Multivariable Calculus 9th Edition by James Stewart solution manual

the ­axis, then about the ­axis.
The point (4−1) on the graph of  corresponds to the
point (−1 · 4−1 · −1) = (−41).
6. The graph of  = () =
√ 3 −  2
has been shifted 2 units to the right and stretched vertically by a factor of 2.
Thus, a function describing the graph is
 = 2( − 2) = 2
 3( − 2) − ( − 2) 2
= 2
 3 − 6 − ( 2
− 4 + 4) = 2
√ − 2
+ 7 − 10
7. The graph of  = () =
√ 3 −  2
has been shifted 4 units to the left, reflected about the ­axis, and shifted downward
1 unit. Thus, a function describing the graph is
 = −1 ·
  
reflect
about ­axis
 ( + 4)
  
shift
4 units left
− 1
  
shift
1 unit left
This function can be written as
 = −( + 4) − 1 = −  3( + 4) − ( + 4) 2 − 1
= −  3 + 12 − ( 2 + 8 + 16) − 1 = − √ − 2 − 5 − 4 − 1
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CHAPTER 1 FUNCTIONSAND MODELS
8. (a) The graph of  = 1 +
√  can be obtained from the graph of  = √  by
shifting it upward 1 unit.
(b) The graph of  = sin can be obtained from the graph of  = sin by compressing horizontally by a factor of , giving
a period of 2 = 2. The graph of  = 5sin is then obtained by stretching vertically by a factor of 5.
9.  = 1 +  2 . Start with the graph of  =  2 and shift 1 unit upward
10.  = ( + 1) 2 . Start with the graph of  =  2 and shift 1 unit to the left.
11.  = | + 2|. Start with the graph of  = || and shift 2 units to the left.
12.  = 1 −  3 . Start with the graph of  =  3 , reflect about the ­axis, and then shift 1 unit upward.
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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33
13.  =
1

+ 2. Start with the graph of  =
1

and shift 2 units upward.
14.  = −
√
 − 1. Start with the graph of  =
√
, reflect about the ­axis, and then shift 1 unit downward.
15.  = sin4. Start with the graph of  = sin and compress horizontally by a factor of 4. The period becomes 24 = 2.
16.  = 1 +
1
 2
. Start with the graph of  =
1
 2
and shift 1 unit upward.
17.  = 2 +

 + 1. Start with the graph of  =
√
, shift 1 unit to the left, and then shift 2 units upward.
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.
34
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CHAPTER 1 FUNCTIONSAND MODELS
18.  = −( − 1) 2 + 3. Start with the graph of  =  2 , shift 1 unit to the right, reflect about the ­axis, and then shift 3 units
upward.
19.  =  2 − 2 + 5 = ( 2 − 2 + 1) + 4 = ( − 1) 2 + 4. Start with the graph of  =  2 , shift 1 unit to the right, and then
shift 4 units upward.
20.  = ( + 1) 3 + 2. Start with the graph of  =  3 , shift 1 unit to the left, and then shift 2 units upward.
21.  = 2 − ||. Start with the graph of  = ||, reflect about the ­axis, and then shift 2 units upward.
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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35
22.  = 2−2cos. Start with the graph of  = cos, reflect about the ­axis, stretch vertically by a factor of 2, and then shift 2
units upward.
23.  = 3sin
1
2  + 1.
Start with the graph of  = sin, stretch horizontally by a factor of 2, stretch vertically by a factor of 3,
and then shift 1 unit upward.
24.  =
1
4
tan   −

4
 .
Start with the graph of  = tan, shift

4
units to the right, and then compress vertically by a
factor of 4.
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.