2.3 Notes.notebook
September 23, 2016
2.3 Polynomial Functions of
Higher Degree with Modeling
Name: ________________
Objective: Students will be able to graph polynomial functions,
predict their end behavior, and find their real zeros using a grapher
or an algebraic method.
In this section, we extend our study to cubic (third degree) and
quartic (fourth degree) polynomials.
Examples Describe how to transform the graph of an appropriate monomial
function into the graph of the given function. Sketch.
2.) h(x) = -(x-2)4+5
1.) g(x) = 4(x + 1)3
Sep 16­9:34 AM
Examples Graph the polys, locate their extrema and zeros, and
explain how it is related to the monomials from which they are built.
1.) f(x) = x3 + x
2.) g(x) = x3 - x
Sep 16­9:45 AM
1
2.3 Notes.notebook
September 23, 2016
Let's explore the groups of graphs below on our calculators.
Group 1:
y1 = x3 + 5x - 2
y2 = 2x3 - x2 + x + 2
y3 = 10x5 + 2x - 4
[-2,2] by [-10,10]
Group 3:
y1 = x4 + 5x - 2
y2 = 2x4 - x2 + x + 2
y3 = 10x6 + 2x - 4
[-5,5] by [-10,10]
Group 2:
y1 = -x3 + 5x - 2
y2 = -2x3 - x2 + x + 2
y3 = -10x5 + 2x - 4
[-5,5] by [-10,10]
Group 4:
y1 = -x4 + 5x - 2
y2 = -2x4 - x2 + x + 2
y3 = -10x6 + 2x - 4
[-5,5] by [-10,5]
Sep 16­9:49 AM
Examples Match the polynomials with their graphs. Don't use a
calculator.
A.) f(x)=x4-3x³+2x²-x+2
C.) h(x)=-x³+2x²-x+2
B.) g(x)=-x5-4x4+3x³+2x²-x+2
D.) k(x) = -x4-3x³+2x²-x+2
3
1
2
4
Sep 16­10:04 AM
2
2.3 Notes.notebook
September 23, 2016
Examples Describe the end behavior using limit notation.
1.) f(x) = -3x4 - 5x3 + 15x2 - 5x + 19
2.) g(x) = (2x-3)(4-x)(x+1)
Example: Find the zeros of the polynomial algebraically.
3.) f(x) = 5x3 - 5x2 - 10x
Sep 16­10:14 AM
Zeros of Polynomial Functions
If (x - c)m is a factor of a polynomial f(x), then c is a zero of f(x)
with multiplicity m.
If m is odd: The graph crosses the x-axis at x = c.
If m is even: The graph touches the x-axis at x = c.
Example State the degree, list the zeros and multiplicities of each.
Graph by hand.
1.) f(x) = (1/2)(x + 2)3(x - 1)2
Sep 16­10:30 AM
3
2.3 Notes.notebook
September 23, 2016
1.) Find a cubic function with the given zeros.
5, -√5 , √5
2.) Use cubic regression to fit a curve through the four points
given in the table.
x
-2
1
4
7
y
2
5
9
26
Sep 16­10:39 AM
Intermediate Value Theorem
Let a and b be real numbers with a < b and let f be continuous on
[a,b]. If f(a) and f(b) have opposite signs, then f(c) = 0 for some
c in [a,b].
Does the IVT guarantee that f(x) = x3 + 2x + 1 has a zero between
x = -1 and x = 0? between x = 2 and x = 3?
Assignment: Pages 209-211: #3, 9-12, 17, 23, 27, 37, 39, 41, 55, 59
Sep 16­10:42 AM
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