1.8: Transformations of Functions

Last updated
Save as PDF

Page ID: 89284

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

Motivating Questions

How is the graph of $y = g(x) = af(x-b) + c$ related to the graph of $y = f(x)\text{?}$
What do we mean by “transformations” of a given function $f\text{?}$ How are translations and vertical stretches of a function examples of transformations?

In our preparation for calculus, we aspire to understand functions from a wide range of perspectives and to become familiar with a library of basic functions. So far, two basic families functions we have considered are linear functions and quadratic functions, the simplest of which are $L(x) = x$ and $Q(x) = x^2\text{.}$ As we progress further, we will endeavor to understand a “parent” function as the most fundamental member of a family of functions, as well as how other similar but more complicated functions are the result of transforming the parent function.

Informally, a transformation of a given function is an algebraic process by which we change the function to a related function that has the same fundamental shape, but may be shifted, reflected, and/or stretched in a systematic way. For example, among all quadratic functions, the simplest is the parent function $Q(x) = x^2\text{,}$ but any other quadratic function such as $g(x) = -3(x-5)^2 + 4$ can also be understood in relation to the parent function. We say that “$g$ is a transformation of $f\text{.}$”

In Preview Activity $\PageIndex{1}$, we investigate the effects of the constants $a\text{,}$ $b\text{,}$ and $c$ in generating the function $g(x) = af(x-b) + c$ in the context of already knowing the function $f\text{.}$

Preview Activity $\PageIndex{1}$

Open a new Desmos graph and define the function $f(x) = x^2\text{.}$ Adjust the window so that the range is for $-4 \le x \le 4$ and $-10 \le y \le 10\text{.}$

In Desmos, define the function $g(x) = f(x) + a\text{.}$ (That is, in Desmos on line 2, enter g(x) = f(x) + a.) You will get prompted to add a slider for $a\text{.}$ Do so.
Explore by moving the slider for $a$ and write at least one sentence to describe the effect that changing the value of $a$ has on the graph of $g\text{.}$
Next, define the function $h(x) = f(x-b)\text{.}$ (That is, in Desmos on line 4, enter h(x) = f(x-b) and add the slider for $b\text{.}$)
Move the slider for $b$ and write at least one sentence to describe the effect that changing the value of $b$ has on the graph of $h\text{.}$
Now define the function $p(x) = cf(x)\text{.}$ (That is, in Desmos on line 6, enter p(x) = cf(x) and add the slider for $c\text{.}$)
Move the slider for $c$ and write at least one sentence to describe the effect that changing the value of $c$ has on the graph of $p\text{.}$ In particular, when $c = -1\text{,}$ how is the graph of $p$ related to the graph of $f\text{?}$
Finally, click on the icons next to $g\text{,}$ $h\text{,}$ and $p$ to temporarily hide them, and go back to Line 1 and change your formula for $f\text{.}$ You can make it whatever you'd like, but try something like $f(x) = x^2 + 2x + 3$ or $f(x) = x^3 - 1\text{.}$ Then, investigate with the sliders $a\text{,}$ $b\text{,}$ and $c$ to see the effects on $g\text{,}$ $h\text{,}$ and $p$ (unhiding them appropriately). Write a couple of sentences to describe your observations of your explorations.

Translations of Functions

We begin by summarizing two of our findings in Preview Activity $\PageIndex{1}$.

Vertical Translation of a Function.

Given a function $y = f(x)$ and a real number $a\text{,}$ the transformed function $y = g(x) = f(x) + a$ is a vertical translation of the graph of $f\text{.}$ That is, every point $(x,f(x))$ on the graph of $f$ gets shifted vertically to the corresponding point $(x,f(x)+a)$ on the graph of $g\text{.}$

As we found in our Desmos explorations in the preview activity, is especially helpful to see the effects of vertical translation dynamically.

Move the slider¹ by clicking and dragging on the red point to see how changing $a$ affects the graph of $y = f(x) + a\text{,}$ which appears in blue. The graph of $y = f(x)$ will appear in grey and remain fixed.

Figure $\PageIndex{4}$ Interactive horizontal translations demonstration (in the HTML version only).

In a vertical translation, the graph of $g$ lies above the graph of $f$ whenever $a \gt 0\text{,}$ while the graph of $g$ lies below the graph of $f$ whenever $a \lt 0\text{.}$ In Figure $\PageIndex{2}$, we see the original parent function $f(x) = |x|$ along with the resulting transformation $g(x) = f(x)-3\text{,}$ which is a downward vertical shift of $3$ units. Note particularly that every point on the original graph of $f$ is moved $3$ units down; we often indicate this by an arrow and labeling at least one key point on each graph.

Figure $\PageIndex{2}$ A vertical translation, $g\text{,}$ of the function $y = f(x) = |x|\text{.}$

Figure $\PageIndex{3}$ A horizontal translation, $h\text{,}$ of a different function $y = f(x)\text{.}$

In Figure $\PageIndex{3}$, we see a horizontal translation of the original function $f$ that shifts its graph $2$ units to the right to form the function $h\text{.}$ Observe that $f$ is not a familiar basic function; transformations may be applied to any original function we desire.

From an algebraic point of view, horizontal translations are slightly more complicated than vertical ones. Given $y = f(x)\text{,}$ if we define the transformed function $y = h(x) = f(x-b)\text{,}$ observe that

\[ h(x+b) = f( (x+b) - b ) = f(x)\text{.} \nonumber \]

This shows that for an input of $x+b$ in $h\text{,}$ the output of $h$ is the same as the output of $f$ that corresponds to an input of simply $x\text{.}$ Hence, in Figure $\PageIndex{3}$, the formula for $h$ in terms of $f$ is $h(x) = f(x-2)\text{,}$ since an input of $x+2$ in $h$ will result in the same output as an input of $x$ in $f\text{.}$ For example, $h(2) = f(0)\text{,}$ which aligns with the graph of $h$ being a shift of the graph of $f$ to the right by $2$ units.

Again, it's instructive to see the effects of horizontal translation dynamically.

Move the slider by clicking and dragging on the red point to see how changing $b$ affects the graph of $y = f(x-b)\text{,}$ which appears in blue. The graph of $y = f(x)$ will appear in grey and remain fixed.

Figure $\PageIndex{4}$ Interactive horizontal translations demonstration (in the HTML version only).

Overall, we have the following general principle.

Horizontal Translation of a Function.

Given a function $y = f(x)$ and a real number $b\text{,}$ the transformed function $y = h(x) = f(x-b)$ is a horizontal translation of the graph of $f\text{.}$ That is, every point $(x,f(x))$ on the graph of $f$ gets shifted horizontally to the corresponding point $(x+b,f(x))$ on the graph of $g\text{.}$

We emphasize that in the horizontal translation $h(x) = f(x-b)\text{,}$ if $b \gt 0$ the graph of $h$ lies $b$ units to the right of $f\text{,}$ while if $b \lt 0\text{,}$ $h$ lies $b$ units to the left of $f\text{.}$

Activity $\PageIndex{2}$

Consider the functions $r$ and $s$ given in Figure $\PageIndex{5}$ and Figure $\PageIndex{6}$

Figure $\PageIndex{5}$ A parent function $r\text{.}$

Figure $\PageIndex{6}$ A parent function $s\text{.}$

On the same axes as the plot of $y = r(x)\text{,}$ sketch the following graphs: $y = g(x) = r(x) + 2\text{,}$ $y = h(x) = r(x+1)\text{,}$ and $y = f(x) = r(x+1) + 2\text{.}$ Be sure to label the point on each of $g\text{,}$ $h\text{,}$ and $f$ that corresponds to $(-2,-1)$ on the original graph of $r\text{.}$ In addition, write one sentence to explain the overall transformations that have resulted in $g\text{,}$ $h\text{,}$ and $f\text{.}$
On the same axes as the plot of $y = s(x)\text{,}$ sketch the following graphs: $y = k(x) = s(x) - 1\text{,}$ $y = j(x) = s(x-2)\text{,}$ and $y = m(x) = s(x-2) - 1\text{.}$ Be sure to label the point on each of $k\text{,}$ $j\text{,}$ and $m$ that corresponds to $(-2,-3)$ on the original graph of $r\text{.}$ In addition, write one sentence to explain the overall transformations that have resulted in $k\text{,}$ $j\text{,}$ and $m\text{.}$
Now consider the function $q(x) = x^2\text{.}$ Determine a formula for the function that is given by $p(x) = q(x+3) - 4\text{.}$ How is $p$ a transformation of $q\text{?}$

Vertical stretches and reflections

So far, we have seen the possible effects of adding a constant value to function output —$f(x)+a$— and adding a constant value to function input — $f(x+b)\text{.}$ Each of these actions results in a translation of the function's graph (either vertically or horizontally), but otherwise leaving the graph the same. Next, we investigate the effects of multiplication the function's output by a constant.

Example $\PageIndex{7}$

Given the parent function $y = f(x)$ pictured in Figure $\PageIndex{8}$, what are the effects of the transformation $y = v(x) = cf(x)$ for various values of $c\text{?}$

Solution

We first investigate the effects of $c = 2$ and $c = \frac{1}{2}\text{.}$ For $v(x) = 2f(x)\text{,}$ the algebraic impact of this transformation is that every output of $f$ is multiplied by $2\text{.}$ This means that the only output that is unchanged is when $f(x) = 0\text{,}$ while any other point on the graph of the original function $f$ will be stretched vertically away from the $x$-axis by a factor of $2\text{.}$ We can see this in Figure $\PageIndex{8}$ where each point on the original dark blue graph is transformed to a corresponding point whose $y$-coordinate is twice as large, as partially indicated by the red arrows.

Figure $\PageIndex{8}$ The parent function $y = f(x)$ along with two different vertical stretches, $v$ and $u\text{.}$

Figure $\PageIndex{9}$ The parent function $y = f(x)$ along with a vertical reflection, $z\text{,}$ and a corresponding stretch, $w\text{.}$

In contrast, the transformation $u(x) = \frac{1}{2}f(x)$ is stretched vertically by a factor of $\frac{1}{2}\text{,}$ which has the effect of compressing the graph of $f$ towards the $x$-axis, as all function outputs of $f$ are multiplied by $\frac{1}{2}\text{.}$ For instance, the point $(0,-2)$ on the graph of $f$ is transformed to the graph of $(0,-1)$ on the graph of $u\text{,}$ and others are transformed as indicated by the purple arrows.

To consider the situation where $c \lt 0\text{,}$ we first consider the simplest case where $c = -1$ in the transformation $z(x) = -f(x)\text{.}$ Here the impact of the transformation is to multiply every output of the parent function $f$ by $-1\text{;}$ this takes any point of form $(x,y)$ and transforms it to $(x,-y)\text{,}$ which means we are reflecting each point on the original function's graph across the $x$-axis to generate the resulting function's graph. This is demonstrated in Figure $\PageIndex{9}$ where $y = z(x)$ is the reflection of $y = f(x)$ across the $x$-axis.

Finally, we also investigate the case where $c = -2\text{,}$ which generates $y = w(x) = -2f(x)\text{.}$ Here we can think of $-2$ as $-2 = 2(-1)\text{:}$ the effect of multiplying by $-1$ first reflects the graph of $f$ across the $x$-axis (resulting in $w$), and then multiplying by $2$ stretches the graph of $z$ vertically to result in $w\text{,}$ as shown in Figure $\PageIndex{9}$

As with vertical and horizontal translation, it's particularly instructive to see the effects of vertical scaling in a dynamic way.

Move the slider by clicking and dragging on the red point to see how changing $c$ affects the graph of $y = cf(x)\text{,}$ which is shown in blue. The graph of $y = f(x)$ will appear in grey and remain fixed.

Figure $\PageIndex{10}$ Interactive vertical scaling demonstration (in the HTML version only).

We summarize and generalize our observations from Example $\PageIndex{7}$ and Figure $\PageIndex{10}$ as follows.

Vertical Scaling of a Function.

Given a function $y = f(x)$ and a real number $c \gt 0\text{,}$ the transformed function $y = v(x) = cf(x)$ is a vertical stretch of the graph of $f\text{.}$ Every point $(x,f(x))$ on the graph of $f$ gets stretched vertically to the corresponding point $(x,cf(x))$ on the graph of $v\text{.}$ If $0 \lt c \lt 1\text{,}$ the graph of $v$ is a compression of $f$ toward the $x$-axis; if $c \gt 1\text{,}$ the graph of $v$ is a stretch of $f$ away from the $x$-axis. Points where $f(x) = 0$ are unchanged by the transformation.

Given a function $y = f(x)$ and a real number $c \lt 0\text{,}$ the transformed function $y = v(x) = cf(x)$ is a reflection of the graph of $f$ across the $x$-axis followed by a vertical stretch by a factor of $|c|\text{.}$

Exercise $\PageIndex{1}$

Figure $\PageIndex{11}$ A parent function $r\text{.}$

Figure $\PageIndex{12}$ A parent function $s\text{.}$

On the same axes as the plot of $y = r(x)\text{,}$ sketch the following graphs: $y = g(x) = 3r(x)$ and $y = h(x) = \frac{1}{3}r(x)\text{.}$ Be sure to label several points on each of $r\text{,}$ $g\text{,}$ and $h$ with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in $g$ and $h$ from $r\text{.}$
On the same axes as the plot of $y = s(x)\text{,}$ sketch the following graphs: $y = k(x) = -s(x)$ and $y = j(x) = -\frac{1}{2}s(x)\text{.}$ Be sure to label several points on each of $s\text{,}$ $k\text{,}$ and $j$ with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in $k$ and $j$ from $s\text{.}$
On the additional copies of the two figures below, sketch the graphs of the following transformed functions: $y = m(x) = 2r(x+1)-1$ (at left) and $y = n(x) = \frac{1}{2}s(x-2)+2\text{.}$ As above, be sure to label several points on each graph and indicate their correspondence to points on the original parent function.

$transformations-act-r-translation-2.svg$

$transformations-act-s-translation-2.svg$
Describe in words how the function $y = m(x) = 2r(x+1)-1$ is the result of three elementary transformations of $y = r(x)\text{.}$ Does the order in which these transformations occur matter? Why or why not?

Combining shifts and stretches: why order sometimes matters

In the final question of Activity $\PageIndex{3}$, we considered the transformation $y = m(x) = 2r(x+1)-1$ of the original function $r\text{.}$ There are three different basic transformations involved: a vertical shift of $1$ unit down, a horizontal shift of $1$ unit left, and a vertical stretch by a factor of $2\text{.}$ To understand the order in which these transformations are applied, it's essential to remember that a function is a process that converts inputs to outputs.

By the algebraic rule for $m\text{,}$ $m(x) = 2r(x+1)-1\text{.}$ In words, this means that given an input $x$ for $m\text{,}$ we do the following processes in this particular order:

add $1$ to $x$ and then apply the function $r$ to the quantity $x+1\text{;}$
multiply the output of $r(x+1)$ by $2\text{;}$
subtract $1$ from the output of $2r(x+1)\text{.}$

These three steps correspond to three basic transformations: (1) shift the graph of $r$ to the left by $1$ unit; (2) stretch the resulting graph vertically by a factor of $2\text{;}$ (3) shift the resulting graph vertically by $-1$ units. We can see the graphical impact of these algebraic steps by taking them one at a time. In Figure $\PageIndex{14}$, we see the function $p$ that results from a shift $1$ unit left of the parent function in Figure $\PageIndex{13}$ (Each time we take an additional step, we will de-emphasize the preceding function by having it appear in lighter color and dashed.)

Figure $\PageIndex{13}$ The parent function $y = r(x)\text{.}$

Figure $\PageIndex{14}$ The parent function $y = r(x)$ along with the horizontal shift $y = p(x) = r(x+1)\text{.}$

Continuing, we now consider the function $q(x) = 2p(x) = 2r(x+1)\text{,}$ which results in a vertical stretch of $p$ away from the $x$-axis by a factor of $2\text{,}$ as seen in Figure $\PageIndex{15}$

Figure $\PageIndex{15}$ The function $y = q(x) = 2p(x) = 2r(x+1)$ along with graphs of $p$ and $r\text{.}$

Figure $\PageIndex{16}$ The function $y = m(x) = q(x)-1 = 2r(x+1) - 1$ along with graphs of $q\text{,}$ $p$ and $r\text{.}$

Finally, we arrive at $y = m(x) = 2r(x+1) - 1$ by subtracting $1$ from $q(x) = 2r(x+1)\text{;}$ this of course is a vertical shift of $-1$ units, and produces the graph of $m$ shown in red in Figure 1.8.16. We can also track the point $(2,-1)$ on the original parent function: it first moves left $1$ unit to $(1,-1)\text{,}$ then it is stretched vertically by a factor of $2$ away from the $x$-axis to $(1,-2)\text{,}$ and lastly is shifted $1$ unit down to the point $(1,-3)\text{,}$ which we see on the graph of $m\text{.}$

While there are some transformations that can be executed in either order (such as a combination of a horizontal translation and a vertical translation, as seen in part (b) of Activity $\PageIndex{2}$), in other situations order matters. For instance, in our preceding discussion, we have to apply the vertical stretch before applying the vertical shift. Algebraically, this is because

\[ 2r(x+1) - 1 \ne 2[r(x+1)-1]\text{.} \nonumber \]

The quantity $2r(x+1) - 1$ multiplies the function $r(x+1)$ by $2$ first (the stretch) and then the vertical shift follows; the quantity $2[r(x+1) - 1]$ shifts the function $r(x+1)$ down $1$ unit first, and then executes a vertical stretch by a factor of $2\text{.}$ In the latter scenario, the point $(1,-1)$ that lies on $r(x+1)$ gets transformed first to $(1,-2)$ and then to $(1,-4)\text{,}$ which is not the same as the point $(1,-3)$ that lies on $m(x) = 2r(x+1) - 1\text{.}$

Activity $\PageIndex{4}$

Consider the functions $f$ and $g$ given in Figure $\PageIndex{17}$ and Figure $\PageIndex{18}$.

Figure $\PageIndex{17}$ A parent function $f\text{.}$

Figure $\PageIndex{18}$ A parent function $g\text{.}$

Sketch an accurate graph of the transformation $y = p(x) = -\frac{1}{2}f(x-1)+2\text{.}$ Write at least one sentence to explain how you developed the graph of $p\text{,}$ and identify the point on $p$ that corresponds to the original point $(-2,2)$ on the graph of $f\text{.}$
Sketch an accurate graph of the transformation $y = q(x) = 2g(x+0.5)-0.75\text{.}$ Write at least one sentence to explain how you developed the graph of $p\text{,}$ and identify the point on $q$ that corresponds to the original point $(1.5,1.5)$ on the graph of $g\text{.}$
Is the function $y = r(x) = \frac{1}{2}(-f(x-1) - 4)$ the same function as $p$ or different? Why? Explain in two different ways: discuss the algebraic similarities and differences between $p$ and $r\text{,}$ and also discuss how each is a transformation of $f\text{.}$
Find a formula for a function $y = s(x)$ (in terms of $g$) that represents this transformation of $g\text{:}$ a horizontal shift of $1.25$ units left, followed by a reflection across the $x$-axis and a vertical stretch by a factor of $2.5$ units, followed by a vertical shift of $1.75$ units. Sketch an accurate, labeled graph of $s$ on the following axes along with the given parent function $g\text{.}$

Summary

The graph of $y = g(x) = af(x-b) + c$ is related to the graph of $y = f(x)$ by a sequence of transformations. First, there is horizontal shift of $|b|$ units to the right ($b \gt 0$) or left ($b \lt 0$). Next, there is a vertical stretch by a factor of $|a|$ (along with a reflection across $y = 0$ in the case where $a \lt 0$). Finally, there's a vertical shift of $c$ units.
A transformation of a given function $f$ is a process by which the graph may be shifted or stretched to generate a new, related function with fundamentally the same shape. In this section we considered four different ways this can occur: through a horizontal translation (shift), through a reflection across the line $y = 0$ (the $x$-axis), through a vertical scaling (stretch) that multiplies every output of a function by the same constant, and through a vertical translation (shift). Each of these individual processes is itself a transformation, and they may be combined in various ways to create more complicated transformations.

Exercises

1.

2.

3.

4.

5.

6.

Let $f(x) = x^2\text{.}$

Let $g(x) = f(x) + 5\text{.}$ Determine $AV_{[-3,-1]}$ and $AV_{[2,5]}$ for both $f$ and $g\text{.}$ What do you observe? Why does this phenomenon occur?
Let $h(x) = f(x-2)\text{.}$ For $f\text{,}$ recall that you determined $AV_{[-3,-1]}$ and $AV_{[2,5]}$ in (a). In addition, determine $AV_{[-1,1]}$ and $AV_{[4,7]}$ for $h\text{.}$ What do you observe? Why does this phenomenon occur?
Let $k(x) = 3f(x)\text{.}$ Determine $AV_{[-3,-1]}$ and $AV_{[2,5]}$ for $k\text{,}$ and compare the results to your earlier computations of $AV_{[-3,-1]}$ and $AV_{[2,5]}$ for $f\text{.}$ What do you observe? Why does this phenomenon occur?
Finally, let $m(x) = 3f(x-2) + 5\text{.}$ Without doing any computations, what do you think will be true about the relationship between $AV_{[-3,-1]}$ for $f$ and $AV_{[-1,1]}$ for $m\text{?}$ Why? After making your conjecture, execute appropriate computations to see if your intuition is correct.

7.

Consider the parent function $y = f(x) = x\text{.}$

Consider the linear function in point-slope form given by $y = L(x) = -4(x-3) + 5\text{.}$ What is the slope of this line? What is the most obvious point that lies on the line?
How can the function $L$ given in (a) be viewed as a transformation of the parent function $f\text{?}$ Explain the roles of $3\text{,}$ $-4\text{,}$ and $5\text{,}$ respectively.
Explain why any non-vertical line of the form $P(x) = m(x-x_0) + y_0$ can be thought of as a transformation of the parent function $f(x) = x\text{.}$ Specifically discuss the transformation(s) involved.
Find a formula for the transformation of $f(x) = x$ that corresponds to a horizontal shift of $7$ units left, a reflection across $y = 0$ and vertical stretch of $3$ units away from the $x$-axis, and a vertical shift of $-11$ units.

8.

We have explored the effects of adding a constant to the output of a function, $y = f(x) + a\text{,}$ adding a constant to the input, $y = f(x+a)\text{,}$ and multiplying the output of a function by a constant, $y = af(x)\text{.}$ There is one remaining natural transformation to explore: multiplying the input to a function by a constant. In this exercise, we consider the effects of the constant $a$ in transforming a parent function $f$ by the rule $y = f(ax)\text{.}$

Let $f(x) = (x-2)^2 + 1\text{.}$

Let $g(x) = f(4x)\text{,}$ $h(x) = f(2x)\text{,}$ $k(x) = f(0.5x)\text{,}$ and $m(x) = f(0.25x)\text{.}$ Use Desmos to plot these functions. Then, sketch and label $g\text{,}$ $h\text{,}$ $k\text{,}$ and $m$ on the provided axes in Figure $\PageIndex{19}$ along with the graph of $f\text{.}$ For each of the functions, label and identify its vertex, its $y$-intercept, and its $x$-intercepts.

Figure $\PageIndex{19}$ Axes for plotting $f\text{,}$ $g\text{,}$ $h\text{,}$ $k\text{,}$ and $m$ in part (a).

Figure $\PageIndex{20}$ Axes for plotting $f\text{,}$ $r\text{,}$ and $s$ from parts (c) and (d).

2. Based on your work in (a), how would you describe the effect(s) of the transformation $y = f(ax)$ where $a \gt 0\text{?}$ What is the impact on the graph of $f\text{?}$ Are any parts of the graph of $f$ unchanged?

3. Now consider the function $r(x) = f(-x)\text{.}$ Observe that $r(-1) = f(1)\text{,}$ $r(2) = f(-2)\text{,}$ and so on. Without using a graphing utility, how do you expect the graph of $y = r(x)$ to compare to the graph of $y = f(x)\text{?}$ Explain. Then test your conjecture by using a graphing utility and record the plots of $f$ and $r$ on the axes in Figure $\PageIndex{20}$

4. How do you expect the graph of $s(x) = f(-2x)$ to appear? Why? More generally, how does the graph of $y = f(ax)$ compare to the graph of $y = f(x)$ in the situation where $a \lt 0\text{?}$

<1> Huge thanks to the amazing David Austin for making these interactive javascript graphics for the text.

Search

Preview Activity \(\PageIndex{1}\)

Horizontal Translation of a Function.

Example \(\PageIndex{7}\)

Vertical Scaling of a Function.

Exercise \(\PageIndex{1}\)

Activity \(\PageIndex{4}\)

Exercises

1.

2.

3.

4.

5.

6.

7.

8.

Motivating Questions

Preview Activity \(\PageIndex{1}\)

Translations of Functions

Vertical Translation of a Function.

Horizontal Translation of a Function.

Activity \(\PageIndex{2}\)

Vertical stretches and reflections

Example \(\PageIndex{7}\)

Vertical Scaling of a Function.

Exercise \(\PageIndex{1}\)

Combining shifts and stretches: why order sometimes matters

Activity \(\PageIndex{4}\)

Summary

Exercises

1.

2.

3.

4.

5.

6.

7.

8.