8.2 Symmetry Graphing Nonlinear Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1...

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8.2 Symmetry Graphing Nonlinear Equations

BobsMathClass.Com Copyright © 2010 All Rights Reserved.

y-axis Symmetry (figure a)

A line or curve drawn on an x-y coordinate system is called a graph. The graph above is a parabola. This parabola is said to be symmetric with respect to the y axis. In other words, the y-axis acts as a line of symmetry for the curve. Each half of the curve is a mirror image of the other half through the y axis.

Symmetry

x-axis Symmetry (figure b) The graph to the right is also a parabola. It is said to be symmetric

with respect to the x axis. The x-axis is referred to as the line ofsymmetry for this graph. Each half of the curve is a mirror imageof the other half through the x-axis.

Origin Symmetry (figure c) The graph above is also a parabola. It is said to be symmetric

with respect to the x axis. The x-axis is referred to as the line ofsymmetry for this graph. Each half of the curve is a mirror imageof the other half through the x-axis.

(figure a)

(figure b)

(figure c)

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The previous slides were examples of graphs that exhibited some form of symmetry.

In our next topic, we will be using algebraic tests on the equations of the graphs to help us determine whether (or not) a graph exhibits one or more of these symmetries. The results of these tests can be used to help us draw the graphs of non-linear equations.

Not all graphs have symmetry and some may have a combination of symmetries.

Y-Axis SymmetryThe graph of an equation is symmetric with respect to the y axis if replacing x with –x results in an equivalent equation.

X-Axis SymmetryThe graph of an equation is symmetric with respect to the x axis if replacing y with –y results in an equivalent equation.

Origin SymmetryThe graph of an equation is symmetric with respect to the origin if replacing x with –x and y with –y results in an equivalent equation.

Tests for Symmetry (Memorize)

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Next, find the point symmetric with respect to the y-axis by replacing x with -x.

Example 1a. Determine the point which will be symmetric to (-2, 3) with respect to the y axis. y axis

5

4

3

2

1 x-axis

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-2

-3

- -4

-5 y axis

5

4

3

2

1 x-axis

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-2

-3

- -4

-5

Solution: First graph the point.

(-2, 3)

Next, find the point symmetric with respect to the origin by replacing x with –x, and replacing y with -y.

Solution: First graph the point.

(-3,4)

1b. Determine the point which will be symmetric to (-3, 4) with respect to the origin.

(3,-4)Answer: (3,-4)

(2, 3)

Answer: (2,3)

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1. Replace x with –x. Does the equation remain the same?

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2y ( x) 3 Since replacing x with x results y x 3 in an equivalent equation, the graph is symmetric to the y axis

2. Replace y with –y. Does the equation remain the same?

2y x 3 Since replacing y with y does not result in an equivalent equation, the graph is not symmetric to the x axis.

3. Replace x with –x and y with – y. Does the equation remain the same?

2

2y ( x) 3 Since replacing x with x and y with y y x +3 does not result in an equivalent equation,

the graph is not symmetric to the origin.

2 The graph of y x 3 is only symmetric to thAnswer: e y axis.

Example 2: a) Determine the type(s) of symmetry exhibited by the graph of y = x2 + 3.

Your Turn Problem #1

Find the points that are symmetric to (-2, 5) with respect to:

a) the x axisb) the y axisc) the origin

Answersa) (-2, -5)b) (2, 5)c) (2, -5)

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Determine the type(s) of Symmetry (with respect to thex axis, y axis and/or origin) exhibited by the graphs of the equations:

Your Turn Problem #2

2a) x y 42

b) y = x 2 2c) 4x 9y 36

1. Replace x with –x. Does the equation remain the same?

( x)y 7 Since replacing x with x does not result in an xy 7 equivalent equation, the graph is not symmetric to

the y axis.

2. Replace y with –y. Does the equation remain the same?

x( y) 7 Since replacing y with y does not result in anxy 7 equivalent equation, the graph is not symmetric to

the x axis.

3. Replace x with –x and y with – y. Does the equation remain the same?( x)( y) 7 Since replacing x with x and y with y

xy 7 results in an equivalent equation, the graph is symmetric to the origin.

The graph of xy=7 is only symmetric to the Answ orier: gin.

Example 2b) Determine the type(s) of symmetry exhibited by the graph of xy=7

Answers.a) symmetry with respect to the x axis only. b) symmetry with respect to the origin only.c) symmetry with respect to the x axis, y axis and origin.

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Graphing Nonlinear Equations – The ParabolaNonlinear equations will be covered more thoroughly in college algebra. For now, we will make use the symmetry information and graph one type of nonlinear equation called a parabola. Basic techniques of graphing is to observe if a graph has symmetry and pick “enough” points to get an accurate sketch of the equation.

Next semester, we will cover the parabola and its properties more in depth. Knowing which values to choose for x is important. If we do not choose the “right” values for x, we will not obtain an accurate depiction of the graph. For the next examples and your turn problems, we will pick the following values for x: 0, 1, 2, –1 , and –2.

Recall the symmetry test.

Y-Axis SymmetryThe graph of an equation is symmetric with respect to the y axis if replacing x with –x results in an equivalent equation.

2 Is y x 1 symmetric to tE hxampl e y-ae: xis?

2 2Replace x with -x: y x 1. This results in y x 1, an equivalent equation.Answer: yes, the graph is symmetric to the y-axis.

Next Slide

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First, consider if this graph has symmetry. Replacing x with –x will result in the same equation. Therefore, this graph is symmetric to the y-axis.

Now we can plot the points and sketch the parabola.

-1 0 3 0 3

2 Graph y x 1 by choosing "enough" valExample 3 ues f. or x.

Second, choose “enough” values for x. For now, use the values given in the table.

x y

0-1-2 1 2

2y (0) 1 y 1

x 0;

2y ( 1) 1 y 0

x 1;

2y ( 2) 1 y 3

x 2;

2y (1) 1 y 0

x 1;

2y (2) 1 y 3

x 2;

•

•

•

•

•

Next Slide

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Your Turn Problem # 3

2Graph y x 2 by choosing points for x.

Answer:

23636

x y

0-1-2 1 2

•

•

•

•

•

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First, consider if this graph has symmetry. Replacing x with –x will result in the same equation. Therefore, this graph is symmetric to the y-axis.

Now we can plot the points and sketch the parabola.

3 1-5 1-5

Second, choose “enough” values for x. For now, use the values given in the table.

2 Graph y 2x 3 by choosing "enough" valExample 4. ues f or x.

x y

0-1-2 1 2

Next Slide

•

•

•

•

•

2y 2(0) 3 y 3

x 0;

2y 2( 1) 3 y 1

x 1;

2y 2( 2) 3 y 5

x 2;

2y 2(1) 3 y 1

x 1;

2y 2(2) 3 y 5

x 2;

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Your Turn Problem # 42Graph y x 4 by choosing points for x.

Answer:

43030

x y

0-1-2 1 2

•

••

•

•

The EndB.R. 11-1-06

8.2 Symmetry Graphing Nonlinear Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1...

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