Graphs and Symmetry

1. Symmetry (Geometry)

 Definition 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.

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.

 Definition 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.

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 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.

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.

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

2. Symmetry (Algebra)

Examples:

1. 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.

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.

2. For

x3 - y2 = 2

we replace with

x3 - (-y)2 = 2

which is equivalent to the original expression, so that

x3 - y2 = 2

is symmetric with respect to the x axis.

2. 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.

1. For

y = x

we replace with

y = (-x)2 =  x

so that

y = x

is symmetric with respect to the y axis.

2. For

y = x3

we replace with

y = (-x)3 = - x3

so that

y = x3

is not symmetric with respect to the y axis.

3. 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.

1. For

y = x3

we replace with

(-y) = (-x)3

so that

-y = -x3 or y = x3

Hence

y = x3

is symmetric with respect to the origin.

3. 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:

Find the x intercepts of

y = x2 + x - 2

We set y = 0 so that

0 =  x2 + 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 fine the y intercepts we set x = 0 and find y.

Example:

Find the y intercepts of

y = x2 + x - 2

Solution:

We set x = 0 to get:

y = 0 + 0 - 2 = -2.

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

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