## Section6.3Gist of Reflections and Vertical Stretches

### Subsection6.3.1Reflections

Thinking again of function notation as language, we know that the notation $f(x)$ indicates that there is an input called $x$ and an output called $f(x)\text{.}$ Therefore, by replacing $x$ with $-x$ we are using the opposite of the input.

Likewise, if we replace $f(x)$ with $-f(x)\text{,}$ we are referring to the opposite of the output.

From a graphical standpoint, both of the transformations $-f(x)$ and $f(-x)$ represent reflections of a graph:

$y = -f(x)$

reflects $y = f(x)$ over the horizontal-axis

$y = f(-x)$

reflects $y = f(x)$ over the vertical-axis

The function $g(x) = x^3 - 5x^2 + 2$ is graphed below. Figure 6.3.2. $g(x) = x^3 - 5x^2 + 2$

To reflect $g(x)$ over the $x$-axis, we would take the opposite of the output by multiplying the formula by $-1\text{.}$

The formula for this reflection is

\begin{align*} y \amp= -g(x)\\ \amp= -1(x^3 - 5x^2 + 2)\\ \amp= -x^3 + 5x^2 - 2 \end{align*}

and it is graphed below:

To reflect $g(x)$ over the $y$-axis, we would use the opposite of the input by evaluating $g(-x)\text{.}$

The formula for this reflection is

\begin{align*} y \amp= g(-x)\\ \amp= (-x)^3 - 5(-x)^2 + 2\\ \amp= -x^3 - 5x^2 + 2 \end{align*}

and it is graphed below: Figure 6.3.3. The reflection $y = -g(x)$ Figure 6.3.4. The reflection $y = g(-x)$

Now practice writing the formula for a function which has been transformed.

One use of the transformation $-f(x)$ is in redefining a variable from positive-to-negative, or vice versa, as shown in the example below.

When deep sea divers swim back to the surface of the water after a dive, they must do so slowly. If a diver ascends too quickly, she may experience decompression sickness. In order to prevent this, divers must ascend at a rate slower than 33 ftmin.

Suppose a diver begins at a depth of 100 ft and ascends at a rate of 30 ftmin.

If we wanted to represent the diver's depth below the water as a function of time, we could write:

\begin{equation*} D = f(t) = 100 - 30t \end{equation*}

Alternatively, we could have written a function for the diver's altitude as a function of time. In this case, the diver's altitude when below the water would be negative, so our function would be:

\begin{equation*} A = g(t) = -100 + 30t \end{equation*}

Notice that the altitude is the opposite of the depth here, so our function $g(t)$ is equal to $-f(t)\text{.}$

The graphs below show these functions as reflections of each other over the horizontal axis. Figure 6.3.7. The diver's depth: $D = f(t)$ Figure 6.3.8. The diver's altitude: $A = g(t) = -f(t)$

### Subsection6.3.2Vertical Stretches and Compressions

In addition to reflecting a function, we can also stretch or compress a function vertically by multiplying the output by a constant number $a\text{.}$

If $a > 1$

The transformation $y = a\cdot f(x)$ will vertically stretch the graph of $y = f(x)$ by a factor of $a\text{.}$

If $0 < a < 1$

The transformation $y = a\cdot f(x)$ will vertically compress the graph of $y = f(x)$ by a factor of $a\text{.}$

For the function

\begin{equation*} f(x) = x^2 - 1\text{,} \end{equation*}

describe the transformations $3f(x)$ and $\frac{1}{2}f(x)\text{.}$

For the first transformation, we have:

\begin{align*} 3f(x) \amp= 3(x^2 - 1)\\ \amp= 3x^2 - 3 \end{align*}

This is just like $f(x) = x^2 - 1\text{,}$ but all of the output values have been multiplied by $3\text{.}$

 $x$ $f(x)$ $3f(x)$ -3 8 24 -2 3 9 -1 0 0 0 -1 -3 1 0 0 2 3 9 3 8 24

Since the output values were all multiplied by $3\text{,}$ this would make all of the $y$-coordinates of the points $3$ times as far from zero (either positive or negative). Figure 6.3.10. $f(x) = x^2 - 1$ Figure 6.3.11. $3f(x) = 3x^2 - 3$

Similarly, if we began with $f(x) = x^2 - 1\text{,}$ then the transformation

\begin{align*} \frac{1}{2}f(x) \amp= \frac{1}{2}(x^2 - 1)\\ \amp= \frac{1}{2}x^2 - \frac{1}{2} \end{align*}

would be just like $f(x)\text{,}$ but all of its output values would be multiplied by $\frac{1}{2}\text{.}$

 $x$ $f(x)$ $\frac{1}{2}f(x)$ -3 8 4 -2 3 1.5 -1 0 0 0 -1 -0.5 1 0 0 2 3 1.5 3 8 4

Since the output values were all multiplied by $\frac{1}{2}\text{,}$ this would make all of the $y$ values of the points only half as far from zero (either positive or negative). Figure 6.3.12. $f(x) = x^2 - 1$ Figure 6.3.13. $\frac{1}{2}f(x) = \frac{1}{2}x^2 - \frac{1}{2}$
##### Even and Odd Functions.

You will notice that some functions, after being reflected over the vertical axis, remain the same as they began. That is, the transformation $f(-x)$ is the same as the original function $f(x)\text{.}$

If the graph of a function is unchanged after reflecting it over the vertical axis, then we say the function is even. In function notation, a function is called even if:

\begin{equation*} f(-x) = f(x) \end{equation*}

Geometrically, an even function is symmetric over the vertical axis.

Two of the functions below are even and two are not. Which ones are the even functions?

1. $a(x) = 3x^4-7x^2+9$

2. $b(x) = \sqrt{x}-9$

3. $c(x) = e^{x^2}+x$

4. $d(x) = e^{x^2} + x^2$

Solution
1. \begin{align*} a(-x) \amp= 3x^4-7x^2+9\\ \amp= 3(-x)^4 - 7(-x)^2 + 9\\ \amp= 3x^4 - 7x^2 + 9\\ \amp= a(x) \end{align*}

Therefore, $a(x)$ is an even function.

2. \begin{align*} b(-x) \amp= \sqrt{-x}-9\\ \amp\neq b(x) \end{align*}

So, $b(x)$ is not an even function.

3. \begin{align*} c(-x) \amp= e^{(-x)^2}+(-x)\\ \amp= e^{x^2} - x\\ \amp\neq c(x) \end{align*}

So, $c(x)$ is not an even function.

4. \begin{align*} d(-x) \amp= e^{(-x)^2} + (-x)^2\\ \amp= e^{x^2} + x^2 \end{align*}

Therefore, $d(x)$ is an even function.

On the other hand, we may try to reflect a function over the vertical axis, and find that this is the same as reflecting it over the horizontal axis. In function notation, this would mean that:

\begin{equation*} f(-x) = -f(x) \end{equation*}

If $f(-x) = -f(x)\text{,}$ then we say $f$ is an odd function. Geometrically, an odd function is symmetric about the origin.

Show that the function $f(x) = 8x^3 + x$ is an odd function.

Solution
\begin{align*} f(-x) \amp= 8(-x)^3 + (-x)\\ \amp= -8x^3 - x\\ \amp= -\left(8x^3 + x\right)\\ \amp= -f(x) \end{align*}

### Subsection6.3.3Conclusion

Test your understanding of this chapter with the following exercises.

Student Learning Outcome: Describe the effect of the transformation of the form $y = a\cdot f(x)$ given the graph of $y = f(x)\text{,}$ and sketch a graph.

Student Learning Outcome: Describe the effects on the graph of a function $f(x)$ when comparing $f(x)$ to $f(-x)$ or $-f(x)\text{.}$

Student Learning Outcome: Use transformations to model situations algebraically (from verbal descriptions, graphs, tables, and data points). Recognize functions as transformations of basic functions.

### Exercises6.3.4Exercises

Interpreting Transformations

Even and Odd

###### 11.

Graphs of Reflections and Stretches

###### 13.

Formula of a Reflection or Vertical Stretch

###### 22.

Geometry of Reflections and Vertical Stretches

###### 33.

Horizontal Stretches