The reflection of the graph of y fx across the xaxis the re

The reflection of the graph of y = f(x) across the x-axis the reflection of the graph of v = f(x) across the y-axis Concept Check Plot each point, and then plot the points that are symmetric to the given point with respect to the (a) x-axis, (b) y-axis, and (c) origin. (5,-3) (-6,1) (-4,-2) (-8,0) Concept Check The graph of y =|x - 2| is symmetric with respect to a vertical line. What is the equation of that line? Concept Check Repeat Exercise 33 for y = -|x + 1|. Without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. See Examples 3 and 4 Determine whether each function is even, odd, or neither. See Example 5

Solution

40 y = x^3 -x

For Symmetry About Y-Axis

For symmetry with respect to the Y-Axis, check to see if the equation is the same when we replace x with x and there is no change in equation.

y = (-x)^3 - (-x) = -x^3 +x = - (x^3 -x) .equation is not symmeteric with respect to x axis.

For Symmetry About X-Axis

Use the same idea as for the Y-Axis, but try replacing y with y and there is no change in equation.

-y = x^3 -x -----> y = -(x^3 -x) equation is not symmeteric with respect to y axis.

Origin Symmetry:

Check to see if the equation is the same when we replace both x with x and y with y.

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

- y = -x^3 +x----> y = x^3 -x

No change in equation

Equation w.r.t is symmteric origin