In setting up the table showing transformations of the graphs of x², |x| and sqrt(x), we saw that y = x², y = |x| are symmetric about the y-axis and the line of symmetry is convenient to use as a locator when translating a graph. What we see when we look at f(-x), where the graph is found by reflecting the graph of f in the y-axis, is that the graph is the same as the original graph if we start with x^2 or |x|. However, we had to do two reflections--the reflection of the graph of y = x³ in the y-axis and the new graph in the x-axis to obtain the y = x³ graph. The graph of y = x³ is symmetric about the origin.

A graph is symmetric with respect to a line iff for every point on one side of the line there is a point on the perpendicular to the line through the point the same distance on the opposite side of the line (in particular, graphs can be symmetric with respect either axis).

Example 1: (a) y = x² and y = |x| are symmetric with respect to the y-axis
(b) x² + y² = 25 is symmetric with respect to both x-axes (and any line through the center)
y = 3(x - 2)² - 4 is symmetric with respect to x = 2

A graph is symmetric with respect to the y-axis iff whenever (x,y) is on the graph (-x, y) is also on the graph. For a function f, the graph of f is symmetric with respect to the y-axis iff f(-x) = f(x). We say f is an even function in this case.

Example 2: Let f(x) = 2x⁴ - 3x² + 1

Problem 1 Part of the graph of a function f is shown below. The function is symmetric with respect to the y-axis. Complete the graph.

  |  ______
  | /      \
__|/________\___
    1     4 5

A graph is symmetric with respect to a point P, iff whenever Q is on the graph there is a point on the line through P and Q (on the opposite side from P) and the same distance from P as Q. If the point is the origin and Q=(x,y), then the symmetric point is (-x,-y)

Example 3 The graphs of y = x, y = x³, and y = x^1/3 are symmetric with respect to the origin.

A graph is symmetric with respect to the origin iff whenever (x,y) in on the graph (-x,-y) is also on the graph. For a function f, the graph of f is symmetric with respect to the origin iff f(-x) = -f(x) (we say that f is an odd function in this case).

Example 4 f(x) = x/(x² + 1)
f(-x) = (-x)/((-x)² + 1) = -[x/(x² + 1)] = -f(x)
Therefore f is an odd function and

Problem 2 Part of the graph of a function f is shown below. The graph is symmetric with respect to the origin. Complete the graph.

  |  ______
  | /      \
__|/________\___
    1      4 5

Remember that symmetry is not altered in translation but the line(s) or point of symmetry is translated.