Topics
4. Modeling with Functions
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United States of America
Grades 9-12

4.03 Function Transformations

Lesson
Practice

Introduction

In the previous lesson, we reviewed the key features of the function families we learned about in Math 1. In Math 1 lesson  Transformations of functions  , we explored the ways we could transform functions. This lesson will review those transformations and prepare us for transforming the new types of functions we will learn about in Math 2.

Function transformations

Exploration

Move the sliders to see how each one affects the graph. Choose different functions to compare the affects across the various graphs.

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  1. How does each slider affect the graph?

A transformation of a function is a change in the position or shape of its graph. The function in any family with the simplest form is known as the parent function, and we frequently consider transformations as coming from the parent function. In the examples of transformations shown below, the parent function is shown as a dashed line.

Reflection

A transformation that produces the mirror image of a figure across a line

A reflection across the x-axis can be represented algebraically by g\left(x\right) = -f\left(x\right)A reflection across the y-axis can be represented algebraically by g\left(x\right) = f\left(-x\right)

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Reflection across the x-axis: {g\left(x\right)=-f\left(x\right)}
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Reflection across the y-axis: {g\left(x\right)=f\left(-x\right)}
Translation

A transformation in which every point in a figure is moved in the same direction and by the same distance

Translations can be categorized as horizontal (moving left or right, along the x-axis) or vertical (moving up or down, along the y-axis), or a combination of the two.

Vertical translations can be represented algebraically by g\left(x\right) = f\left(x\right) + kwhere k > 0 translates upwards and k < 0 translates downwards.

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Vertical translation of 4 units upwards: {g\left(x\right) = f\left(x\right) + 4}
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Vertical translation of 4 units downwards: {g\left(x\right)=f\left(x\right)-4}

Similarly, horizontal translations can be represented by g\left(x\right) = f\left(x - h\right) where h > 0 translates to the right and h < 0 translates to the left.

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Horizontal translation of 3 units left: {g\left(x\right) = f\left(x+3\right)}
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Horizontal translation of 3 units to the right: {g\left(x\right) = f\left(x - 3\right)}
Vertical compression

A transformation that scales all of the y-values of a function by a constant factor towards the x-axis

Vertical stretch

A transformation that scales all of the y-values of a function by a constant factor away from the x-axis

A vertical compression or stretch can be represented algebraically by g\left(x\right) = af\left(x\right)where 0 < \left|a\right| < 1 corresponds to a compression and \left|a\right| > 1 corresponds to a stretch.

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Vertical compression with scale factor of 0.5: {g\left(x\right) = 0.5f\left(x\right)}
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Vertical stretch with a scale factor of 2: {g\left(x\right) = 2f\left(x\right)}
Horizontal compression

A transformation that scales all of the x-values of a function by a constant factor toward the y-axis

Horizontal stretch

A transformation that scales all of the x-values of a function by a constant factor away from the y-axis

A horizontal compression or stretch can be represented algebraically by g\left(x\right) = f\left(bx\right)where \left|b\right| > 1 corresponds to a compression and 0 < \left|b\right| < 1 corresponds to a stretch.

For horizontal stretches and compressions, b=\dfrac{1}{\text{scale factor}}.

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Horizontal compression with a scale factor of 0.5: {g\left(x\right)=f\left(2x\right)}
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Horizontal stretch by a scale factor of 2: {g\left(x\right)=f\left(0.5x\right)}

When performing multiple transformations at once, we use the standard function notation a\cdot f\left[b\left(x-h\right)\right]+k with the correct values of a,b,h and k to apply transformations to f\left(x\right). When given a transformed function, we must convert it back to standard notation to correctly identify the transformations applied to the parent function.

Examples

Example 1

A function is shown in the graph below. Determine an equation for the function after it has been reflected across the x-axis and translated 4 units to the left.

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Worked Solution
Create a strategy

There are two main approaches we can use here:

  • Apply the transformations to the graph, then determine the equation of the final graph
  • Determine the equation of the graph shown, then apply the transformations algebraically
Apply the idea

Choosing to apply the transformations to the graph first results in the following:

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Looking at the final graph, we can see that it is a straight line which passes through the origin and has a slope of -\dfrac{1}{2}. So, an equation for this function is g\left(x\right) = -\frac{x}{2}

Reflect and check

The original function has an equation of f\left(x\right) = \dfrac{x}{2} - 2.

A reflection across the x-axis is represented by changing the signs of all the function values (i.e. multiplying throughout by -1), so this transformation results in h\left(x\right) = -\dfrac{x}{2} + 2.

A horizontal translation of 4 units to the left is represented by adding 4 directly to the x-values. So, this transformation results in

\displaystyle g\left(x\right)\displaystyle =\displaystyle -\frac{x + 4}{2} + 2
\displaystyle =\displaystyle -\frac{x}{2} - \frac{4}{2} + 2
\displaystyle =\displaystyle -\frac{x}{2} - 2 + 2
\displaystyle =\displaystyle -\frac{x}{2}

which is the same result.

Example 2

Point A\left(-3, 9\right) lies on the graph of f\left(x\right). Determine the coordinates of the corresponding point on the graph of g\left(x\right) = \dfrac{1}{3}\cdot f\left(x + 4\right).

Worked Solution
Create a strategy

We can use the given expression to determine the transformations from f\left(x\right) to g\left(x\right), then apply these transformations to the point A.

Apply the idea

In the expression \dfrac{1}{3}\cdot f\left(x + 4\right), the +4 inside of function f indicates a horizontal translation of 4 units to the left, while the \dfrac{1}{3} outside of function f indicates a vertical shrink by a factor of \dfrac{1}{3}.

Shifting the point left 4 units takes the point to \left(-7, 9\right), and shrinking it vertically by 3 results in the point \left(-7, 3\right).

Reflect and check

We can also think about these transformations algebraically.

The only point we know on the graph of f is A, which tells us that f\left(-3\right) = 7. We can rewrite this to be in the correct form for g\left(x\right) as follows:

\displaystyle f\left(-3\right)\displaystyle =\displaystyle 9Known point
\displaystyle f\left(-7 + 4\right)\displaystyle =\displaystyle 9Rewrite -3 in the form ⬚ + 4
\displaystyle \dfrac{1}{3}\cdot f\left(-7 + 4\right)\displaystyle =\displaystyle 3Multiply both sides by \dfrac{1}{3}
\displaystyle g\left(-7\right)\displaystyle =\displaystyle 3Use definition of g\left(x\right)

So, the corresponding point is \left(-7, 3\right).

Example 3

The graph of a function f\left(x\right) is shown below.

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a

Determine the equation after the function has been translated 6 units right and horizontally stretched by a factor of 2.

Worked Solution
Create a strategy

The graph has a v-shape, which tells us that it is an absolute value function. The parent absolute value function is f\left(x\right)=\left|x\right|. We need to give the transformed function a new name, like g\left(x\right), since we are creating a new function.

A horizontal translation and a horziontal stretch are represented by b and h in the function notation g\left(x\right)=f\left[b\left(x-h\right)\right]. Remember b=\dfrac{1}{\text{stretch factor}} for horizontal stretches and compressions.

Apply the idea

In this case, the stretch factor is 2, so b=\dfrac{1}{2}. To translate the function 6 units right, h=6. To find the equation after it has been horizontally translated and stretched, we need to find g\left(x\right)=f\left[\dfrac{1}{2}\left(x-6\right)\right].

\displaystyle g\left(x\right)\displaystyle =\displaystyle f\left[\dfrac{1}{2}\left(x-6\right)\right]
\displaystyle =\displaystyle \left|\dfrac{1}{2}\left(x-6\right)\right|
\displaystyle =\displaystyle \left|\dfrac{1}{2}x-3\right|

The equation of the transformed function is g\left(x\right)=\left|\dfrac{1}{2}x-3\right|.

Reflect and check

After we simplified the equation, the transformations that we applied to the graph are no longer easy to identify. If one is not careful, it may seem as if this function was only translated 3 units to the right. However, that is not the case as we know this function was translated 6 units to the right.

In order to identify the transformations when given an equation, we must convert it to standard form first by factoring out the coefficient of x:g\left(x\right)=a\cdot f\left[b\left(x-h\right)\right]+k

The standard form of the equation in this example is g\left(x\right)=\left|\frac{1}{2}\left(x-6\right)\right| which shows that the parent absolute value function was translated 6 units to the right and stretched horizontally by a factor of 2.

b

Graph g\left(x\right) and f\left(x\right) on the same coordinate plane.

Worked Solution
Create a strategy

A horizontal stretch will make the graph wider, and horizontal translation of 6 units to the right will move the vertex to \left(6,0\right). We will stretch the graph horizontally to determine the correct shape of the graph, then we will shift it to the right.

Apply the idea

Using a table of values for f\left(x\right) can help us stretch the function horizontally. A table of values for the parent absolute value function is:

x-4-3-2-101234
f\left(x\right)432101234

We can double the x-values and leave the y-values the same in order to stretch the graph horizontally by a factor of 2.

x-8-6-4-202468
y432101234

Using the tables to graph both functions, we can see we have stretched f\left(x\right) horizontally by 2.

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Now, we simply need to shift all the points on the stretched graph to the right 6 units.

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Reflect and check

In general, when the transformations are both horizontal transformations (or a reflection across the y-axis), apply the reflections, stretches, and compressions first. Apply the translations last. The same rule applies when the transformations are both vertical (or a reflection across the x-axis).

Example 4

Describe how g\left(x\right)=3^{x-5}-6 has been transformed from its parent function, f\left(x\right)=3^{x}.

Worked Solution
Create a strategy

When we compare f\left(x\right) and g\left(x\right), we can see that there are numbers that have been added and subtracted. Addition and subtraction represent translations.

Apply the idea

5 is subtracted from the input values and 6 is substracted from the output values. In function notation, it can be represented as:

g\left(x\right)=f\left(x-5\right)-6

The graph of g\left(x\right) is the graph of f\left(x\right) after it has been translated right 5 units and down 6 units.

Reflect and check

When we graph the functions, we can see that f\left(x\right) has been shifted right 5 units and down 6 units to obtain the graph of g\left(x\right).

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Idea summary

The reflections and translations can be summarized as follows:

A figure showing a summary of reflections and translation. The expression a f of left bracket b left parenthesis x minus h right parenthesis right bracket plus k is shown. Under the column labeled vertical: for a: when a is greater than 0, no reflection; when a is less than 0, reflection across the x-axis; when absolute value of a is greater than 1, vertical stretch; and when absolute value of a is between 0 and 1, vertical compression; for k: when k is greater than 0, vertical translation up; and when k is less than 0, vertical translation down. Under the column labeled horizontal: for b: when b is greater than 0, no reflection; when b is less than 0, reflection across the y-axis; when absolute value of b is greater than 1, horizontal compression; and when absolute value of b is between 0 and 1, horizontal stretch; for h: when h is greater than 0, horizontal translation right; and when h is less than 0, horizontal translation left.

Outcomes

F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

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