Gifted Precalculus

1
Gifted Precalculus

• Section 2.2 Polynomial Functions of Higher Degree

2
Compare the graphs of these functions.

• f(x) x3
• g(x) -x3
• h(x) x4
• j(x) -x4
• k(x) x5
• m(x) -x5
• n(x) x6
• p(x) -x6

3
Compare the graphs of these functions.

• f(x) x3 2x2 x 2
• g(x) -x3 4x2 3x 1
• h(x) x4 3x3 - 4x2 5x 2
• j(x) -x4 - x3 2x2 3x 2

4
Leading Coefficient Test

• See page 149

5
Use the Leading Coefficient Test to determine the
right-hand and left-hand behavior of the graph of
the polynomial function.

• f(x) (1/3)x3 5x
• g(x) 1 x6

6
Sketch the graphs using transformations

• f(x) (1/2)(x 2)5 -3
• g(x) -3x4 2
• h(x) -2(x 3)3

7
Find all the real zeros and relative extrema of
the function. Graph the function.

• g(t) t5 6t3 9t
• f(x) x4 4x2

8
Zeros of Polynomial Functions

• For a polynomial function f of degree n, the
  following statements are true.
• The graph of f has at most n real zeros.
• The function f has at most n 1 relative
  extrema.

9
Real Zeros of Polynomial Functions

• If f is a polynomial function and a is a real
  number, the following statements are equivalent.
• x a is a zero of the function f.
• x a is a solution of the polynomial function
  f(x) 0.
• (x a) is a factor of the polynomial function
  f(x)
• (a,0) is an x-intercept of the graph of f

10
Multiplicity

• A factor (x a)k of a polynomial function has a
  multiplicity of k.
• If k is odd, the graph crosses the x-axis at a.
• If k is even, the graph touches the x-axis at a.

11
Find a polynomial function that has the given
zeros.

• -4, 5
• ,

12
Use a graphing utility to graph. Identify any
symmetry with respect to the x-axis, y-axis, or
origin. Determine the x-intercepts.