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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

Example6.3.1

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

Figure6.3.2\(g(x) = x^3 - 5x^2 + 2\)

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:

Figure6.3.3The reflection \(y = -g(x)\)
Figure6.3.4The reflection \(y = g(-x)\)

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

Exercise6.3.5
WeBWorK Exercise

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.

Example6.3.6

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.

Figure6.3.7The diver's depth: \(D = f(t)\)
Figure6.3.8The 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 \(a\text{.}\)

Example6.3.9

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).

Figure6.3.11\(f(x) = x^2 - 1\)
Figure6.3.12\(3f(x) = 3x^2 - 3\)

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).

Figure6.3.14\(f(x) = x^2 - 1\)
Figure6.3.15\(\frac{1}{2}f(x) = \frac{1}{2}x^2 - \frac{1}{2}\)
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.

Example6.3.16

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.

Example6.3.17

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*}

Subsection6.3.3Exercises

Interpreting Transformations

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Even and Odd

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Graphs of Reflections and Stretches

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Formula of a Reflection or Vertical Stretch

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Geometry of Reflections and Vertical Stretches

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Additional Problems

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Horizontal Stretches

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