Mathematics 265 Introduction to Calculus I

Study Guide :: Unit 2

Functions

Transformations

In this course, we study four basic transformations:

• shifts (upward, downward, to the left or to the right).
• stretches (vertical or horizontal).
• compressions (vertical or horizontal).
• reflections (with respect to the $x$-axis or the $y$-axis).

Shifts

Shifts may be vertical or horizontal. Let us consider vertical shifts first.

A vertical shift moves the graph of a function up or down without modifying the graph, as in Figures 2.14 and 2.15, and 2.16 and 2.17, below.

Figure 2.14. Basic graph 1

Figure 2.15. Basic graph 1 shifted 4 units upward

The graph in Figure 2.15 is the shift 4 units upward of the graph in Figure 2.14.

Figure 2.16. Basic graph 2

Figure 2.17. Basic graph 2 shifted 1/3 units downward

The graph in Figure 2.17 is the shift $1/3$ units downward of the graph in Figure 2.16.

We can recognize a shift from the formula of a function $f(x)$ by the constant that is added to it (see the definition below).

Example 2.42. The graph of $f(x) = {x^2}$ is shown in Figure 2.14; the graph in Figure 2.15 is the graph of the function $F(x) = {x^2} + 4$, which is the shift $4$ units up of the graph of $f(x)$. You can see that $F(x) = f(x) + 4$.

Example 2.43. The graph of ${\displaystyle f\left(x\right)=\left|x\right|}$ is shown in Figure 2.16 above. The graph in Figure 2.17 is the graph of the function

${\displaystyle F\left(x\right)=f\left(x\right)-\frac{1}{3}=\left|x\right|-\frac{1}{3},}$

which is the shift $1/3$ units downward of the graph of $f(x)$.

Example 2.44. If we analyse the function $F(x) = \cos x + 2$, we see that it is the sum of the cosine function and the constant $2$. Hence, the graph of $F\left(x\right)$ is the shift $2$ units up of the graph of the cosine function (see Figure 2.18).

Figure 2.18. Graph of the cosine function shifted $2$ units upward

Shifts to the left or right are obtained by adding a constant to the independent variable.

Example 2.45. The graph of $f(x) = {x^2}$ is shown in Figure 2.14, and the graph of the function $F\left(x\right)={\left(x+4\right)}^2\text{,}$ which is the shift $4$ units to the left of the graph of $f\left(x\right)\text{,}$ is shown in Figure 2.19, below. As you can see, $F\left(x\right)=f\left(x+4\right)\text{;}$ hence, $F\left(-4\right)=f\left(-4+4\right)=f\left(0\right)=0\text{.}$

Figure 2.19. Graph of $F\left(x\right)={\left(x+4\right)}^2\text{;}$ that is, $f(x) = {x^2}$ shifted $4$ units to the left

Figure 2.20. Graph of $F\left(x\right)=|x-3|$; that is, $f(x) = |x|$ shifted $3$ units to the right

Example 2.46. The graph of $f(x) = |x|$ is shown in Figure 2.16. The graph in Figure 2.20, above, is the graph of the function $F(x) = |x - 3|$, which is the shift to the right of the graph of $f(x)$. We also have $F(x) = f(x - 3)$. As you can see, $F(3) = f(3 - 3) = f(0) = 0$.

Stretches and Compressions

As can shifts, stretches and compressions can be vertical or horizontal. Figures 2.21 to 2.24, below, show vertical stretches and compressions.

Figure 2.21. Basic graph

Figure 2.22. Basic graph stretched vertically by $2$ units

The graph in Figure 2.22 is a stretch $2$ units vertically of the graph in Figure 2.21. Observe that the points of intersection in the $x$-axis are the same in both graphs.

Figure 2.23. Basic graph

The graph in Figure 2.24, below, is a compression $1∕2$ units vertically of the graph in Figure 2.23.

Figure 2.24. Basic graph compressed vertically by $\frac{1}{2}$ units

To stretch the graph of a function vertically, we multiply the function by a constant greater than 1; for a vertical compression, we multiply the function by a number smaller than 1.

Example 2.47. The graph of $f(x) = \sin x$ is shown in Figure 2.21. The graph in Figure 2.22 is the graph of the function $F(x) = 2\sin x$, which is the vertical stretch of the graph of $f(x)$ by a factor of $2$. As you can see, $F(x) = 2f(x)$.

Example 2.48. The graph of $f(x) = |x|$ is shown in Figure 2.23. The graph in Figure 2.24 is the graph of the function

$F(x) = \dfrac{1}{2}|x| = \dfrac{1}{2}f(x)$,

which is the vertical compression of the graph of $f(x)$ by a factor of $2$.

Example 2.49. The graph of

${\displaystyle F\left(x\right)=\frac{x^2}{3}}$

is a vertical compression of the function $f(x) = {x^2}$ by a factor of $3$:

${\displaystyle F\left(x\right)=\frac{x^2}{3}=\frac{1}{3}{x}^2=\frac{1}{3}f\left(x\right)}$

(see Figure 2.25, below).

Figure 2.25. Graph of $f\left(x\right)=x^2\text{,}$ and graph of $f(x) = {x^2}$ compressed vertically by a factor of $3$

To compress the graph of a function horizontally, we multiply the independent variable of the function by a constant greater than $1$; to stretch it horizontally, we multiply it by a number smaller than $1\text{.}$

Example 2.50. The graph of $f\left(x\right)=sin x$ is shown in Figure 2.26, below.

Figure 2.26. Graph of $f\left(x\right)=sin x$

The graph in Figure 2.27 is the graph of the function $F(x)=\sin (2x)$, which is the horizontal compression of the graph of $f(x)$ by a factor of $2$. As you can see, $F\left(x\right)=f\left(2x\right)\text{.}$ Observe that the range values of the dependent variable $F$ are the same, from $-1$ to $1\text{.}$

Figure 2.27. Graph of $F\left(x\right)=sin \left(2x\right)$

The points of intersection with the $x$-axis of $sin x$ are

$0,\pi ,2\pi ;$

and the points of intersection with the $x$-axis of $sin \left(2x\right)$ are

${\displaystyle 0=\left(\frac{1}{2}\right)0,\frac{\pi }{2}=\left(\frac{1}{2}\right)\pi ,\pi =\left(\frac{1}{2}\right)2\pi .}$

Example 2.51. The graph of $f\left(x\right)=sin x$ is shown in Figure 2.26. The graph in Figure 2.28, below, is the graph of the function

${\displaystyle F\left(x\right)=sin \left(\frac{x}{2}\right),}$

which is the horizontal stretch of the graph of $f(x)$ by a factor of $2$. As you can see,

${\displaystyle F\left(x\right)=f\left(\frac{x}{2}\right).}$

Figure 2.28. Graph of $F\left(x\right)=sin \left(\frac{x}{2}\right)$

The points of intersection with the $x$-axis of $sin x$ are

$0,\pi ,2\pi ;$

and those of ${\displaystyle sin \left(\frac{x}{2}\right)}$ are

$0=2\left(0\right),2\pi =2\left(\pi \right),4\pi =2\left(2\pi \right).$

Reflections

To reflect a graph with respect to the $x$-axis—that is, to have the $x$-axis act as a mirror—we multiply the function by $-1$; to reflect the function with respect to the $y$-axis, we multiply the independent variable by $-1$.

Example 2.52. The reflection of the graph of the sine function with respect to the $x$-axis corresponds to the function $-\sin x\text{,}$ as is shown in Figure 2.29, below.

Figure 2.29. Graph of $f\left(x\right)=-sin x$

Example 2.53. The graph the function $f\left(x\right)=\sqrt{x}$ is shown in Figure 2.30, below.

Figure 2.30. Graph of $f\left(x\right)=\sqrt{x}$

The reflection of $f\left(x\right)=\sqrt{x}$ with respect to the $y$-axis corresponds to the function $F\left(x\right)=\sqrt{-x}$, as is shown in Figure 2.31, below. Observe that the domain of the function $f$ is $\left(0,\infty \right)$ and the domain of $F$ is $\left(-\infty ,0\right)\text{.}$

Figure 2.31. Graph of $F\left(x\right)=\sqrt{-x}$

Multiple Transformations

We can apply several transformations to one basic graph.

Example 2.54. If the basic function is $f$, then a vertical stretch by a factor of $3$ is the function $3f\left(x\right)\text{.}$ Continuing with a shift up by a factor of $2$ gives the function $3f\left(x\right)+2\text{.}$

Let the basic function be $f\left(x\right)=x^2\text{,}$ then the graph of $F\left(x\right)=3x^2+2$ is obtained after two transformations: a vertical stretch first, and a shift second, in that order (see Figure 2.32, below).

Figure 2.32. Deriving the graph of $F\left(x\right)=3x^2+2$ from that of $f(x) = {x^2}$

The challenge in sketching a new graph from a basic one is to identify the transformations in the order in which they are applied. Observe that the order matters: if we shift by $2$ units up and then stretch vertically by $3$ units, we go to $f\left(x\right)+2$ first, then to $3\left(f\left(x\right)+2\right)=3f\left(x\right)+6\text{.}$ If the basic function is $f\left(x\right)=x^2\text{,}$ under these transformations in this order, $F\left(x\right)=3f\left(x\right)+\mathrm{6,}$ and the graph of this function, is as shown in Figure 2.33, below. [What would the function be if we applied the vertical stretch first? What would the graph look like?]

Figure 2.33. Graph of $F\left(x\right)=3f\left(x\right)+6$

Example 2.55. Consider the function $F\left(x\right)=3{\left(x+5\right)}^2-2\text{.}$ If we were to evaluate this function at a number $x$, we would proceed as follows.

Conclusion: The function $F(x) = 3{(x + 5)^2} - 2$ is obtained from the basic function $f(x) = {x^2}$ with the three transformations listed in the third column, in the order given. Figures 2.34 and 2.35, below, show the first two stages of these transformations.

Figure 2.34. Stage 1 of the transformation of $f(x) = {x^2}$ to $F(x)=3{(x+5)}^2-2$

Figure 2.35. Stage $2$ of the transformation of $f(x) = {x^2}$ to $F\left(x\right)=3{\left(x+5\right)}^2-2$

The final graph of the function is shown in Figure 2.36:

Figure 2.36. Graph of $F\left(x\right)=3{\left(x+5\right)}^2-2$

Example 2.56. To sketch the graph of

${\displaystyle F\left(x\right)=2cos\left(\frac{x}{2}\right),}$

we make the following analysis:

The graph is shown in Figure 2.37, below.

Figure 2.37. Graph of $F\left(x\right)=2cos \left(\frac{x}{2}\right)$