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Lesson 3-1 Symmetry and Coordinate Graphs

Lesson 3-1

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Lesson 3-1. Symmetry and Coordinate Graphs. Symmetry with respect to the origin. Two Steps: Find f(-x) and –f(x) If f(-x)=-f(x), the graph is symmetric with respect to the origin. Symmetry with respect to the x-axis , y-axis , the line y=x, and the line y=-x. - PowerPoint PPT Presentation

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Page 1: Lesson 3-1

Lesson 3-1Symmetry and Coordinate

Graphs

Page 2: Lesson 3-1

Symmetry with respect to the origin

Two Steps:1. Find f(-x) and –f(x)2. If f(-x)=-f(x), the graph is symmetric

with respect to the origin.

Page 3: Lesson 3-1

Symmetry with respect to the x-axis, y-axis, the line y=x, and the line y=-x.

1. Substitute (a,b) into the equation.2. x-axis, substitute (a,-b)3. y-axis, substitute (-a,b)4. y=x, substitute (b,a)5. y=-x, substitute (-b,-a)6. Check to see which test produces equivalent

equations.

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Vocabulary

• Image Point – When applying point symmetry to a set of points, each point P in the set must have an image point P′ which is also in the set.

• Point Symmetry - Two distinct points P and P ′are symmetric with respect to a point M if and only if M is the midpoint of segment PP . ′Point M is symmetric with respect to itself.

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• Line Symmetry – Two distinct points P and P ′are symmetric with respect to a line l if and only if l is the perpendicular bisector of segment PP . A point P is symmetric to itself ′with respect to line l if and only if P is on l.

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A figure that is symmetric with respect to a given point can be rotated 180° about that point and appear

unchanged.

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The origin is a common point of symmetry.

The values in the tables suggest that f(-x)=-f(x) whenever the graph of a function is symmetric with respect to the origin.

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Determine whether the graph is symmetric with respect to the origin.

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We can verify by following two steps:

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Symmetric to Origin

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Determine whether the graph is symmetric with respect to the origin.

The graph appears to be symmetric with respect to the origin.

1. Find f(-x) and - f(x).2. If f(-x) = - f(x), the graph has point symmetry.

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Determine whether the graph is symmetric with respect to the origin.

The graph is not symmetric with respect to the origin.

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• Line Symmetry – Two distinct points P and P ′are symmetric with respect to a line l if and only if l is the perpendicular bisector of segment PP . A point P is symmetric to itself ′with respect to line l if and only if P is on l.

Graphs that have line symmetry can be folded along the line of symmetry so that the two halves match exactly. Some graphs have more than one line of symmetry.

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Line Symmetry

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Line Symmetry

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Some common lines of symmetry are the x-axis, the y-axis, the line y=x , and the line y=-x.

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Determine whether the graph of x²+y= 3 is symmetric with respect to the x-axis, y-axis, the line y = x, the line y = -x, or none of these.

Substituting (a,b) into the equation yields a²+b=3 x axis (a,-b) a²-b=3y axis (-a,b) (-a)²+b=3 a²+b=3 y=x (b,a) b²+a=3y=-x (-b,-a) (-b)²+(-a)=3 b²-a=3

The graph is symmetric to the y axis

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The graph is symmetric to both the x and y axis.

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Even and Odd Functions

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Classwork

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