Lesson 3-1Symmetry and Coordinate

Graphs

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.

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.

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.

• 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.

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

unchanged.

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.

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

We can verify by following two steps:

Symmetric to Origin

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.

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

The graph is not symmetric with respect to the origin.

• 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.

Line Symmetry

Line Symmetry

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

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

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

Classwork