Lesson 3.1 Graph Cubic Functions Goal Graph and analyze cubic functions.

Vocabulary Page 126 A cubic function is a nonlinear function that can be written in the standard form y = ax 3 + bx 2 + cx + d where a ≠ 0. A function f is an odd function if f (-x) =-f (x). –The graphs of odd functions are symmetric about the origin. A function f is an even function if f (-x) = f (x). –The graphs of even functions are symmetric about the y-axis.

Y – Axis Symmetry Fold the y -axis (x, y)  (-x, y) Even Function (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

Tests for Even and Odd Functions Even f(-x) = f(x) Odd f(-x) = -f(x) Both begin with f(-x)

End Behavior The end behavior of a function’s graph is the behavior of the graph as x approaches positive infinity (+∞) or negative infinity (-∞). If the degree is odd and the leading coefficient is positive: f (x) → - ∞ as x → - ∞ and f (x) → +∞ as x → +∞. If the degree is odd and the leading coefficient is negative: f (x) → + ∞ as x → - ∞ and f (x) → - ∞ as x → -∞. DownLeft Up Right Up Left Right Left Down

If the degree is odd and the leading coefficient is positive: f (x) → - ∞ as x → - ∞ and f (x) → +∞ as x → +∞. Down Left Up Right

If the degree is odd and the leading coefficient is negative: f (x) → + ∞ as x → - ∞ and f (x) → - ∞ as x → -∞. Down Left Up Right

Homework Lesson 3.1 Page 128 # 10 – 15 all Lesson 3.1 Page 129 # 5-11