WARM UP Zeros: Domain: Range: Relative Maximum: Relative Minimum: Intervals of Increase: Intervals of Decrease:

Symmetry Essential Question: How do you determine the shape and symmetry of the graph by the polynomial equation?

Even, Odd, or Neither Functions Not to be confused with End behavior To determine End Behavior, we check to see if the leading degree is even or odd With Functions, we are determining symmetry (if the entire function is even, odd, or neither)

Even and Odd Functions (algebraically) A function is even if f(-x) = f(x) If you plug in x and -x and get the same solution, then it’s even. Also: It is symmetrical over the y-axis. The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then it’s Odd. If you get different y’s, then it’s Neither. A function is odd if f(-x) = -f(x) If you plug in x and -x and get opposite solutions, then it’s odd. Also: It is symmetrical over the origin

Y – Axis Symmetry Fold the y-axis Even Function (x, y)  (-x, y) -5 1 -4 2 -1 3 4 4 11 -1 -4 -2 -3 (x, y)  (-x, y)

Test for an Even Function A function y = f(x) is even if , for each x in the domain of f. f(-x) = f(x) Symmetry with respect to the y-axis

Symmetry with respect to the origin (x, y)  (-x, -y) (2, 2)  (-2, -2) (1, -2)  (-1, 2) Odd Function

Test for an Odd Function A function y = f(x) is odd if , for each x in the domain of f. f(-x) = -f(x) Symmetry with respect to the Origin

Ex. 1 Even, Odd or Neither? Graphically Algebraically EVEN

Ex. 2 Even, Odd or Neither? Graphically Algebraically ODD

Ex. 3 Even, Odd or Neither? Graphically Algebraically EVEN

Ex. 4 Even, Odd or Neither? Graphically Algebraically Neither

Even, Odd or Neither? EVEN ODD

Even functions are symmetric about the y-axis What do you notice about the graphs of even functions? Even functions are symmetric about the y-axis

Odd functions are symmetric about the origin What do you notice about the graphs of odd functions? Odd functions are symmetric about the origin

EVEN

ODD

Neither

Neither

EVEN

ODD

Neither

EVEN