Transcript No Slide Title

4.2
Polynomial Functions
and Models
♦ Understand the graphs of polynomial functions.
♦ Evaluate and graph piecewise-defined functions
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Graphs of Polynomial Functions
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Constant Polynomial Function
Has no x-intercepts or turning points
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Linear Polynomial Function
Degree 1 and one x-intercept and no turning
points.
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Quadratic Polynomial Functions
Degree 2, parabola that opens up or down. Can
have zero, one or two x-intercepts. Has exactly
one turning point, which is also the vertex.
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Cubic Polynomial Functions
Degree 3, can have zero or two turning points.
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Quartic Polynomial Functions
Degree 4, can have up to four x-intercepts and
three turning points.
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Quintic Polynomial Functions
Degree 5, may have up to five x-intercepts and
four turning points.
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Degree, x-intercepts, and turning points
The graph of a polynomial function of degree n  1 has
at most n x-intercepts and at most n  1 turning points.
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Example
Use the graph of the polynomial
function shown.
a) How many turning points and
x-intercepts are there?
b) Is the leading coefficient a
positive or negative? Is the
degree odd or even?
c) Determine the minimum
degree of f.
Solution
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Group Work
Graph f(x) = 2x3  5x2  5x + 7, and then complete the
following.
a) Identify the x-intercepts.
b) Approximate the coordinates of any turning
points to the nearest hundredth.
c) Use the turning points to approximate any local
extrema.
Solution
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Example
Let f(x) = 3x4 + 5x3  2x2.
a) Give the degree and leading coefficient.
b) State the end behavior of the graph of f.
Solution
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Piecewise-Defined Polynomial Functions
Example Evaluate f(x) at 6, 0, and 4.
5 x
 3
f ( x)  x  1
3  x 2

Solution
if x  5
if  4  x  2
if x  2
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Example
Complete the following.
a) Sketch the graph of f.
b) Determine if f is continuous on its domain.
c) Evaluate f(1).
4

f ( x) 4  x 2
2 x  6

if  4  x  0
if 0  x  2
if 2  x  4
Solution
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