2.8 Analyze Graphs of Polynomial Functions

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2.8 Analyze Graphs of Polynomial Functions
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Example 1
Graph the function. Identify the x-intercepts
(zeors), the points where the local maximums and
local minimums occur, and the turning points of
the function.
h (x) 0.5x3 x2 x 2
x-intercept 3.074local minimum (0.387,
1.792)local maximum (1.721, 4.134) 2 turning
points
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Example 2
Graph the function. Identify the x-intercepts
(zeros), the points where the local maximums and
local minimums occur, and the turning points of
the function.
 
x-intercepts 1, 4local minimum (3.25,
17.056)local maximum none 2 turning points
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Turning Points of a Polynomial Function The graph
of every polynomial function of degree n has at
most n-1 turning points. If a polynomial
function has n distinct real zeros, then its
graph has exactly n-1 turning points.
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1. Multiply (x 2)(3x 1)
ANSWER
3x2 7x 2
2. Find the intercepts of y (x 6)(x 5)
6 and 5
ANSWER
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3. An object is projected vertically upward. Its
distance D in feet above the ground after t
seconds is given by D 16t2 144t 100.
Find its maximum distance above the ground.
424 ft
ANSWER
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EXAMPLE 1
Use x-intercepts to graph a polynomial function
SOLUTION
STEP 1
Plot the intercepts. Because 3 and 2 are
zeros of f, plot (3, 0) and (2, 0).
STEP 2
Plot points between and beyond the x-intercepts.
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EXAMPLE 1
Use x-intercepts to graph a polynomial function
STEP 3
STEP 4
Draw the graph so that it passes through the
plotted points and has the appropriate end
behavior.
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EXAMPLE 2
Find turning points
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EXAMPLE 2
Find turning points
SOLUTION
a. f (x) x3 3x2 6
a. Use a graphing calculator to graph the
function.
Notice that the graph of f has one x-intercept
and two turning points.
You can use the graphing calculators zero,
maximum, and minimum features to approximate the
coordinates of the points.
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EXAMPLE 2
Find turning points
SOLUTION
b. g (x) 5 x4 6x3 3x2 10x 3
a. Use a graphing calculator to graph the
function.
Notice that the graph of g has four x-intercepts
and three turning points.
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EXAMPLE 3
Maximize a polynomial model
How long should you make the cuts?
What is the maximum volume?
What will the dimensions of the finished box be?
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EXAMPLE 3
Maximize a polynomial model
SOLUTION
Write a verbal model for the volume. Then write a
function.
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EXAMPLE 3
Maximize a polynomial model
(320 72x 4x2)x
Multiply binomials.
4x3 72x2 320x
Write in standard form.
To find the maximum volume, graph the
volume function on a graphing calculator.
Consider only the interval 0 lt x lt 8 because this
describes the physical restrictions on the size
of the flaps.
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EXAMPLE 3
Maximize a polynomial model
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for Examples 1, 2 and 3
GUIDED PRACTICE
Graph the function. Identify the x-intercepts and
the points where the local maximums and local
minimums occur.
4. f (x) x4 3x3 x2 4x 5
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for Examples 1, 2 and 3
GUIDED PRACTICE
5. WHAT IF? In Example 3, how do the answers
change if the piece of cardboard is 10
inches by 15 inches?
(150 50x 4x2)x
Multiply binomials.
4x3 50x2 150x
Write in standard form.
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Daily Homework Quiz

2. You are making a rectangular box out of a 22-
inch by 30-inch piece of cardboard, as shown in
the diagram. You want the box to have the
greatest possible volume. How long should you
make the cuts? What is the maximum volume?