Solved Examples on the Symmetry of Functions

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(1.) Find the point that is symmetric to the point $(3, 7)$ with respect to:
(a.) x-axis
(b.) y-axis
(c.) origin


Symmetric about the x-axis
For each $(x, y)$ on the graph; there is also $(x, -y)$ on the same graph
Same $x$, Opposite $y$
For each $(3, 7)$; there should be $(3, −7)$
Point symmetric to $(3, 7)$ = $(3, −7)$

Symmetric about the y-axis
For each $(x, y)$ on the graph; there is also $(−x, y)$ on the same graph
Same $y$, Opposite $x$
For each $(3, 7)$; there should be $(−3, 7)$
Point symmetric to $(3, 7)$ = $(−3, 7)$

Symmetric about the origin
For each $(x, y)$ on the graph; there is also $(−x, −y)$ on the same graph
Opposite $x$, Opposite $y$
For each $(3, 7)$; there should be $(−3, −7)$
Point symmetric to $(3, 7)$ = $(−3, −7)$
(2.) Find the point that is symmetric to the point $(-3, 7)$ with respect to:
(a.) x-axis
(b.) y-axis
(c.) origin


Symmetric about the x-axis
For each $(x, y)$ on the graph; there is also $(x, -y)$ on the same graph
Same $x$, Opposite $y$
For each $(-3, 7)$; there should be $(-3, -7)$
Point symmetric to $(-3, 7)$ = $(-3, -7)$

Symmetric about the y-axis
For each $(x, y)$ on the graph; there is also $(-x, y)$ on the same graph
Same $y$, Opposite $x$
For each $(-3, 7)$; there should be $(3, 7)$
Point symmetric to $(-3, 7)$ = $(3, 7)$

Symmetric about the origin
For each $(x, y)$ on the graph; there is also $(-x, -y)$ on the same graph
Opposite $x$, Opposite $y$
For each $(-3, 7)$; there should be $(3, -7)$
Point symmetric to $(-3, 7)$ = $(3, -7)$
(3.) Find the point that is symmetric to the point $(3, -7)$ with respect to:
(a.) x-axis
(b.) y-axis
(c.) origin


Symmetric about the x-axis
For each $(x, y)$ on the graph; there is also $(x, -y)$ on the same graph
Same $x$, Opposite $y$
For each $(3, -7)$; there should be $(3, 7)$
Point symmetric to $(3, -7)$ = $(3, 7)$

Symmetric about the y-axis
For each $(x, y)$ on the graph; there is also $(-x, y)$ on the same graph
Same $y$, Opposite $x$
For each $(3, -7)$; there should be $(-3, -7)$
Point symmetric to $(3, -7)$ = $(-3, -7)$

Symmetric about the origin
For each $(x, y)$ on the graph; there is also $(-x, -y)$ on the same graph
Opposite $x$, Opposite $y$
For each $(3, -7)$; there should be $(-3, 7)$
Point symmetric to $(3, -7)$ = $(-3, 7)$
(4.) Find the point that is symmetric to the point $(-3, -7)$ with respect to:
(a.) x-axis
(b.) y-axis
(c.) origin


Symmetric about the x-axis
For each $(x, y)$ on the graph; there is also $(x, -y)$ on the same graph
Same $x$, Opposite $y$
For each $(-3, -7)$; there should be $(-3, 7)$
Point symmetric to $(-3, -7)$ = $(-3, 7)$

Symmetric about the y-axis
For each $(x, y)$ on the graph; there is also $(-x, y)$ on the same graph
Same $y$, Opposite $x$
For each $(-3, -7)$; there should be $(3, -7)$
Point symmetric to $(-3, -7)$ = $(3, -7)$

Symmetric about the origin
For each $(x, y)$ on the graph; there is also $(-x, -y)$ on the same graph
Opposite $x$, Opposite $y$
For each $(-3, -7)$; there should be $(3, 7)$
Point symmetric to $(-3, -7)$ = $(3, 7)$
(5.) Determine whether the function is even, odd, or neither even nor odd.
$f(x) = x\sqrt{16 - x^2}$
In terms of symmetry, what is the nature of the graph of the function?


A function is even if $f(-x) = f(x)$
A function is odd if $f(-x) = -f(x)$
If a function is not even, and is not odd; then the function is neither even nor odd

$ f(x) = x\sqrt{16 - x^2} \\[3ex] f(x) = x * \sqrt{16 - x^2} \\[3ex] $ First: Use f(1) to make it easy.

$ f(x) = x * \sqrt{16 - x^2} \\[3ex] f(1) = 1 * \sqrt{16 - 1^2} \\[3ex] f(1) = \sqrt{16 - 1} \\[3ex] f(1) = \sqrt{15} \\[3ex] $ Second: Test for even

$ f(x) = x * \sqrt{16 - x^2} \\[3ex] f(-1) -1 * \sqrt{16 - (-1)^2} \\[3ex] f(-1) = -1 * \sqrt{16 - 1} \\[3ex] f(-1) = -1 * \sqrt{15} \\[3ex] f(-1) = -\sqrt{15} \\[3ex] f(-1) \ne f(1) \\[3ex] -\sqrt{15} \ne \sqrt{15} \\[3ex] $ Function is not even.

Third: Test for odd

$ f(x) = x * \sqrt{16 - x^2} \\[3ex] -f(1) = -1 * f(1) \\[3ex] -f(1) = -1 * \sqrt{15} \\[3ex] -f(1) = -\sqrt{15} \\[3ex] f(-1) = -f(1) \\[3ex] -\sqrt{15} = -\sqrt{15} \\[3ex] $ Function is odd.
The graph of the function is symmetric about the origin
(6.) Determine whether the function is even, odd, or neither even nor odd.
$f(x) = |16x|$
In terms of symmetry, what is the nature of the graph of the function?


A function is even if $f(-x) = f(x)$
A function is odd if $f(-x) = -f(x)$
If a function is not even, and is not odd; then the function is neither even nor odd

$ f(x) = |16x| \\[3ex] $ First: Use f(1) to make it easy.

$ f(x) = |16x| \\[3ex] f(1) = |16 * 1| \\[3ex] f(1) = |16| \\[3ex] f(1) = 16 \\[3ex] $ Second: Test for even

$ f(x) = |16x| \\[3ex] f(-1) = |16 * -1| \\[3ex] f(-1) = |-16| \\[3ex] f(-1) = 16 \\[3ex] f(-1) = f(1) \\[3ex] 16 = 16 \\[3ex] $ Function is even.
The graph of the function is symmetric about the y-axis
(7.) Determine whether the function is even, odd, or neither even nor odd.
$f(x) = \dfrac{9x}{5x^2 - 6}$
In terms of symmetry, what is the nature of the graph of the function?


A function is even if $f(-x) = f(x)$
A function is odd if $f(-x) = -f(x)$
If a function is not even, and is not odd; then the function is neither even nor odd

$ f(x) = \dfrac{9x}{5x^2 - 6} \\[3ex] $ First: Use f(1) to make it easy.

$ f(x) = \dfrac{9x}{5x^2 - 6} \\[5ex] f(1) = \dfrac{9 * 1}{5(1)^2 - 6} \\[5ex] f(1) = \dfrac{9}{5(1) - 6} \\[5ex] f(1) = \dfrac{9}{5 - 6} \\[5ex] f(1) = \dfrac{9}{-1} \\[5ex] f(1) = -9 \\[3ex] $ Second: Test for even

$ f(x) = \dfrac{9x}{5x^2 - 6} \\[5ex] f(-1) = \dfrac{9 * -1}{5(-1)^2 - 6} \\[5ex] f(1) = \dfrac{-9}{5(1) - 6} \\[5ex] f(1) = \dfrac{-9}{5 - 6} \\[5ex] f(1) = \dfrac{-9}{-1} \\[5ex] f(1) = 9 \\[3ex] f(-1) \ne f(1) \\[3ex] 9 \ne -9 \\[3ex] $ Function is not even.

Third: Test for odd

$ f(x) = x * \sqrt{16 - x^2} \\[3ex] -f(1) = -1 * f(1) \\[3ex] -f(1) = -1 * -9 \\[3ex] -f(1) = 9 \\[3ex] f(-1) = -f(1) \\[3ex] 9 = 9 \\[3ex] $ Function is odd odd.
The graph of the function is symmetric about the origin.
(8.) Determine whether the function is even, odd, or neither even nor odd.
$f(x) = 3x - |3x|$
In terms of symmetry, what is the nature of the graph of the function?


A function is even if $f(-x) = f(x)$
A function is odd if $f(-x) = -f(x)$
If a function is not even, and is not odd; then the function is neither even nor odd

$ f(x) = 3x - |3x| \\[3ex] $ First: Use f(1) to make it easy.

$ f(x) = 3x - |3x| \\[3ex] f(1) = 3(1) - |3 * 1| \\[3ex] f(1) = 3 - |3| \\[3ex] f(1) = 3 - 3 \\[3ex] f(1) = 0 \\[3ex] $ Second: Test for even

$ f(x) = 3x - |3x| \\[3ex] f(-1) = 3(-1) - |3 * -1| \\[3ex] f(1) = -3 - |-3| \\[3ex] f(1) = -3 - 3 \\[3ex] f(1) = -6 \\[3ex] f(-1) \ne f(1) \\[3ex] -6 \ne 0 \\[3ex] $ Function is not even.

Third: Test for odd

$ f(x) = x * \sqrt{16 - x^2} \\[3ex] -f(1) = -1 * f(1) \\[3ex] -f(1) = -1 * 0 \\[3ex] -f(1) = 0 \\[3ex] f(-1) \ne -f(1) \\[3ex] -6 \ne 0 \\[3ex] $ Function is not odd.
The function is neither even nor odd.
(9.) Determine whether the graph is even, odd, or neither even nor odd.
$9x^2 - 8y^2 = 4$
In terms of symmetry, what is the nature of the graph of the function?


A graph is symmetrical about the y-axis if $f(-x) = f(x)$
A graph is symmetrical about the x-axis if $f(x) = -f(x)$
A graph is symmetrical about the origin if $f(-x) = -f(x)$

First: Let us isolate $y$

$ 9x^2 - 8y^2 = 4 \\[3ex] 9x^2 - 4 = 8y^2 \\[3ex] 8y^2 = 9x^2 - 4 \\[3ex] y^2 = \dfrac{9x^2 - 4}{8} \\[5ex] y = \pm \dfrac{9x^2 - 4}{8} \\[5ex] y = f(x) \\[3ex] \therefore f(x) = \pm \sqrt{\dfrac{9x^2 - 4}{8}} \\[5ex] $
This is not a function.

Teacher: Why is it not a function?
Student: It is not a function because an input value can have two output values due to the $\pm$ sign
Teacher: Correct!

Second: Determine f(1) to make it easy.

$ f(x) = \pm \sqrt{\dfrac{9x^2 - 4}{8}} \\[5ex] f(1) = \pm \sqrt{\dfrac{9(1)^2 - 4}{8}} \\[5ex] f(1) = \pm \sqrt{\dfrac{9(1) - 4}{8}} \\[5ex] f(1) = \pm \sqrt{\dfrac{9 - 4}{8}} \\[5ex] f(1) = \pm \sqrt{\dfrac{5}{8}} \\[5ex] f(1) = \sqrt{\dfrac{5}{8}} \:\:\;\;OR\;\;\:\: f(1) = -\sqrt{\dfrac{5}{8}} \\[5ex] $ Third: Test for symmetry about the y-axis

$ f(x) = \pm \sqrt{\dfrac{9x^2 - 4}{8}} \\[5ex] f(-1) = \pm \sqrt{\dfrac{9(-1)^2 - 4}{8}} \\[5ex] f(-1) = \pm \sqrt{\dfrac{9(1) - 4}{8}} \\[5ex] f(-1) = \pm \sqrt{\dfrac{9 - 4}{8}} \\[5ex] f(-1) = \pm \sqrt{\dfrac{5}{8}} \\[5ex] f(-1) = \sqrt{\dfrac{5}{8}} \;\;\:\:OR\;\;\:\: f(1) = -\sqrt{\dfrac{5}{8}} \\[5ex] f(-1) = f(1) \\[3ex] $ The graph is symmetric about the y-axis

Fourth: Test for symmetry about the x-axis

$ f(x) = \pm \sqrt{\dfrac{9x^2 - 4}{8}} \\[5ex] -f(1) = -1 * f(1) \\[3ex] -f(1) = -1 * \pm \sqrt{\dfrac{5}{8}} \\[5ex] -f(1) = \mp \sqrt{\dfrac{5}{8}} \\[5ex] -f(1) = -\sqrt{\dfrac{5}{8}} \:\:\;\;OR\;\;\:\: -f(1) = \sqrt{\dfrac{5}{8}} \\[5ex] f(1) = -f(1) \\[3ex] $ The graph is symmetric about the x-axis

Fifth: Test for symmetry about the origin

$ f(x) = \pm \sqrt{\dfrac{9x^2 - 4}{8}} \\[5ex] -f(1) = -1 * f(1) \\[3ex] -f(1) = -1 * \pm \sqrt{\dfrac{5}{8}} \\[5ex] -f(1) = \mp \sqrt{\dfrac{5}{8}} \\[5ex] -f(1) = -\sqrt{\dfrac{5}{8}} \:\:\;\;OR\;\;\:\: f(1) = \sqrt{\dfrac{5}{8}} \\[5ex] f(-1) = -f(1) \\[3ex] $ The graph is symmetric about the origin.
(10.) Determine the symmetries (if any) of the graph of the given relation: $6y = 2x^2 - 5$


A graph is symmetrical about the:
y-axis if $f(-x) = f(x)$
x-axis if $f(x) = -f(x)$
origin if $f(-x) = -f(x)$

First: Isolate $y$

$ 6y = 2x^2 - 5 \\[3ex] y = \dfrac{2x^2 - 5}{6} \\[5ex] y = \dfrac{2x^2}{6} - \dfrac{5}{6} \\[5ex] y = \dfrac{x^2}{3} - \dfrac{5}{6} \\[5ex] $ Second: Determine f(1) to make it easy.

$ f(x) = \dfrac{x^2}{3} - \dfrac{5}{6} \\[5ex] f(1) = \dfrac{1^2}{3} - \dfrac{5}{6} \\[5ex] = \dfrac{1}{3} - \dfrac{5}{6} \\[5ex] = \dfrac{2}{6} - \dfrac{5}{6} \\[5ex] = -\dfrac{3}{6} \\[5ex] = -\dfrac{1}{2} \\[5ex] $ Third: Test for symmetry about the y-axis

$ f(x) = \dfrac{x^2}{3} - \dfrac{5}{6} \\[5ex] f(-1) = \dfrac{(-1)^2}{3} - \dfrac{5}{6} \\[5ex] = \dfrac{1}{3} - \dfrac{5}{6} \\[5ex] = \dfrac{2}{6} - \dfrac{5}{6} \\[5ex] = -\dfrac{3}{6} \\[5ex] = -\dfrac{1}{2} \\[5ex] = f(1) \\[3ex] $ The graph is symmetric about the y-axis

Fourth: Test for symmetry about the x-axis

$ f(x) = \dfrac{x^2}{3} - \dfrac{5}{6} \\[5ex] -f(1) = -1 * f(1) \\[3ex] -f(1) = -1 * -\dfrac{1}{2} \\[5ex] -f(1) = \dfrac{1}{2} \\[5ex] -\dfrac{1}{2} \ne \dfrac{1}{2} \\[5ex] f(1) \ne -f(1) \\[3ex] $ The graph is not symmetric about the x-axis

Fifth: Test for symmetry about the origin

$ f(x) = \dfrac{x^2}{3} - \dfrac{5}{6} \\[5ex] f(-1) = -\dfrac{1}{2} \\[5ex] -f(1) = \dfrac{1}{2} \\[5ex] -\dfrac{1}{2} \ne \dfrac{1}{2} \\[5ex] f(-1) = -f(1) \\[3ex] $ The graph is not symmetric about the origin
(11.)


A graph is symmetrical about the y-axis if $f(-x) = f(x)$
A graph is symmetrical about the x-axis if $f(x) = -f(x)$
A graph is symmetrical about the origin if $f(-x) = -f(x)$

(12.) ACT The point (a, b) in the standard (x, y) coordinate plane lies on a curve that is symmetric with respect to the y-axis.
Which of the following points must also lie on that curve?

$ A.\;\; (-a, -b) \\[3ex] B.\;\; (-a, b) \\[3ex] C.\;\; (a, -b) \\[3ex] D.\;\; (-b, a) \\[3ex] E.\;\; (b, a) \\[3ex] $

A graph is symmetrical about the y-axis if $f(-x) = f(x)$
This means that: For each (x, y) on the graph, there is also (−x, y) on the same graph.
Similarly, for each (a, b) on the graph, there is also (−a, b) on the same graph.