Lesson 3.1 Graph Cubic Functions

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

2. Vocabulary Page 126 • A cubic function is a nonlinear function that can be written in the standard form y =ax3 + bx2 + 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.

3. Even Function Y – Axis SymmetryFold the y-axis (x, y)  (-x, y) (x, y)  (-x, y)

4. 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

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

6. 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

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

8. 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 → -∞ . Right Left Down Left Right Up Up Left Down Right

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

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

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