(b) The power model in part (a) is approximately = 15 . Squaring both sides gives us
2
= 3 , so the model matches
Kepler’s Third Law,
2
= 3 .
35. (a) If = 60, then = 07 03 239, so you would expect to find 2 species of bats in that cave.
(b) = 4 ⇒ 4 = 07 03 ⇒
40
7
= 310 ⇒ =
40
7
103
3336, so we estimate the surface area of the cave
to be 334 m 2 .
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
30
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CHAPTER 1 FUNCTIONSAND MODELS
36. (a) Using a computing device, we obtain a power function = , where 31046 and 0308.
(b) If = 291, then = 178, so you would expect to find 18 species of reptiles and amphibians on Dominica.
37. We have =
4 2
=
4
1
2
=
(4)
2
. Thus, =
2
with =
4
.
1.3 New Functions from Old Functions
1. (a) If the graph of is shifted 3 units upward, its equation becomes = () + 3.
(b) If the graph of is shifted 3 units downward, its equation becomes = () − 3.
(c) If the graph of is shifted 3 units to the right, its equation becomes = ( − 3).
(d) If the graph of is shifted 3 units to the left, its equation becomes = ( + 3).
(e) If the graph of is reflected about the axis, its equation becomes = −().
(f) If the graph of is reflected about the axis, its equation becomes = (−).
(g) If the graph of is stretched vertically by a factor of 3, its equation becomes = 3().
(h) If the graph of is shrunk vertically by a factor of 3, its equation becomes =
1
3 ().
2. (a) To obtain the graph of = () + 8 from the graph of = (), shift the graph 8 units upward.
(b) To obtain the graph of = ( + 8) from the graph of = (), shift the graph 8 units to the left.
(c) To obtain the graph of = 8() from the graph of = (), stretch the graph vertically by a factor of 8.
(d) To obtain the graph of = (8) from the graph of = (), shrink the graph horizontally by a factor of 8.
(e) To obtain the graph of = −() − 1 from the graph of = (), first reflect the graph about the axis, and then shift it
1 unit downward.
(f) To obtain the graph of = 8( 1
8 ) from the graph of = (), stretch the graph horizontally and vertically by a factor
of 8.
3. (a) Graph 3: The graph of is shifted 4 units to the right and has equation = ( − 4).
(b) Graph 1: The graph of is shifted 3 units upward and has equation = () + 3.
(c) Graph 4: The graph of is shrunk vertically by a factor of 3 and has equation =
1
3 ().
(d) Graph 5: The graph of is shifted 4 units to the left and reflected about the axis. Its equation is = −( + 4).
(e) Graph 2: The graph of is shifted 6 units to the left and stretched vertically by a factor of 2. Its equation is
= 2( + 6).
4. (a) = () − 3: Shift the graph of 3 units down. (b) = ( + 1): Shift the graph of 1 unit to the left.
c ° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
SECTION 1.3 NEW FUNCTIONSFROM OLD FUNCTIONS
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31
(c) =
1
2 (): Shrink the graph of vertically by a
factor of 2.
(d) = −(): Reflect the graph of about the axis.
5. (a) To graph = (2) we shrink the graph of
horizontally by a factor of 2.
The point (4−1) on the graph of corresponds to the
point
1
2
· 4−1 = (2−1).
(b) To graph = 1
2
we stretch the graph of
horizontally by a factor of 2.
The point (4−1) on the graph of corresponds to the
point (2 · 4−1) = (8−1).
(c) To graph = (−) we reflect the graph of about
the axis.
The point (4−1) on the graph of corresponds to the
point (−1 · 4−1) = (−4−1).
(d) To graph = −(−) we reflect the graph of about