roots and end behavior of polynomials
Module 5 : Investigation 5
MAT 170 | Precalculus
October 21, 2016
questions 1 & 3
(1) The function f is defined by f(x) = (x + 1)(x − 3)2 .
(a) Use algebraic methods to find the roots and x-intercepts of f.
(b) What do the roots of a polynomial function represent ?
(c) Explain why the zeros of a polynomial function occur where
each factor is equal to zero.
(d) Draw a number line and indicate where the roots are, where
the output of f is positive, where the outputs of f are negative.
(e) Construct a rough graph of f.
(3) How do the functions f and k(x) = −(x + 1)(x − 3)2 compare ?
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end behavior
Definition
The end behavior of a function describes the behavior of the
function outputs when the input values increase or decrease
without bound.
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end behavior
For example, let f(x) = −x3 + 5x2 − 3x − 9.
x
f(x)
1
−8
10
−539
100
−950309
10000
−999500030009
1000000
−9.99995 × 1017
As x increases without bound (x → ∞) the outputs of f decrease
without bound (f(x) → −∞).
x
f(x)
-1
0
-10
1521
-100
1050291
-10000
1.0005 × 1012
-1000000
1.000005 × 1018
As x decreases without bound (x → −∞) the outputs of f increase
without bound (f(x) → ∞).
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end behavior
We can see the end behavior of f(x) = −x3 + 5x2 − 3x − 9 by
considering its graph.
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question 4
A function of the form f(x) = axn for real numbers a and n is called a
power function.
(a) Let f(x) = x3 , g(x) = x5 , h(x) = x2 , and j(x) = x8
As x increases without bound
(x → ∞) determine :
As x decreases without bound
(x → −∞) determine :
f(x) →
f(x) →
g(x) →
g(x) →
h(x) →
j(x) →
h(x) →
j(x) →
(b) What general statements can you make about how the exponent
on a power function impacts the behavior of the function ?
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question 7
Answer the following questions given this graph of the function g.
(a) What are the
roots of g ?
(c) & (d) On what
interval(s) is the
graph of g concave
up and concave
down ?
(e) & (f) Describe the end behavior of g.
(g) Does the function have odd or even degree ?
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