Chapter 3 – The Nature of Graphs Section 1 – Symmetry and Coordinate Graphs

Basic Symmetry The most basic type of symmetry is point symmetry. A graph has point symmetry if it is symmetric about a SPECIFIC POINT. For example, the origin (0,0) is a common point of symmetry. f(x)=1/x and g(x)=x3 are both symmetric with respect to the origin.

Algebraically Determine Symmetry To determine whether or not a function is symmetric with respect to the origin, we simply need to see if –f(x)=f(-x). If –f(x)=f(-x), then the function is symmetric about the origin.

Line Symmetry If a function is symmetric with respect to the x-axis, then for some point (a,b) in the function, there should also be (a,-b) in the function. If a function is symmetric with respect to the y-axis, then for some point (a,b) in the function, there should also be (-a,b) in the function.

More Line Symmetry If a function is symmetric with respect to the line y=x, then for some point (a,b) in the function, there should be a point (b,a) also in the function. If a function is symmetric with respect to the line y=-x, then for some point (a,b) in the function, there should be a point (-b,-a) also in the function.

SYMMETRY WITH RESPECT TO THE LINE Line Symmetry Table SYMMETRY WITH RESPECT TO THE LINE DEFINITION x-axis If point (a,b) exists, then point (a,-b) must also exist. y-axis If (a,b) exists, then (-a,b) must also exist. y=x If (a,b) exists, then (b,a) must also exist. y=-x (-b,-a) must also exist.

Figuring Lines of Symmetry EX 1: Determine whether the graph of xy=-2 is symmetric with respect to the x-axis, y-axis, the line y=x, the line y=-x, or none of these. First step, simply substitute (a,b) into our equation, and then substitute the rest of the points to see if they work out to be -2.

Even or Odd Functions A function whose graph is symmetric with respect to the y-axis is an even function. A function whose graph is symmetric with respect to the x-axis is an odd function. It is possible for functions to be neither even nor odd.

Assignment Chapter 3, Section 1 pgs 133-135 #3,4,6-14,20,23,27,39, 42