Today in Pre-Calculus• Go over homework• Notes: (need calculator & book)

– Graphs of Polynomial Functions•End Behavior•Zeros

• Homework

End Behavior of Polynomial Functions

Page 203 Exploration 1

If degree of polynomial is odd

If degree of polynomial is even

If leading coefficient is negative: graph reflects over x-axis so the end behavior changes signs.

lim ( ) lim ( )x x

f x and f x

lim ( ) lim ( )x x

f x and f x

End Behavior of Polynomial Functions

Examples: Describe the end behavior of the following functions without graphing them.

1) f(x) = x3 + 2x2 – 11x – 12

2) g(x) =–2x4 + 2x3 – 22x2 – 18x + 35

lim ( ) lim ( )x x

f x and f x

lim ( ) lim ( )x x

f x and f x

Finding Zeros of Polynomial Functions

Example: Find the zeros of f(x) = 5x3 – 5x2 – 30x

5x3 – 5x2 – 30x = 0 set equal to zero

5x(x2 – x – 6) = 0 factor GCF

5x(x – 3)(x + 2) = 0 factor

5x = 0 x – 3 = 0 x + 2 = 0 set EVERY term = 0

x = 0, 3, -2 solve for x

Multiplicity of a ZeroIf f is a polynomial function and (x – c)m is a factor of f, then c is a zero of multiplicity m. (c is a repeated zero).

Example: f(x) = (x – 2)3(x+1)2

If the multiplicity is odd, then the graph crosses the x-axis at (c,0) and the value of f changes sign at x = c

If the multiplicity is even, then the graph touches (but does not cross) the x-axis at (c,0) and the value of f does NOT change sign at x = c

Homework

• Pg. 209: 9-12, 26-28, 34, 36, 39-41

• Bring books tomorrow

• Quiz: Friday, October 24