1.2: Graphs and Symmetry

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Oct 6, 2021
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- 1.1: Functions and Graphs
- 1.3: Shifting and Reflecting

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Symmetry (Geometry)

Definition: Symmetric with respect to the y-axis

We say that a graph is symmetric with respect to the y-axis if for every point $(a,b)$ on the graph, there is also a point $(-a,b)$ on the graph; hence $f(x,y) = f(-x,y).$

Visually we have that the y-axis acts as a mirror for the graph. We will demonstrate several functions to test for symmetry graphically using the graphing calculator.

$ysym.gif$

Definition: Symmetric with respect to the x-axis

We say that a graph is symmetric with respect to the x axis if for every point $(a,b)$ on the graph, there is also a point $(a,-b)$ on the graph; hence $f(x,y) = f(x,-y).$

Visually we have that the x-axis acts as a mirror for the graph. We will demonstrate several functions to test for symmetry graphically using the graphing calculator.

Definition: Symmetry with respect to Origin

We say that a graph is symmetric with respect to the origin if for every point $(a,b)$ on the graph, there is also a point $(-a,-b)$ on the graph; hence $f(x,y) = f(-x,-y).$

Visually we have that given a point $P$ on the graph if we draw a line segment $PQ$ through $P$ and the origin such that the origin is the midpoint of $PQ$ , then $Q$ is also on the graph.

$alt$

We will use the graphing calculator to test for all three symmetries.

Symmetry (Algebra)

x-axis Symmetry

To test algebraically if a graph is symmetric with respect the x-axis, we replace all the $y$ 's with $-y$ and see if we get an equivalent expression.

Example $\PageIndex{1}$

For

$x - 2y = 5$

we replace with

$x - 2(-y) = 5.$

Simplifying we get

$x+2y=5.$

Which is not equivalent to the original expression. So

$x - 2y = 5$

Is not symmetric with respect to the x-axis.

Example $\PageIndex{2}$

For

$x^3 - y^2 = 2$

we replace with

$x^3 - (-y)^2 = 2$

which is equivalent to the original expression, so that

$x^3 - y^2 = 2$

is symmetric with respect to the x-axis.

y-axis Symmetry

To test algebraically if a graph is symmetric with respect to the y axis, we replace all the x's with -x and see if we get an equivalent expression.

Example $\PageIndex{3}$

For

$y = x^2$

we replace with

$y = (-x)^2 = x^2$

so that

$y = x^2$

is symmetric with respect to the y-axis.

Example $\PageIndex{4}$

For

$y = x^3$

we replace with

$y = (-x)^3 = - x^3$

so that

$y = x^3$

is not symmetric with respect to the y-axis.

Origin Symmetry

To test algebraically if a graph is symmetric with respect to the origin we replace both $x$ and $y$ with $-x$ and $-y$ and see if the result is equivalent to the original expression.

Example $\PageIndex{5}$

For

$y = x^3$

we replace with

$(-y) = (-x)^3$

so that

$-y = -x^3 \text{ or } y=x^3.$

Hence

$y = x^3$

is symmetric with respect to the origin.

Intercepts

We define the x intercepts as the points on the graph where the graph crosses the x axis. If a point is on the x axis, then the y coordinate of the point is 0. Hence to find the x intercepts, we set $y = 0$ and solve.

Example $\PageIndex{6}$ : x intercepts

Find the x intercepts of

$y = x^2+ x - 2$

Solution

We set $y = 0$ so that

$0 = x^2+ x - 2 = (x + 2)(x - 1)$

Hence that x intercepts are at $(-2,0)$ and $(1,0)$

We define the y intercepts of a graph to be the points where the graph crosses the y-axis. At these points the $x$ coordinate is 0 hence to find the y intercepts we set $x = 0$ and find $y$ .

Example $\PageIndex{7}$ : y intercepts

Find the y intercepts of

$y = x^2+ x - 2$

Solution

We set $x = 0$ to get:

$y = 0 + 0 - 2 = -2.$

Hence the y intercept is at $(0,-2)$ .

Contributors and Attributions

Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.

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Symmetry (Geometry)

Symmetry (Algebra)

x-axis Symmetry

y-axis Symmetry

Origin Symmetry

Intercepts

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