Power functions and polynomials
Math 102 Section 106
Cole Zmurchok
September 9, 2016
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From Figure 1.1a, we see that the power functions (y = xn for po
intersect at x = 0 and x = 1. This is true for all integer powers. The
demonstrates another extremely important fact: the greater the power
graph near the origin and the steeper the graph beyond x > 1. This can b
Small
powers
dominate
close
to xWe
= say
0; that
large
of the
relative size
of the power
functions.
close to the or
with lower
powers dominate,
whilex.
far from the origin, the higher power
powers
dominate
for large
Last time: asymptotic thinking
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y
y
x5
x4
x3
x2
Even and odd functions
Definitions
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An even function f (x) is symmetric about the
y axis:
f (x) = f (−x).
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An odd function f (x) is symmetric about the
origin:
f (x) = −f (−x).
Even and odd functions
Example
The function y = f (x) = C for C constant is an
even function.
I Analytic argument: Since f (x) = C, we know
that f (−x) = C. This implies that
f (x) = f (−x), from which we conclude that
f (x) is an even function.
I Geometric argument: a constant function is
symmetric about the y-axis.
Even and odd functions
Q1.
A.
B.
C.
D.
E.
The product of two odd functions is
an odd function
an even function
both even and odd
neither even nor odd
not enough information to tell
Even and odd functions
2
x
Q2. The function f (x) = 1+x
2 (the quotient of two
polynomials is called a rational function) is
A. an odd function
B. an even function
C. both even and odd
D. neither even nor odd
E. not enough information to tell
Even and odd functions
Q3.
A.
B.
C.
D.
E.
3
x
The function g(x) = 1+x
3 is
an odd function
an even function
both even and odd
neither even nor odd
not enough information to tell
Even and odd functions
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If f (x) and g(x) are both even functions then
f + g, f − g, f g, and f /g are all even
functions.
If f (x) and g(x) are both odd functions then
f + g and f − g are odd functions, but f g and
f /g are even functions.
For you to think about: Why is this true?
Power functions and curve sketching
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Goal: to be able to easily sketch the graph of a
simple polynomial function
f (x) = axn + bxm .
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A polynomial is a sum of power functions
multiplied by constants.
Key idea:
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Lower powers dominate near x = 0.
Higher powers dominate for x far from 0.
Power functions and curve sketching
Example
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y = x3 + ax is in pre-lecture video and the
course notes:
y = 2x2 − x3 is in the video
Sketch a graph of the polynomial y = x5 + ax3 .
Power functions and curve sketching
Example (Sketch y = x5 + ax3.)
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Near x = 0, y ≈ ax3 . a < 0, a = 0, a > 0:
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Far from x = 0, y ≈ x5 :
Power functions and curve sketching
Example (Sketch y = x5 + ax3.)
a < 0:
a = 0:
a > 0:
Power functions and curve sketching
Example (Sketch y = x5 + ax3.)
a < 0:
a = 0:
a > 0:
Power functions and curve sketching
Q4. Suppose a = −4, with y = x5 + ax3 . The
zeros of y = x5 − 4x3 are:
A. x = 2
B. x = ±4
C. x = ±2
D. x = 0, ±2
√
E. x = 0, ± 2
Power functions and curve sketching
Q5. Which of the functions below has this graph?
A.
B.
C.
D.
E.
x3 − x5
x5 − x3
x4 + x2
x4 − x2
x2 − x4
Homework Tip: Use a spreadsheet
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Average rate of change calculations from data.
Today...
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Even functions vs. odd functions
Polynomials: f (x) = axn + bxm
n
+bxm
Rational functions: g(x) = ax
`
cx +dxk
Easily sketching the graph of simple
polynomials:
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Large powers away from x = 0
Small powers near x = 0
Check the last slides for related exam problems.
Answers
1.
2.
3.
4.
5.
B
B
D
D
E
Related Exam Questions
1. When x = 1000, the function
4
2
+64x−87
g(x) = 6x +12x
is closet to
2x3 −6x2 +x
A.
B.
C.
D.
E.
0.003
3000
1000000
6
3
2. Which is the following graphs is Sketch the
graph of f (x) = 8x2 − x5 .