Section1.8Transformations of Functions

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 1.8.1, 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 Activity1.8.1.

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 $-10 \le y \le 10\text{.}$

1. 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 $g\text{.}$

2. 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 $h\text{.}$

3. 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 $f\text{?}$

4. 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.

Subsection1.8.1Translations of Functions

We begin by summarizing two of our findings in Preview Activity 1.8.1.

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.

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

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 1.8.2, 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 1.8.2. A vertical translation, $g\text{,}$ of the function $y = f(x) = |x|\text{.}$ Figure 1.8.3. A horizontal translation, $h\text{,}$ of a different function $y = f(x)\text{.}$

In Figure 1.8.3, 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

\begin{equation*} h(x+b) = f( (x+b) - b ) = f(x)\text{.} \end{equation*}

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 1.8.3, 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.

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 $f\text{.}$

Activity1.8.2.

Consider the functions $r$ and $s$ given in Figure 1.8.5 and Figure 1.8.6. Figure 1.8.5. A parent function $r\text{.}$ Figure 1.8.6. A parent function $s\text{.}$
1. 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 $f\text{.}$

2. 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 $m\text{.}$

3. 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 $q\text{?}$

Subsection1.8.2Vertical 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.

Example1.8.7.

Given the parent function $y = f(x)$ pictured in Figure 1.8.8, 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 1.8.8 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 1.8.8. The parent function $y = f(x)$ along with two different vertical stretches, $v$ and $u\text{.}$ Figure 1.8.9. 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 1.8.9 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 1.8.9.

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

We summarize and generalize our observations from Example 1.8.7 and Figure 1.8.10 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 $|c|\text{.}$

Activity1.8.3.

Consider the functions $r$ and $s$ given in Figure 1.8.11 and Figure 1.8.12. Figure 1.8.11. A parent function $r\text{.}$ Figure 1.8.12. A parent function $s\text{.}$
1. 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 $r\text{.}$

2. 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 $s\text{.}$

3. 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.  4. 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?

Subsection1.8.3Combining shifts and stretches: why order sometimes matters

In the final question of Activity 1.8.3, 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:

1. add $1$ to $x$ and then apply the function $r$ to the quantity $x+1\text{;}$

2. multiply the output of $r(x+1)$ by $2\text{;}$

3. 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 1.8.14, we see the function $p$ that results from a shift $1$ unit left of the parent function in Figure 1.8.13. (Each time we take an additional step, we will de-emphasize the preceding function by having it appear in lighter color and dashed.) Figure 1.8.13. The parent function $y = r(x)\text{.}$ Figure 1.8.14. 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 1.8.15. Figure 1.8.15. The function $y = q(x) = 2p(x) = 2r(x+1)$ along with graphs of $p$ and $r\text{.}$ Figure 1.8.16. 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 1.8.2), 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

\begin{equation*} 2r(x+1) - 1 \ne 2[r(x+1)-1]\text{.} \end{equation*}

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 $m(x) = 2r(x+1) - 1\text{.}$

Activity1.8.4.

Consider the functions $f$ and $g$ given in Figure 1.8.17 and Figure 1.8.18. Figure 1.8.17. A parent function $f\text{.}$ Figure 1.8.18. A parent function $g\text{.}$
1. 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 $f\text{.}$

2. 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 $g\text{.}$

3. 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 $f\text{.}$

4. 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 $g\text{.}$ Subsection1.8.4Summary

• 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.

Exercises1.8.5Exercises

6.

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

1. 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?

2. 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?

3. 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?

4. 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 $y = f(x) = x\text{.}$

1. 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?

2. 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.

3. 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.

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

1. 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 1.8.19 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 1.8.19. Axes for plotting $f\text{,}$ $g\text{,}$ $h\text{,}$ $k\text{,}$ and $m$ in part (a). Figure 1.8.20. 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 1.8.20.

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 $a \lt 0\text{?}$