Section 2.2: Power Functions with Modeling
**Power Functions & Variation
DEFINITION P o w e r F unction
Any function that can be written in the fonn
f(x)
=
k • x a, where k and a are nonzero constants,
is a power flmction. The constant a is the power, and k is the constant of
variation, or constant of proportion. We say I(x) varies as the a th power of x, or
f(x) is proportional to the a th power of x.
~ rower
[ex)
constant
In Exercises 1-10, determine whether the function is a power function,
given that c, g, k, and 'Tr represent constants. For those that are power
functions, state the power and constant of variation.
@ f(X)
= _ !x 5
2. f(x)
2
(j)r(x) = 3·2 x
\. fowe r ~
= 9x5/3
t)
I
Con~1ont·,
.. i
4.f(x ) =13
Noi u power
fuo(tlon x In tx p.
@ E(m) = me2
_ 1 '2
7. d - '2 gt
5 . Gon~l·Qn1 ·. c
'f ower : \
1
Kd.,~
k
. @I = d
q. ~onstant : K fOWex - -1 Direct Variation:
2
"0
6. KE(v) =
8. V =
4
1 kv5
2
3 '1T? 10. F(a) =
In • a
Example 1: Write the statement as a power function equation . Use k for the constant of variation if one
is not given.
a) The area A of an equilateral triangle va ries direct lY j§ the square of the length s of its
sides. F- K-Q~
~;s
~
A~ Ks?
ns
b) The volume V of a circular cylinder with fixed height[[g(lli)portional to the square of
4VQYles dlrec-tl'j
radius r.
c)
The current I in an electrical circuit is inversely proportional to the resistance R, with
constant of variation V.
Example 2: Graph Analysis: State the power and constant of variation for the function, graph it, and
analyze it.
a) [(x)
=
2x- 3
Constant:
1.
Constant:
Power:
Power: - 3
0 : fK X1\ (.1
R:
rR
D: 4
t
\R
R: () po?)
X,-O
Increasingi Decreasing:) ( .·<Y ,o ) u (o , ex;; )
Continuous? '~eS Increasing/Decreasing: (c>,()C)
Even/Odd? 01)D
Even/Odd?
Continuous?
Bounded?
Extrema?
1\\0 .
Bounded? ~\o~
NO
Extrema? LOj 0 ) NONE
Asymptotes? ~ -=End Behavior. 0
E.\lE.t-J 0
J
X'" 0
Asymptotes? (N ONE:.) End Behavior. ~
I
(-'cY'/ 0 ) **Monomial Functions & Their Graphs
A single-term polynomial function is a power function that is also called a monomial function.
In Exercises 11-16, determine whether the function is a monomial
function, given that I and 'TT represent constants. For those that are
monomial functions state the degree and leading coefficient. For those
that are not, explain why not.
@ f(X)
=
~oeff. ~
12. f(x) = 3x- 5
-4
@ Y= -6x
@ S= 4rrr2
7
de~ree
= -2' 5 x
A =M
14. Y
16.
1'3
·4
.;: 0
--1
d~9ree -: 1
~oeff. ~
**Graphs of Power Functions
55. Keeping Warm For mammals and other warm-blooded animals
to stay warm requires quite a bit of energy. Temperature loss is
related to surface area, which is related to body \'/eight, and tem­
perature gain is related to circulation, which is related to pulse
rate . In the final analysis, scientists have concluded that the pulse
rate r of mammals is a power function of their body weight w.
(a) Draw a scatter plot of the data in Table 2.12.
i'5 , coeff.
d~Bree
~
40
-; 2
.,
(b) Find the power regression moclel.
(c) Superimpose the regression curve on the scatter plot.
(d) Use the regression model to predict the pulse rate for a 450-kg
horse . Is the result close to the 38 beats/min repolted by
A. J. Clark in 1927?
: Table 2.12 Weight and Pulse Rate of
Selected Manunals
Mammal
~
Rat
Guinea pig
Rabbit
Small dog
Large dog
Sheep
Human
Body \""eight (kg)
0.2
0.3
2
5
30
50
70
Pulse rate
(beats/min)
420
300
205
120
dJ ij,23i . 2 (450-·1Ql)
~ 3'1 ~, Ut,cits 'm; r)
85
70
72
Homework: 1,4,7,10,13,16,17,19,24,27,32,35,38.42.47.50,52,55, 58, 62, 63