the Least Common Denominator (LCD) must be found.

    • If the denominator of a fraction is 1, then the fraction simplifies to an integer.

    • If the numerator and denominator of a fraction are equal, then the fraction simplifies to 1.

    • Rule of equivalent fractions:

      • Two fractions and are equal and can be written as if

    • Remember the reciprocal. If we flip a fraction upside down, we get its reciprocal.

      • Multiplying a fraction by its reciprocal always gives us 1, provided the numerator of the original fraction is not zero.

  • Properties:

    • Commutativity of addition with fractions:

    • Commutativity of multiplication with fractions:

    • Associativity of addition with fractions:

    • Associativity of multiplication with fractions:

    • Distributivity of multiplication over addition with fractions:

  • Techniques:

    • We can use multiplication or division to make equivalent fractions:

    • We can write integers as fractions:

    • Adding fractions:

      • , for fractions with the same denominator

      • , for fractions with different denominators

    • Subtracting fractions:

      • , for fractions with the same denominator

      • , for fractions with different denominators

    • Multiplying fractions:

    • Dividing fractions:

    • Simplifying (also called canceling) fractions:

      • It is important to remember that we can only cancel in a fraction when it has an unbroken chain of multiplication throughout the entire numerator and denominator.

      • can be simplified to , but is in its simplest form (cannot cancel here).

Absolute Values

Taking the absolute value of a number means turning a negative number positive, and doing nothing to a positive number or zero. For example:

The straight lines surrounding the number means “the absolute value of”. So when we see this notation, we know to take the absolute value of the number or variable inside.

Manipulating Powers

The rules of powers (also known as exponents) come in very handy in our study of AP Calculus. They make solving problems much easier. And don’t forget, these rules work in reverse, too! But first, let's recap a couple of definitions.

Let's take the number 2 and multiply it by itself 4 times. We get:

This gets tedious to write out, especially if we want to multiply a number by itself many times. So, we use exponents or powers to write this more concisely:

This is read as "two to the fourth power equals sixteen" or "two to the power of four equals sixteen."

Exponents or Powers are used to show the repeated multiplication of a number (or variable) by itself. In our example above, 4 is the exponent (or power). It tells us how many times to multiply the number 2 times itself.

The Base is the number (or variable) that is being multiplied. In our example above, 2 is the base.

Now, let's recap the power rules. For these, we need to make a couple of assumptions:

  1. We assume X and Y are nonzero real numbers.

  2. We assume that m and n are integers.

The power rules are listed as follows:

  • Product rule:

  • Quotient rule:

  • Negative exponent rule:

  • Zero power rule:

  • Power of a power rule:

  • Product to a power rule:

  • Quotient to a power rule:

  • Fractional power rule:

But be careful! powers do not distribute in this way. This type of power must be multiplied out the long way. Does the FOIL method ring a bell?

Manipulating Roots

As you might have noticed from the power rules, powers and roots are related. This relationship is extremely useful when solving and simplifying AP Calculus problems. Basically, any root can be converted to a power, and vice versa. The root rules are summarized below.

  • The number under an even root (2, 4, 6, etc.) can’t be negative!

  • But the number under an odd root (3, 5, 7, etc.) can be negative.

  • For roots where n is an even number:

  • For roots where n is an odd number:

  • Zero root rule:

  • Identity root rule:

  • Product rule:

  • Quotient rule:

  • Root to a root rule:

  • Rationalizing the denominator:

    • By convention, we don't want roots in the denominator of a fraction. We can remove them by a process called rationalization, where we multiply the fraction with a root in its denominator by another fraction (that otherwise would simplify to 1) with that same root in its denominator so that the root is canceled out. For example:

Roots can be rewritten as powers. It is common practice in AP Calculus to first rewrite a problem with roots in it as a problem with powers instead because powers are more visually appealing and a bit easier to understand.

Manipulating Logarithms

A logarithmic expression is another way of writing an exponential expression. The base of a logarithmic expression can be any positive number, except for 1. If we want to use e as the base of a logarithmic expression, we write ln instead of log. If we want to use 10 as the base of a logarithmic expression, by convention, we don’t write it. The rules for logarithmic expressions are summarized below.

  • Zero log rule:

  • Identity log rule:

  • Product rule:

  • Quotient rule:

  • Power rule:

  • Log of a power rule:

  • Power of a log rule:

Logarithms and powers (or exponents) are inverses of each other. It is also common practice in AP Calculus to rewrite logarithms as powers, again because powers are more visually appealing and a bit easier to understand.

The formula to convert logs to powers (and vise versa) is:

Factoring

Another super helpful tool for AP calculus is factoring. Factoring is like breaking a number into its prime factors, but for algebraic expressions. Factoring an expression involves rewriting a sum of terms as a product of terms. We do this by pulling out - or factoring - the Greatest Common Factor (GCF) from all the terms of the expression.

Let's look at the expression:

Each term in the expression has as a factor. So, this GCF can be pulled out of the expression, and it can be rewritten as:

Once we pull out the GCF, the next step is to look for one of the following patterns:

  • Difference of squares - it is absolutely necessary to know how to factor this!

    • A difference of squares can be factored, but a sum of squares cannot.

  • Sum of cubes

  • Difference of cubes

  • Square of the sum of two numbers

  • Square of the difference of two numbers

  • Cube of the sum of two numbers

  • Cube of the difference of two numbers

  • Square of the sum of three numbers

  • Square of the difference of three numbers

Solving Quadratic Equations

Going back to algebra days, a quadratic equation is a second-degree polynomial. Quadratic equations can be solved by one of the following methods.

Factoring a Quadratic Equation

If it is possible, factoring a quadratic equation is perhaps the easiest and quickest way to solve a quadratic equation. We can test if a quadratic equation is factorable by finding its discriminant. If the discriminant is a perfect square, then the quadratic equation is factorable.

Solve the quadratic equation:

Solution:

  • Move all the terms to the left side of the equals sign.

  • Factor, using the FOIL (First, Outer, Inner, Last) method to double-check they are correct.

  • Set each factor equal to zero and solve for x.

The Quadratic Formula

Regardless if a quadratic equation is factorable, we can always solve it using the quadratic formula. For a quadratic equation of the form:

We can find the solutions using the quadratic formula:

Using the same equation as the previous example, solve the quadratic using the quadratic formula.

Solution:

  • Move all the terms to the left side of the equals sign.

  • Plug the coefficients into the formula and solve.
    • In our case, a is 2, b is -5, and c is -12.

Completing the Square

This method for solving a quadratic equation is aptly named because it involves taking the quadratic formula in question and making it a perfect square so that it is solvable by taking the square root. We can always use this method to solve a quadratic equation as well, regardless of if it is factorable.

Solve the quadratic equation:

Solution:

  • Move all terms with an x in them to the left side of the equals sign, and the constant on the right side.

  • Divide both sides of the equation by the coefficient of if it isn't 1.

  • Now for the tricky part: take the coefficient of x (not ) , divide it in half, square it, and then add the result to both sides of the equation.
    • In our case, the coefficient of x is 8.
      • Half of -8 is -4. -4 squared is 16. So, we add 16 to both sides of the equation:

  • Factor the left side of the equation into a squared binomial.

    Note that the factor is always the coefficient of x, divided in half.

  • Finally, take the square root of both sides of the equation and simplify.

    Remember that the right side needs a plus/minus sign.

Manipulating Functions: A Refresher on Trig Functions

Many of the problems we will encounter in AP Calculus involve manipulating trig functions. So, for those of us who don't want to have to relearn trigonometry at the same time as learning AP Calculus, this refresher is key.

Trigonometry starts with triangles - specifically right triangles. The image and table below summarizes key terms of right triangles and the six main trig functions.

Manipulating functions right triangle with sides labeled StudySmarterA right triangle with its sides labeled - StudySmarter Originals

The 6 main trig functions

A way to remember Sin, Cos, and Tan ratios

The main trig functions are defined by ratios of the sides of a right triangle in a specific order. The side labeled Hypotenuse (or just H for short) is always the triangle's longest side. The side labeled Adjacent (or just A for short) is the side that is touching the angle of the triangle we are considering (which is θ in the image above). The side labeled Opposite (or just O for short) is the side that is across from the angle. SohCahToa (pronounced "So-Kah-Toe-Ah") is a mnemonic device used to help us remember the ratios that make up the sine, cosine, and tangent trig functions.

SohCahToa mnemonic device
SohCahToa

From there, we can remember cosecant, secant, and cotangent as the reciprocals of sine, cosine, and tangent.

SohCahToa reciprocals

With that in mind, there are two special right triangles that we deal with in a lot of AP Calculus problems that deserve mention.

Special Right Triangles

Also called an isosceles right triangle, the 45-45-90 triangle is exactly what it sounds like: a right triangle whose other two angles are 45 degrees. What is special about this triangle is that the two legs of the triangle that touch the right angle are always the same size, and the hypotenuse is always times longer than the two legs. It's a good idea to memorize the trig functions for this triangle.

The 30-60-90 right triangle is also aptly named. What is special about this triangle is that the hypotenuse is always twice as long as the shortest leg, and the longer leg is always times longer than the shorter leg. It's also good to memorize the trig functions for this triangle for both the and angles.

A more convenient way to view the trig functions of these angles (plus angles of 0 and 90 degrees) is with a table.

angle (θ) in degreesangle (θ) in radianssin(θ)cos(θ)tan(θ)csc(θ)sec(θ)cot(θ)
0010undefined1undefined
30°π/62
45°π/411
60°π/32
90°π/210undefined1undefined0

The Unit Circle

While the SohCahToa mneumonic device is great for right triangles, it doesn't work for angles at or greater than 90 degrees. For those, we need the unit circle. The unit circle allows us to find trig values for any angle, not just acute ones. The unit circle has a radius of one unit (hence the name) and is set in the x-y coordinate plane.

The above image tells us quite a lot. So let's walk through it a bit. To start, when we measure angles using the unit circle, we start at the positive x-axis and rotate counterclockwise (see the angle in the image). If we rotated clockwise, we would get a negative angle (see the angle in the image). This is true regardless of whether we are measuring the angle in degrees or radians. In AP Calculus, radians are almost always the preferred way to measure angles. Remember, the radian measure of an angle is the length of the arc along the circumference of the unit circle (see the bolded edge of the circle labeled ). The easiest way to convert between degrees and radians is the formula:

A closer look at the unit circle above reveals the two 30-60-90 right triangles. One is in the first quadrant of the circle, the other is in the second quadrant. Based on these two triangles, we can gather that the coordinates on the unit circle tell us the angle's cosine and sine values.

The endpoint of an angle on the unit circle gives us, in order, the angle's cosine and sine values. The x-coordinate is the cosine value, and the y-coordinate is the sine value. We can remember the order by remembering x and y are in alphabetical order just like cosine and sine.

With this in mind, let's see the unit circle with all of the cosine and sine values plotted for the 30-60-90 and 45-45-90 triangles.

It is important to note that the values of all the main trig functions are positive in the first quadrant, only sine and cosecant are positive in the second quadrant, only tangent and cotangent are positive in the third quadrant, and only cosine and secant are positive in the fourth quadrant of the unit circle. Another helpful mnemonic device to remember which functions are positive in what quadrants is All Students Take Calculus.

  • All - All 6 trig functions are positive in the first quadrant.

  • Students - only Sine (and its reciprocal, cosecant) are positive in the second quadrant.

  • Take - only Tangent (and its reciprocal, cotangent) are positive in the third quadrant.

  • Calculus - only Cosine (and its reciprocal, secant) are positive in the fourth quadrant.

Graphing Trig Functions

The main trig functions - sine, cosine, and tangent - and their reciprocals - cosecant, secant, and cotangent - are periodic functions. This means their graphs are the same shape repeated over and over again, indefinitely. The period of these functions is the length of one of their cycles.

Manipulating functions graphs of all 6 trig functions StudySmarterThe graphs of all 6 trig functions - StudySmarter Originals

Now that we have refreshed the unit circle, we can sketch the graphs of sine, cosine, and tangent easily by noting some important facts:

  • For sine and cosine:

    • The graphs of sine and cosine are the same shape.

      • The cosine graph is just shifted to the left 90 degrees.

    • The sine and cosine graphs have a max of 1 and a min of -1.

    • The sine and cosine graphs repeat their shapes indefinitely to the left and right.

      • Their shapes repeat every 360 degrees, so their period is 360 degrees (which is the same number of degrees a circle has)

    • The unit circle tells us that:

    • Using these 5 points, we can sketch one cycle of the sine and cosine functions.

  • For tangent:

    • The basic shape of the graph is a backward S

      • This shape is repeated indefinitely to the left and right.

        • It repeats every 180 degrees, so its period is 180 degrees.

    • Because , we can use the unit circle to sketch the graph:

      • Because , we draw vertical asymptotes at those points.

Trig Identities and Formulas

The most common way we will want to solve AP Calculus problems that involve trig functions is by using trig identities. Several "gotcha" moments in AP Calculus revolve around us remembering that these identities exist! Plus, trig identities help us solve more complex calculus problems much more quickly and easily. So, let's list them right here.

Ratio Trig Identities

First, recall that the tangent is a ratio of sine/cosine, and therefore cotangent is the ratio of cosine/sine.

Reciprocal Trig Identities

Remember which functions are reciprocals of each other.

Pythagorean Trig Identities

The Pythagorean trig identities are probably some of the most useful trig identities. These are based on the Pythagorean Theorem.

Identity 2 is derived by dividing identity 1 by .

Identity 3 is derived by dividing identity 1 by .

Opposite Angle Trig Identities

These are also known as even/odd identities. As you can see, cosine and secant are the only two even functions, the rest of the trig functions are odd functions.

Periodic Trig Identities

If n is any integer, then the following identities hold true.

These come from the fact that the period of sine and cosine and their reciprocals is , and the period of tangent and cotangent is , so shifting these by this much left or right results in the same function.

Cofunction Trig Identities

These show which trig functions are complements of each other.

  • Sine and cosine are complements, so they arecofunctions of each other.

  • Tangent and cotangent are complements, so they are cofunctions of each other.

  • Secant and cosecant are complements, so they are cofunctions of each other.

Double Angle Trig Identities

The double angle trig identities are also very useful to remember.

Half-Angle Trig Identities

The half-angle trig identities are also very useful to remember. The plus or minus depends on in which quadrant the original given value exists.

Sum and Difference of Angles Trig Identities

The sum and difference of angles trig identities are helpful for simplifying complex-looking expressions.

Sum to Product Trig Identities

The sum-to-product trig identities take a sum of two trig functions and convert them to a product of trig functions.

Product to Sum Trig Identities

The product-to-sum trig identities take a product of two trig functions and convert them to a sum of trig functions.

The Laws of Sines, Cosines, and Tangents

For triangles that aren't right triangles, we also have the Law of Sines, the Law of Cosines, and the Law of Tangents.

Manipulating functions triangle with no right angle StudySmarterA triangle with no right angle - StudySmarter Originals

Law of SinesLaw of CosinesLaw of TangentsMollweide's Formula

Manipulating Functions: An Introduction

Now that we've got all our review out of the way, let's introduce the ways we can manipulate functions. There are four main ways we can manipulate functions:

Algebra of Functions, or, Algebraic Manipulation

The first way we can manipulate functions is by algebraic manipulation. This usually means adding, subtracting, multiplying, and/or dividing a value from a function, or adding, subtracting, multiplying, and/or dividing two or more functions. It can also mean taking a function to a power, root, log, etc. Algebraic manipulation can even mean simplifying an expression, equation, or function. We can algebraically manipulate algebraic equations, exponential functions, logarithmic functions, and trig functions.

Manipulating Algebraic Equations

Before we get into manipulating algebraic equations, let's review what exactly an algebraic equation is.

An Algebraic Equation is like a scale: two expressions, or quantities, sit on either side of an equals sign. The equals sign means that the expression on the left and the expression on the right must be equal to each other.

In other words, "equation" means "equality". Algebraic equations involve equating one quantity with another.

Essentially what we have to do here is make sure we keep both sides of the equals sign equal, as we would with a balance scale. What we do to one side of the equation, we must do to the other side to keep the balance!

Manipulating Exponential Functions

Before we get into manipulating exponential functions, let's review what exactly an exponential function is.

An Exponential Function is a function that looks very similar to a quadratic function, but with a very important twist: the independent variable and exponent are switched!

If b is any real number greater than 0, but not 1, then an exponential function is a function of the form,

where b is called the base and x is any real number.

Again, what we need to do here is make sure we treat the equals sign like a scale, just like we did with the algebraic equations.

Manipulating Logarithmic Functions

Before we get into manipulating logarithmic functions, let's review what exactly a logarithmic function is.

A Logarithmic Function is the inverse of the exponential function and therefore has a similar definition.

If b is any real number greater than 0, but not 1, then a logarithmic function is a function of the form,

where b is called the base, and x is any real number.

Again, what we need to do here is make sure we treat the equals sign like a scale, just like we did with the algebraic equations.

Manipulating Trig Functions

Before we get into manipulating trig functions, let's review what exactly a trig function is.

A Trig Function (also called a circular function) is a function of a circle (or just an arc) or angle that, at its core, is most simply expressed in terms of the ratios of two sides of a right triangle.

The 6 main trig functions are:

  • Sine and its reciprocal Cosecant
  • Cosine and its reciprocal Secant
  • Tangent and its reciprocal Cotangent

Again, what we need to do here is make sure we treat the equals sign like a scale, just like we did with the algebraic equations.

For more information, check out our Algebra of Functions article! There, we cover these topics much more in-depth, and with plenty of examples.

Function Transformations

All the function transformations you have learned up until now still apply in AP Calculus. Any function can be transformed, both horizontally and/or vertically, by shifting, reflecting, stretching, or shrinking it. Horizontal transformations only change the x-coordinates of points. Vertical transformations only change the y-coordinates of points.

Horizontal Transformations of Graphs

Horizontal transformations are made when we either add/subtract a number from a function's input variable, usually x, or multiply x by a number. Horizontal transformations, except reflection, work in the opposite way we'd expect them to. Here is a summary of how horizontal transformations work:

  • Shifts - Adding a number to x shifts the function to the left, subtracting shifts it to the right.

  • Shrinks - Multiplying x by a number greater than 1 shrinks the function horizontally (think horizontal compression).

  • Stretches - Multiplying x by a number less than 1 stretches the function horizontally.

  • Reflections - Multiplying x by -1 reflects the function horizontally (over the y-axis).

Vertical Transformations of Graphs

Vertical transformations are made when we either add/subtract a number from the entire function, or multiply the entire function by a number. Unlike horizontal transformations, vertical transformations work the way we expect them to (yay!). Here is a summary of how vertical transformations work:

  • Shifts - Adding a number to the entire function shifts it up, subtracting shifts it down.

  • Shrinks - Multiplying the entire function by a number less than 1 shrinks the function vertically (think vertical compression).

  • Stretches - Multiplying the entire function by a number greater than 1 stretches the function vertically.

  • Reflections - Multiplying the entire function by -1 reflects it vertically (over the x-axis).

For more information, check out our Function Transformations article! There, we cover the topic much more in-depth, and with plenty of examples.

Combining Functions

Combining functions is another way to manipulate functions in AP Calculus. There are two main ways to combine functions:

  1. Using algebraic manipulation (as described above) to add/subtract/multiply/divide two or more functions.

  2. Substituting the independent variable of one function with another function in a process called function composition.

Since we talked about algebraically manipulating functions already, let's discuss function composition here. Function composition involves taking one function, say , and plugging it into another function, say , and then solving, usually for a value of x.

This also comes with its own notation: both of which are read as "f of g of x".

It is important to note that when combining functions, the order matters!

In other words, .

For more information, check out our Combining Functions article! There, we cover the topic much more in-depth, and with plenty of examples.

Symmetry of Functions

Certain functions display properties of symmetry that help us understand them and the shapes of their graphs. There are two types of symmetry when we talk about functions and their graphs:

  • Even - If a function has even symmetry, that means it is symmetrical with respect to the y-axis.
  • Odd - If a function has odd symmetry, that means it is symmetrical with respect to the origin.

For more information, check out our Symmetry of Functions article! There, we cover the topic much more in-depth, and with plenty of examples.

Manipulating Functions - Key takeaways

  • To be able to manipulate functions, we need to review algebra and trig concepts.
  • There are 4 main ways to manipulate functions.
    1. Algebraic manipulation (algebra of functions):
      • The addition, subtraction, multiplication, and division of functions, or
      • Raising a function to a power, taking it to a root or absolute value, or
      • Any other algebraic operation we can think of.
    2. Function transformations (manipulating the graphs of functions):
      • This happens in two ways:
        • Horizontal transforms of a graph.
        • Vertical transforms of a graph.
    3. Combining functions (function composition):
      • Plugging one function into another and solving for a variable.
    4. Symmetry of functions:
      • Knowing how the graph of a function should look based on a function's properties of symmetry.
      • There are two types of symmetry when we talk about functions and their graphs:
        • Even - the graph is symmetrical with respect to the y-axis.
        • Odd - the graph is symmetrical with respect to the origin.
  • For AP Calculus, it is important to know how to:
    • Manipulate algebraic equations

    • Manipulate exponential functions

    • Manipulate logarithmic functions

    • Manipulate trig functions

Frequently Asked Questions about Manipulating Functions

There are three main ways to manipulate a function.

  1. Algebraic manipulation
  2. Function composition
  3. Transformations

Manipulating functions using algebra could mean adding, subtracting, multiplying, or dividing a value from a function, or it could mean adding, subtracting, multiplying, or dividing two or more functions together.

Manipulating functions using function composition, or combining functions, means using one function as an input to another and then solving.

Manipulating functions using transformations means algebraically manipulating either the entire function or just the independent variable to change how the graph of the function looks.

You can manipulate exponential functions by using algebraic manipulation, function composition, and/or transformations.

You can manipulate rational functions by using algebraic manipulation, function composition, and/or transformations.

But be careful! You must avoid having 0 in the denominator of a rational function!

You can manipulate log functions by using algebraic manipulation, function composition, and/or transformations.

Final Manipulating Functions Quiz

Question

Can the denominator of a fraction be zero?

Show answer

Answer

No, if the denominator of a fraction is zero, then it is undefined.

  • BUT, the numerator of a fraction can be zero.

    • Any fraction with a zero in the numerator is equal to zero, as long as zero is not in the denominator.

Show question

Question

What is the key to adding/subtracting fractions?

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Answer

When adding or subtracting fractions, the denominators must be equal.

  • To add or subtract fractions, the Least Common Denominator (LCD) must be found.

Show question

Question

What happens when a fraction is multiplied by its reciprocal?

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Answer

Multiplying a fraction by its reciprocal always gives us 1, provided the numerator of the original fraction is not zero.

Show question

Question

What does it mean to take an absolute value of a number or variable?

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Answer

Taking the absolute value of a number (or variable) means turning a negative number (or variable) positive, and doing nothing to a positive number (or variable) or zero.

Show question

Question

What does the endpoint of an angle on the unit circle give us?

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Answer

The endpoint of an angle on the unit circle gives us, in order, the angle's cosine and sine values. The x-coordinate is the cosine value, and the y-coordinate is the sine value. We can remember the order by remembering x and y are in alphabetical order, just like cosine and sine.

Show question

Question

What is a helpful mnemonic device to remember which functions are positive in what quadrants of the unit circle?

Show answer

Answer

All Students Take Calculus

  • All - All 6 trig functions are positive in the first quadrant.

  • Students - only Sine (and its reciprocal, cosecant) are positive in the second quadrant.

  • Take - only Tangent (and its reciprocal, cotangent) are positive in the third quadrant.

  • Calculus - only Cosine (and its reciprocal, secant) are positive in the fourth quadrant.

Show question

Question

What are the 3 main ways to manipulate functions?

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Answer

  1. Function composition
  2. Function transformations
  3. Algebraic manipulation

Show question

Question

What does function composition involve?

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Answer

Function composition involves taking one function, plugging it into another function, and then solving, usually for a value of x.

Show question

Question

How can a graph be transformed?

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Answer

Horizontally and Vertically

Show question

Question

What are the 4 ways a graph can be horizontally and/or vertically transformed?

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Answer

  1. Shifts
  2. Shrinks
  3. Stretches
  4. Reflections

Show question

Question

How do shifts, shrinks, stretches, and reflections work for horizontal transformations?

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Answer

  • Shifts - Adding a number to x shifts the function to the left, subtracting shifts it to the right.

  • Shrinks - Multiplying x by a number greater than 1 shrinks the function.

  • Stretches - Multiplying x by a number less than 1 stretches the function.

  • Reflections - Multiplying x by -1 reflects the function horizontally (over the y-axis).

Show question

Question

How do shifts, shrinks, stretches, and reflections work for vertical transformations?

Show answer

Answer

  • Shifts - Adding a number to the entire function shifts it up, subtracting shifts it down.

  • Shrinks - Multiplying the entire function by a number less than 1 shrinks the function.

  • Stretches - Multiplying the entire function by a number greater than 1 stretches the function.

  • Reflections - Multiplying the entire function by -1 reflects it vertically (over the x-axis).

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Question

How can functions be manipulated using algebraic manipulation?

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Answer

Functions can be algebraically manipulated by:

  • Adding, subtracting, multiplying, and/or dividing a value from a function.
  • Adding, subtracting, multiplying, and/or dividing two or more functions.
  • Taking a function to a power, root, log, etc.
  • Simplifying an expression, equation, or function.

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Question

What is the Axis (or Line) of Symmetry?

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Answer

The Axis (or Line) of Symmetry is the line that passes through the center of a shape and divides it into identical halves, so that one half is the mirror-image of the other half.

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Question

What is an even function?

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Answer

If a function has even symmetry, that means it is symmetrical with respect to the y-axis.

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Question

What is an odd function?

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Answer

If a function has odd symmetry, that means it is symmetrical with respect to the origin.

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Question

What are the two ways to check if a function is even or odd?

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Answer

Graphically and Algebraically

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Question

True or False?

Most functions are neither even nor odd.

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Answer

True

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Question

Which of the following is the only function that is both even and odd?

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Answer

The zero function

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Question

How do you algebraically prove a function is even or odd?

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Answer

By replacing the input value (x) with a negative input (-x) and considering the output.

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Question

How do you graphically prove a function is even or odd?

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Answer

By checking for reflective symmetry with respect to the y-axis or for rotational symmetry with respect to the origin.

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Question

Which of the following results in an even function?

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Answer

Adding two even functions.

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Question

Which of the following results in an odd function?

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Answer

Adding two odd functions.

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Question

What is the axis of symmetry equation for a parabola?

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Answer

  • For a vertical parabola whose vertex is (h, k), the axis of symmetry equation is: x = h
  • For a horizontal parabola whose vertex is (h, k), the axis of symmetry equation is: y = k

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Question

Which trig functions are even functions?

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Answer

sine and cosecant

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Question

True or False?

All trig functions are either even or odd.

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Answer

True

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Question

What is combining functions?

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Answer

Combining functions is a way to create new functions from existing ones. This can be done using basic arithmetic (addition, subtraction, multiplication, division) or by function composition.

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Question

What is function composition?

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Answer

Function composition is a way to combine functions outside basic arithmetic. To compose functions, one function is substituted into another and then evaluated.

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Question

When arithmetically combining functions (using addition, subtraction, multiplication, and/or division), which is the domain of the combined function?

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Answer

The domain of the first function

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Question

What else do we need to do to the domain of a combined function if we divide two functions?

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Answer

Nothing

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Question

When we evaluate arithmetically combined functions, we can either...

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Answer

combine the functions, then evaluate

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Question

Can a function be composed with itself?

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Answer

Yes!

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Question

What is function decomposition?

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Answer

Function decomposition is the process of taking a function and breaking it down into multiple, simpler functions.

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Question

What needs to be considered when combining piecewise functions?

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Answer

When combining piecewise functions, the domain of each case needs to be considered, since it is possible that only one piece of the function could be defined at a point.

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Question

What are function transformations?

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Answer

Function transformations are the processes used on an existing function and its graph to give us a modified version of that function and its graph that has a similar shape to the original function.

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Question

We can break down the transformations into two major categories:

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Answer

Horizontal transformations

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Question

Any function can be transformed, true or false?

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Answer

True

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Question

There are four main types of transformations:

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Answer

Horizontal and vertical shifts (or translations)

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Question

Horizontal transformations only change the _-coordinates of functions

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Answer

x

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Question

Vertical transformations only change _-coordinates of functions.

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Answer

y

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Question

Horizontal transformations are made when we either add/subtract a number from ___ or multiply it by a number.

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Answer

A function's input variable (usually x)

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Question

Horizontal transformations, except ___, work in the opposite way we'd expect them to.

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Answer

Reflections

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Question

Summarize how horizontal transformations work.

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Answer

  • Horizontal Shifts - Adding a number to x shifts the function to the left, subtracting shifts it to the right.

  • Horizontal Shrinks - Multiplying x by a number whose magnitude is greater than 1 shrinks the function.

  • Horizontal Stretches - Multiplying x by a number whose magnitude is between 0 and 1 stretches the function.

  • Horizontal Reflections - Multiplying x by -1 reflects the function horizontally (over the y-axis).

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Question

Vertical transformations are made when we either add/subtract a number from ___, or multiply it by a number.

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Answer

The entire function

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Question

Vertical transformations work the way we expect them to, true or false?

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Answer

True

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Question

Summarize how vertical transformations work.

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Answer

  • Vertical Shifts - Adding a number to the entire function shifts it up, subtracting shifts it down.

  • Vertical  Shrinks - Multiplying the entire function by a number whose magnitude is between 0 and 1 shrinks the function.

  • Vertical  Stretches - Multiplying the entire function by a number whose magnitude is greater than 1 stretches the function.

  • Vertical  Reflections - Multiplying the entire function by -1 reflects it vertically (over the x-axis).

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Question

What is a common mistake when identifying whether a transformation is horizontal or vertical?

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Answer

When identifying whether a transformation is horizontal or vertical, keep in mind that transformations are only horizontal if they are applied to x when it has a power of 1.

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Question

To determine when the order of transformations affects the graph:

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Answer

  • The order of transformations matters when there are two (or more) of the same type, that is, horizontal or vertical, of transformations.
  • The order of transformations does not matter if there is one horizontal and one vertical transformation.

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Question

The correct order in which to perform function transformations is as follows:

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Answer

  1. Start with the parentheses (horizontal shifts)

    • If the power of x is not 1, this is a vertical shift!

  2. Apply the multiplication (stretches and/or shrinks)

  3. Apply the negations (reflections)

  4. Apply the addition/subtraction (vertical shifts)

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