Supplemental Videos.
The main topics of this section are also presented in the following videos:
Section 5.3 Reflections and Even and Odd Functions
In the previous section we discussed shifting a function horizontally and vertically. Now, we will take a look at what happens as we reflect a function across the \(x\)-axis or \(y\)-axis.
Subsection 5.3.1 Reflection About y-axis
Let \(f(x)=(x+1)^2+1\text{,}\) and set
\begin{equation*}
h(x)=f(-x)=(-x+1)^2+1.
\end{equation*}
Compared to \(f(x)\text{,}\) the graph of \(h(x)\) is flipped, or reflected, about the \(y\)-axis as shown below.
y-axis reflection
| \(x\) | \(f(x)\) | \(h(x)=f(-x)\) |
| \(-2\) | \(2\) | \(10\) |
| \(-1\) | \(1\) | \(5\) |
| \(0\) | \(2\) | \(2\) |
| \(1\) | \(5\) | \(1\) |
| \(2\) | \(10\) | \(2\) |
Looking at the graph and table above, we notice a relationship between the outputs of \(f(x) \) and \(f(-x)\) in this particular case, and as it so happens, this relationship holds in general.
Notice that if you take the column of output values for \(f(x)\) and flip them upside down you get the outputs of \(f(-x)\text{.}\) This makes sense as this is saying, for example,
\begin{equation*}
f(-2)=h(\alert{2})=f(-(\alert{2}))
\end{equation*}
as \(h(x)=f(-x)\text{.}\) In general, we have the following definition:
Reflection about y-axis.
Compared with the graph of \(y=f(x)\text{,}\) the graph of \(f(-x)\) is reflected about the \(y\)-axis.
There are some functions which do not change when we apply a reflection about the \(y\)-axis. The most common example of this is the function \(y=x^2\text{.}\) Looking at a graph of this function, it is easy to see that when you reflect it across the \(y\)-axis it remains unchanged. These sorts of functions have a particular name:
Even Function.
A function \(f\) is called an even function if
\begin{equation*}
f(x)=f(-x)
\end{equation*}
for all \(x \) in the domain of \(f \text{.}\) In other words, a function is even if performing a reflection about the \(y\)-axis does not change the graph of the function.
To help remember the definition of an even function, notice that the example of an even function we gave was of \(y=x^2\text{.}\) Other examples are \(y= x^4 \text{,}\) \(y=x^6 \text{,}\) \(y=x^8\text{,}\) etc. Notice that the exponent of each of these functions is an even number.
Subsection 5.3.2 Reflection About x-axis
Compared to \(y=x^2\text{,}\) the graph of \(h(x) = -x^2\) is flipped, or reflected, about the \(x\)-axis. The \(y\)-coordinate of each point on the graph of \(y = x^2\) is replaced by its additive inverse.
negative x-sq and basic parabola
| \(x\) | \(y=x^2\) | \(h(x)=-x^2\) |
| \(-2\) | \(4\) | \(-4\) |
| \(-1\) | \(1\) | \(-1\) |
| \(0\) | \(0\) | \(0\) |
| \(1\) | \(1\) | \(-1\) |
| \(2\) | \(4\) | \(-4\) |
In general, we have the following definition:
Reflection about x-axis.
Compared with the graph of \(y=f(x)\text{,}\) the graph of \(-f(x)\) is reflected about the \(x\)-axis.
We now know that a function satisfying \(f(x)=f(-x)\) is an even function. While there do not exist any functions satisfying \(f(x)=-f(x)\text{,}\) we do have another type of symmetry to discuss here.
Example 5.30.
Consider the function \(f(x)=x^3 \text{.}\) If we reflect \(f \) about the \(y \)-axis, we get the function \(f(-x)\) as shown below:
odd function
If we now reflect the function \(f(-x)\) about the \(x\) axis, we can see from the above picture that we end up back with the original function. As an equation, this says
\begin{equation*}
f(x)=-f(-x).
\end{equation*}
This type of function is said to have symmetry about the origin, or called an odd function.
Odd Function.
A function \(f\) is called an odd function if
\begin{equation*}
f(x)=-f(-x)
\end{equation*}
for all \(x \) in the domain of \(f \text{.}\) In other words, a function is odd if performing a reflection about the \(y\)-axis and \(x\)-axis (doesn’t matter which is performed first) does not change the graph of the function.
To help remember the definition of an odd function, we have a similar strategy as for even functions. Some examples of odd functions are \(y=x^3\text{,}\) \(y=x^5\text{,}\) \(y=x^7 \text{,}\) etc. Each of these examples have exponents which are odd numbers, and they are odd functions.
Exercises 5.3.3 Exercises
1. Review of Shifts.
Let \(C(t)\) represent the number of croissants sold at a small cafe on a given day, \(t\text{.}\) Use shifts of the function \(C(t)\) to find a formula for each of the following functions in terms of \(C(t)\text{.}\) Be sure to use capital letters where appropriate.
a) \(M(t)\text{,}\) the number of muffins sold at the cafe on day \(t\text{,}\) is remarkably always 5 more than the number of croissants sold on day \(t\text{.}\)
\(M(t)=\)
b) \(B(t)\text{,}\) the number of croissants the bakers plan to make on day \(t\text{,}\) is equal to the number of croissants sold on the same day of the week the week before.
\(B(t)=\)
Hint.
a) Since the number of muffins sold on day \(t\) is always 5 more than the number of croissants sold on the day \(t\text{,}\) we want to add 5 to \(C(x)\text{.}\) Write the formula that gives this.
b) Since the number of croissants the bakers plan to make on day \(t\) is the same as the number of croissants sold on the same day of the week the week before, we know the output of \(B(t)\) is the same as the output of \(C(t-7)\text{.}\) Write the formula that gives this.
2. Graphs of Reflections.
Consider the function \(f(x)=x^3-2x^2+1\) graphed below, with values in the table:\\
| \(x\) | -1 | 0 | 1 | 2 |
| \(f(x)\) | -2 | 1 | 0 | 1 |
a) Consider the graph of \(f(x)\) reflected over the \(x\)-axis. Call this new function \(g(x)\text{.}\)
i) Which of the following is a graph of \(g(x)\text{?}\)
-
A
-
B
-
C
-
D
 |
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| A | B |
 |
|
| C | D |
(Click on a graph to enlarge it)
ii) Fill in the corresponding table with values for \(g(x)\text{.}\)
| \(x\) | -1 | 0 | 1 | 2 |
| \(g(x)\) |
b) Consider the graph of \(f(x)\) reflected over the \(y\)-axis. Call this new function \(h(x)\text{.}\)
i) Which of the following is a graph of \(h(x)\text{?}\)
-
A
-
B
-
C
-
D
|
|
| A | B |
|
|
| C | D |
(Click on a graph to enlarge it)
ii) Fill in the corresponding table with values for \(h(x)\text{.}\)
| \(x\) | -2 | -1 | 0 | 1 |
| \(h(x)\) |
Hint.
a) We first want to reflect the function \(f(x)\) over the \(x\)-axis.
i) The graph will look like it’s been mirrored over the \(x\)-axis.
ii) Since we are reflecting over the \(x\)-axis, we see that each each output has been negated. Fill in the table appropriately.
b) We first want to reflect the function \(f(x)\) over the \(y\)-axis.
i) The graph will look like it’s been mirrored over the \(y\)-axis.
ii) Since we are reflecting over the \(y\)-axis, we see that each each input has been negated. Fill in the table accordingly.
3. Points on Reflected Graphs.
The graph of \(y=f(x)\) contains the point \((-6,1)\text{.}\) What point must lie on the resulting graph if the graph is
a) reflected about the \(y\)-axis?
b) reflected about the \(x\)-axis?
c) translated down 2 unit(s)?
d) translated right by 1 unit(s)?
e) reflected about the \(x\)-axis, then shifted up by 4 unit(s)?
f) translated up by 4 unit(s), then reflected about the \(x\)-axis?
Hint.
On each of these, it may be useful to sketch a pair of axes and plot the point. Then consider what would happen to that point for each transformation.
The graph of \(y=f(x)\) contains the point \((-6,1)\text{.}\)
a) If \(f(x)\) is reflected about the \(y\)-axis, we would negate the input value.
b) If \(f(x)\) is reflected about the \(x\)-axis, we would negate the output value.
c) If \(f(x)\) is translated down 2 unit(s), we would subtract 2 from the output value.
d) If \(f(x)\) is translated right by 1 unit(s), then we add 1 to the input value.
e) Consider if \(f(x)\) is reflected about the \(x\)-axis, then shifted up by 4 unit(s). Make sure you apply the transformations in the correct order. It may be useful to go step-by-step here.
f) Consider if \(f(x)\) is translated up by 4 unit(s), then reflected about the \(x\)-axis. Make sure you apply the transformations in the correct order. It may be useful to go step-by-step here.
4. Table of an Even Function.
Suppose \(f(x)\) is an even function. Fill in as many missing values in the following table as possible. If it is not possible to fill in a value, write "NONE". You can assume that \(f\) is defined on all values listed in the table.
| \(x\) | -6 | -4 | -2 | 0 | 2 | 4 | 6 |
| \(f(x)\) | -2 | -3 | 3 |
Hint.
Since \(f(x)\) is an even function, we know that \(f(x)=f(-x)\) for all values of the input \(x\text{.}\)
What does this mean about the value of \(f(0)\text{?}\)
5. Table of an Odd Function.
Suppose \(f(x)\) is an odd function. Fill in as many missing values in the following table as possible. If it is not possible to fill in a value, write "NONE". You can assume that \(f\) is defined on all values listed in the table.
| \(x\) | -6 | -4 | -2 | 0 | 2 | 4 | 6 |
| \(f(x)\) | -1 | 4 | -4 |
Hint.
Since \(f(x)\) is an odd function, we know that \(-f(x)=f(-x)\) for all values of the input \(x\text{.}\)
What does this mean about the value of \(f(0)\text{?}\)
6. Reflecting Graph.
The function \(g(x)\) is graphed below.
a) Which of the following graphs is the graph of \(g(-x)\text{?}\)
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A
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B
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C
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D
 |
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| A | B |
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|
| C | D |
(Click on a graph to enlarge it)
Hint.
Notice we have negated the input values. Does this mean we reflect over the \(x\)-axis or the \(y\)-axis?
7. Reflecting Graph.
The function \(g(x)\) is graphed below.
a) Which of the following graphs is the graph of \(-g(x)+2\text{?}\)
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A
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B
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C
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D
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| A | B |
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| C | D |
(Click on a graph to enlarge it)
Hint.
First, determine what transformations have occurred. Then choose the appropriate graph with those transformations. It may be useful to think about individual points to find the answer.
