Transcript Slide 1

Homework, Page 196
Determine whether the function is a power
function, given that c, g, k, and π are constants.
For those that are power functions, state the
power and constant of variation.
1. f  x    1 x 5
2
The function is a power function of power 5 and
constant of variation negative one-half.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 1
Homework, Page 196
Determine whether the function is a power
function, given that c, g, k, and π are constants.
For those that are power functions, state the
power and constant of variation.
2
5. E  m  mc
The function is a power function of power 1 and
constant of variation c2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 2
Homework, Page 196
Determine whether the function is a monomial
function, given that c, g, k, and π are constants.
For those that are power functions, state the
power and constant of variation.
1 2
9. d  gt
2
The function is a power function of power 2 and
constant of variation g/2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 3
Homework, Page 196
Determine whether the function is a monomial
function, given that l and π are constants. For
those that are monomial functions, state the
degree and leading coefficient. If not, why not.
13. y  6 x7
The function is a monomial function of degree 7
and leading coefficient –6.
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Slide 2- 4
Homework, Page 196
Write the statement as a power function
equation. Use k for the constant of variation if
one is not given.
17. The area A of an equilateral triangle varies
directly as the square of the length s of its sides.
A  ks 2
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Slide 2- 5
Homework, Page 196
Write the statement as a power function
equation. Use k for the constant of variation if
one is not given.
21. The energy E produced in a nuclear
reaction is proportional to the mass m, with the
constant of variation being c2, the square of the
speed of light.
E  mc 2
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Slide 2- 6
Homework, Page 196
Write a sentence that expresses the relationship in the
formula, using the language of variation or proportion.
25. n = c/v, where n is the refractive index of a
medium, v is the velocity of light in the medium, and c
is the constant velocity of light in free space.
The refractive index of a medium n varies directly as the
velocity of light in free space, and inversely as the
velocity of light v in the medium.
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Slide 2- 7
Homework, Page 196
State the power and constant of variation for the
function, graph it, and analyze it as in Example 2.
14
29. f  x   x
2
Power – 1/4
Constant of variation: 1/2
Domain: All nonnegative reals
Range: All nonnegative reals
Continuous
Increasing on [0, ∞)
Neither even nor odd
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Slide 2- 8
Homework, Page 196
29. Cont’d
Not symmetric
Bounded below
Local minimum at the origin
No asymptotes
14
x 
End Behavior: lim
x  2
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Slide 2- 9
Homework, Page 196
Describe how to obtain the graph of the given monomial
function from the graph of g (x) = xn with the same
power n. State whether f is even or odd. Sketch the
graph by hand and support your answer with a grapher.
33. f  x   1.5x5
To obtain f from g, vertically
stretch g by a factor of 1.5
and reflect about the y-axis.
y
x
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Slide 2- 10
Homework, Page 196
Match the equation to one of the curves labeled in the
y
figure.
a
b
2 4
c
37. f  x    x  g.
3
d
x
e
h
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g
f
Slide 2- 11
Homework, Page 196
Match the equation to one of the curves labeled in the
y
figure.
a
b
2
c
41. f  x   2x  h.
d
x
e
h
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g
f
Slide 2- 12
Homework, Page 196
State the values of the constants k and a for a function f (x) = kxa.
Describe the portion of the curve that lies in quadrant I or IV,
Determine whether f is even, odd, or undefined for x < 0.
Describe the rest of the curve, if any. Graph the function to see
whether it matches the description.
y

45. f  x   2 x .
4

3


x
k  2; a  4










3



The portion of the curve in Quadrant IV has the general shape of
a parabola. The function is even, the rest of the graph in Quadrant
III mirroring the Quadrant IV portion.
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Slide 2- 13
Homework, Page 196
Data are given for y as a power function of x. Write an equation
for the power function, and state its power and constant of
variation.
49. x
y
2
2
4
0.5
6
8
10
0.2222…. 0.125 0.08
y  kx a  2  k 2a  0.5  k 4a  0.2  k 6a  0.125  k 8a
0.08  k10a  y  8 x 2
Power : 2
Constant : 8
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Slide 2- 14
Homework, Page 196
53. Diamonds have an extremely high refraction index
of n = 2.42 on average over the range of visible light.
Determine the speed of light through a diamond.
c
n   n  2.42, c  3.00  108 m/s
v
c 3.00 108
v 
 1.240  108 m/s
n
2.42
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Slide 2- 15
Homework, Page 196
57. The data in the table show the intensity of light
from a 100-W light bulb at varying distances.
Distance (m) Intensity (W/m2)
1.0
7.95
1.5
3.53
2.0
2.01
2.5
1.27
3.0
0.90
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Slide 2- 16
Homework, Page 196
57.
a. Draw a scatter plot of the data.
b. Find the power regression model. Is the
power close the theoretical value of –2?
y  7.932 x
The power is close to the
theoretical –2.
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1.987
Slide 2- 17
Homework, Page 196
57. c. Superimpose the regression curve on the
scatter plot.
d. Use the regression model to predict the light
intensity at 1.7 and 3.4 m.
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Slide 2- 18
Homework, Page 196
Let f  x   3x
1
61.
statements is true?
a. f  0  0
c.
f 1  1
e.
f  0  is undefined.
3
. Which of the following
b.
f  1  3
d.
f  3  3
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f  x   3x
1
3
3
3
x
Slide 2- 19
2.3
Polynomial Functions of Higher
Degree with Modeling
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
What you’ll learn about





Graphs of Polynomial Functions
End Behavior of Polynomial Functions
Zeros of Polynomial Functions
Intermediate Value Theorem
Modeling
… and why
These topics are important in modeling and can be used to
provide approximations to more complicated functions, as you
will see if you study calculus.
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Slide 2- 21
The Vocabulary of Polynomials
 Each monomial in the sum : an x n  an1 x n1  ...  a0 is a
term of the polynomial.
 A polynomial function written in this way, with
terms in descending degree, is written in standard form.
 The constants an , an1 ,..., a0  are the coefficients of the
polynomial.
 The term an x n is the leading term, and a0 is the
constant term.
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Slide 2- 22
Example Graphing Transformations of
Monomial Functions
Describe how to transform the graph of an appropriate
monomial function f ( x)  an x n into the graph of
h( x)  ( x  2) 4  5. Sketch h( x) and compute the
y -intercept.
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Slide 2- 23
Cubic Functions
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Slide 2- 24
Quartic Function
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Slide 2- 25
Local Extrema and Zeros of Polynomial
Functions
A polynomial function of degree n has at most
n – 1 local extrema and at most n zeros.
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Slide 2- 26
Leading Term Test for Polynomial End
Behavior
n
For any polynomial function f ( x)  an x  ...  a1 x  a0 ,
the limits lim f ( x) and lim f ( x) are determined by the
x 
x
degree n of the polynomial and its leading coefficient an .
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Slide 2- 27
Leading Term Test for Polynomial End
Behavior
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Slide 2- 28
Example Applying Polynomial Theory
Describe the end behavior of g ( x)  2 x  3x  x  1
using limits.
4
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3
Slide 2- 29
Example Finding the Zeros of a
Polynomial Function
Find the zeros of f ( x)  2 x  4 x  6 x.
3
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2
Slide 2- 30
Multiplicity of a Zero of a Polynomial
Function
If f is a polynomial function and  x  c  is a factor of f
m
but  x  c 
m 1
is not, then c is a zero of f of multiplicity m.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 31
Zeroes of Odd and Even Multiplicity


If a polynomial function f has a real zero c of
odd multiplicity, then the graph crosses the xaxis at (c, 0) and the value of f changes sign at
x = c.
If a polynomial function f has a real zero c of
even multiplicity, then the graph does not cross
the x-axis at (c, 0) and the value of f does not
change sign at x = c.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 32
Example Sketching the Graph of a
Factored Polynomial
Sketch the graph of f ( x)  ( x  2) ( x 1) .
3
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2
Slide 2- 33
Intermediate Value Theorem
If a and b are real numbers with a < b and if f is
continuous on the interval [a,b], then f takes on
every value between f (a) and f (b). Therefore, if
y0 is between f (a) and f (b), then y0 = f (c) for
some number c in [a,b].
If f (a) and f (b) have opposite signs, then
f (c) = 0 for some number c in [a, b].
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Slide 2- 34
Another Box Problem

Squares of width x are cut from each corner of a 10-cm by 25cm sheet of cardboard. The resulting tabs are then folded up to
form a box with no top. Determine all the values of x that will
result in the box having a volume of, at most, 175 cm3.
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Slide 2- 35
Homework


Review Section 2.3
Page 209, Exercises: 1 – 65 (EOO)
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Slide 2- 36