Chapter 2
Functions and
Graphs
Section 4
Polynomial and
Rational Functions
Learning Objectives for Section 2.4
Polynomial and Rational Functions
 The student will be able to graph and identify properties of
polynomial functions.
 The student will be able to calculate polynomial regression
using a calculator.
 The student will be able to graph and identify properties of
rational functions.
 The student will be able to solve applications of
polynomial and rational functions.
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Polynomial Functions
A polynomial function is a function that can be written in
the form
an x n  an 1 x n 1    a1 x  a0
for n a nonnegative integer, called the degree of the
polynomial. The domain of a polynomial function is the
set of all real numbers.
A polynomial of degree 0 is a constant. A polynomial of
degree 1 is a linear function. A polynomial of degree 2 is a
quadratic function.
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Odd Polynomials
 A polynomial is called odd if it only contains odd powers of x.
 The graphs of all odd functions are symmetric with respect to
the origin. (Test: If you can rotate the graph around the origin
and get the same picture in less than 360, then it’s odd)
Odd degree, but NOT an
odd polynomial.
𝑦 = 𝑥3
𝑦 = −0.2𝑥 5 + 4𝑥
Barnett/Ziegler/Byleen Business Calculus 12e
𝑦 = 𝑥 3 + 2𝑥 2
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Even Polynomials
 A polynomial is called even if it only contains even
powers of x.
 The graphs of all even functions are symmetric with
respect to the y-axis.
Even degree, but NOT an
even polynomial.
𝑦 = 𝑥2
𝑦 = 𝑥4
𝑦 = −𝑥 4 + 3𝑥 2
Barnett/Ziegler/Byleen Business Calculus 12e
𝑦 = 𝑥 2 − 4𝑥 + 3
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Shapes of Polynomials
As we look at the shapes of some polynomials of various
degrees, observe some of the following properties:
• Number of x-intercepts
• Number of local maxima/minima
• End behavior
o What happens as x approaches +∞ or -∞?
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Graphs of Polynomials
f (x)  x  2
•
•
•
•
Degree = 1
One x-intercept
No maxima/minima
End behavior:
• As x  + , y  + 
• As x  - , y  - 
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Graphs of Polynomials
g(x)  x  2x
3
•
•
•
•
Degree = 3
Three x-intercepts
1 local max, 1 local min.
End behavior:
• As x  + , y  + 
• As x  - , y  - 
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Graphs of Polynomials
h(x)  x  5x  4x  1
5
•
•
•
•
3
Degree = 5
Five x-intercepts
2 local max, 2 local min.
End behavior:
• As x  + , y  + 
• As x  - , y  - 
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Observations
Odd Degree Polynomials
 For an odd degree polynomial:
• The graph starts negative, ends positive, or vice versa,
depending on whether the leading coefficient is positive or
negative
• Either way, a polynomial of degree n crosses the x axis at
least once, at most n times.
𝑦 = −𝑥 3 + 4𝑥 2
f (x)  x  2
g(x)  x 3  2x
Barnett/Ziegler/Byleen Business Calculus 12e
h(x)  x 5  5x 3  4x  1
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Graphs of Polynomials
F(x)  x  2x  2
2
•
•
•
•
Degree = 2
Zero x-intercepts
1 absolute min.
End behavior:
• As x  + , y  + 
• As x  - , y  + 
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Graphs of Polynomials
G(x)  2x 4  4x 2  x  1
• Degree = 4
• Two x-intercepts
• 1 local max, 1 local min., 1
absolute min.
• End behavior:
• As x  + , y  + 
• As x  - , y  + 
Barnett/Ziegler/Byleen Business Calculus 12e
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Graphs of Polynomials
H (x)  x  7x  14x  x  5
6
4
2
• Degree = 6
• Four x-intercepts
• 2 local max, 2 local min., 1
absolute min.
• End behavior:
• As x  + , y  + 
• As x  - , y  + 
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Observations
Even Degree Polynomials
 For an even degree polynomial:
• the graph starts negative, ends negative, or starts and
ends positive, depending on whether the leading
coefficient is positive or negative
• either way, a polynomial of degree n crosses the x axis
at most n times. It may or may not cross at all.
𝑦 = −𝑥 2 + 2𝑥 − 2
G(x)  2x 4  4x 2  x  1
Barnett/Ziegler/Byleen Business Calculus 12e
H (x)  x 6  7x 4  14x 2  x  5
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Characteristics of Polynomials
 Graphs are continuous. Their graphs can be sketched without
lifting up your pencil.
 Graphs have no sharp corners.
 Graphs usually have turning points, which is a point that
separates an increasing portion of the graph from a decreasing
portion.
 A polynomial of degree n can have at most n linear factors.
Therefore, the graph of a polynomial function of positive
degree n will have at most n roots/zeros. The real roots are the
x-intercepts of the graph. The imaginary roots come in pairs
and are not visible on its graph.
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Example 1
 State the degree of the polynomial, determine the y-intercept,
determine the x-intercepts.
𝑓(𝑥) = 𝑥 3 − 2𝑥 2 − 24𝑥
𝐴) 𝐷𝑒𝑔𝑟𝑒𝑒 = 3
𝐵) 𝑓 0 = 0
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 0
𝐶) 0 = 𝑥(𝑥 2 − 2𝑥 − 24)
0 = 𝑥(𝑥 + 4)(𝑥 − 6)
𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 𝑎𝑟𝑒: 0, −4, & 6
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Example 2
 State the degree of the polynomial, determine the y-intercept,
determine the x-intercepts.
𝑓 𝑥 = (2 − 4𝑥)(𝑥 + 5)(3𝑥 + 9)(𝑥 − 1)
𝐴) 𝐷𝑒𝑔𝑟𝑒𝑒 = 4
𝐵) 𝑓 0 = 2 ∙ 5 ∙ 9 ∙ −1
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = −90
𝐶) 0 = (2 − 4𝑥)(𝑥 + 5)(3𝑥 + 9)(𝑥 − 1)
1
𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 𝑎𝑟𝑒: , −5, −3, & 1
2
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 3
 State the degree of the polynomial, determine the y-intercept,
determine the x-intercepts.
𝑓 𝑥 = 𝑥2 + 9
𝐴) 𝐷𝑒𝑔𝑟𝑒𝑒 = 2
𝐵) 𝑓 0 = 9
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 9
𝐶) 0 = 𝑥 2 + 9
𝑥 2 = −9
𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠: 𝑁𝑜𝑛𝑒;
𝑟𝑜𝑜𝑡𝑠 𝑎𝑟𝑒 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦
𝑥 = ±3𝑖
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 4
 Write the lowest-degree polynomial that has x-intercepts
2, -3, and -1.
𝑓 𝑥 = (𝑥 − 2)(𝑥 + 3)(𝑥 + 1)
𝑓 𝑥 = (𝑥 2 + 𝑥 − 6)(𝑥 + 1)
𝑓 𝑥 = 𝑥 3 + 𝑥 2 − 6𝑥 + 𝑥 2 + 𝑥 − 6
𝑓 𝑥 = 𝑥 3 + 2𝑥 2 − 5𝑥 − 6
Barnett/Ziegler/Byleen Business Calculus 12e
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Determining the Minimum Degree
of a Polynomial
 A polynomial of degree n will have, at most, n – 1
“bumps”.
g(x)  x 3  2x
f (x)  x  2
𝑦 = 𝑥4
h(x)  x 5  5x 3  4x  1
4
2
𝑦 = −𝑥 2 + 2𝑥 − 2 G(x)  2x  4x  x  1
H (x)  x 6  7x 4  14x 2  x  5
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Example 5
 What is the minimum degree of a polynomial that could
have these graphs? Is the leading coefficient positive or
negative?
Odd behavior, 4 𝑏𝑢𝑚𝑝𝑠
→ 𝑀𝑖𝑛. 𝑑𝑒𝑔𝑟𝑒𝑒 5
and leading coeff. is positive
Even behavior, 3 𝑏𝑢𝑚𝑝𝑠
→ 𝑀𝑖𝑛. 𝑑𝑒𝑔𝑟𝑒𝑒 4
and leading coeff. is negative
Barnett/Ziegler/Byleen Business Calculus 12e
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Cubic Regression
Lake Trout
Length (in.)
Weight (oz.)
10
5
14
12
18
26
22
56
26
96
30
152
34
226
38
326
44
536
Using cubic regression,
predict the weight of a lake
trout that is 46 inches long.
Barnett/Ziegler/Byleen Business Calculus 12e
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Cubic Regression
Lake Trout
Length (in.)
Weight (oz.)
10
5
14
12
18
26
22
56
26
96
30
152
34
226
38
326
44
536
Barnett/Ziegler/Byleen Business Calculus 12e
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Question
 Estimate the weight of a lake trout that is 46 inches long.
• Select TRACE
• Press Up Arrow to select the graph of y1
• Enter 46
• f(46) = 619.6 oz.
 The weight of a lake trout that is 46 inches long is about
619.6 oz.
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