Graphing Exercise
1. For the given function f . Find the domain of f . Find the x− and y− intercept(s) if possible. Find any
horizontal, vertical, and slant asymptote(s). Determine the interval where f is increasing, the interval where
f is decreasing. Find all critical numbers of f and classify them as local max, local min, or neither. Find the
interval where f concaves up, and the interval where f concaves down. Find all inflection points. Use any
other knowledge you have about f to draw the graph of f .
a. f (x) = sin(2x)
Ans:
f is defined everywhere.
x−intercepts where x =
π
n, n an integer.
2
y− intercept at (0, 1)
No asymptote of any kind.
f 0 (x) = 2 cos(2x), f 00 (x) = −4 sin(2x),
π
local maxima at x = + πn, n any integer.
4
3π
local minima at x =
+ πn, n any integer.
4
3π
5π
increasing
on intervals
+ πn,
+ πn , n any integer.
(231,211)
4
4
y
6
3π
π
+ πn,
+ πn , n any integer.
decreasing on intervals
4
4
5
π
inflection points at x = n, n any integer.
2
4
π
concave up at
+ πn, π + πn , n any integer.
2
3
π
concave down at πn, + πn , n any integer.
2
2
1
-6
-5
-4
-3
-2
-1
o
1
2
3
4
5
6
x
-1
-2
Graphing...
f(x) = sin(2x) domain = (-∞, ∞)
0.17s
b. f (x) = x3 − x2 − 8x + 1
Ans:
f is defined everywhere.
y−intercept (0, 1)
No asymptote of any kind.
f 0 (x) = 3x2 − 2x − 8, f 00 (x) = 6x − 2
local maximum at x = −
4
3
local minimum at x = 2
4
increasing on −∞, −
∪ (2, ∞)
3
4
decreasing on − , 2
3
1
inflection point at x =
3
1
concave up on
,∞
3
1
concave down on −∞,
3
(231,171)
y
14
13
12
11
10
9
8
7
6
5
4
3
2
1
-20
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9-8-7-6-5-4-3-2-1 o-1
x
-21 2 3 4 5 6 7 8 9 1011121314151617181920
-3
-4
-5
-6
-7
-8
-9
-10
-11
Graphing...
f(x) = x^3-x^2-8x+1 domain = (-∞, ∞)
0.12s
c. f (x) = ex + 1
Ans:
f is defined everywhere.
No x− intercept.
y−intercept (0, 2)
f 0 (x) = ex , f 00 (x) = ex
horizontal asymptote y = 1
No local max or min., always increasing. No inflection point, always concave up.
(231,171)
y
14
13
12
11
10
9
8
7
6
5
4
3
2
1
-20
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9-8-7-6-5-4-3-2-1 o-1
x
-21 2 3 4 5 6 7 8 9 1011121314151617181920
-3
-4
-5
-6
-7
-8
-9
-10
-11
Graphing...
f(x) = e^x+1 domain = (-∞, ∞)
0.09s
d. f (x) =
x2 − 1
x+2
Ans:
Domain of f : All real numbers except −2
x−intercept x = −1 and x = 1
y−intercept 0, − 12
x2 + 4x + 1 00
6
f (x) =
,
f
(x)
=
(x + 2)2
(x + 2)3
0
vertical asymptote x = −2, slant asymptote y = x − 2
√
√
local max x = −2 − 3, local min x = −2 + 3
√ √
increasing −∞, −2 − 3 ∪ (−2 + 3, ∞)
√
√
decreasing (−2 − 3, −2) ∪ (−2, −2 + 3)
No inflection point but f changes concavity at the vertical asymptote x = −2.
concave up (−2, ∞)
concave down (−∞, −2)
(231,164)
y 23
22
21
20
19
18
17
16
15
14
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11
10
9
8
7
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5
4
3
2
1
-1
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-9-8-7-6-5-4-3-2-1o-2
12345678910
1112131415161718292021223242526272839303132x34
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-20
Graphing...
f(x) = (x^2-1)/(x+2) domain = (-∞, ∞)
0.14s
e. f (x) = ln(x − 1)
Ans:
x−intercept (2, 0). No y−intercept.
f is defined only for x > 1. Vertical asymptote at x = 1.
1
1
f 0 (x) =
, f 00 (x) = −
x−1
(x − 1)2
No local max or min. Function always increasing.
No inflection point. Function always concave down.
(231,180)
y
7
6
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
o
-1 1
2
3
4
5
6
7
8
9
10
x
-2
-3
-4
-5
Graphing...
f(x) = ln(x-1) domain = (0,∞)
0.05s
f. f (x) =
√
3
x
Ans:
f always defined. x− and y− intercept (0, 0)
2
1
f 0 (x) = √
, f 00 (x) = − √
3
3
2
3 x
9 x5
Critical number at x = 0, neither local max or min. Always increasing.
Inflection point at x = 0. Concave up on (−∞, 0), concave down on (0, ∞)
(231,180)
y
7
6
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
o
-1 1
2
3
4
5
6
7
8
9
10
x
-2
-3
-4
-5
Graphing...
f(x) = x^(1/3) domain = (-∞, ∞)
0.12s
g. f (x) = (x − 3)(x + 1)(x + 2)
Ans:
f defined everywhere. x− intercept x = −1, x = −2, x = 3
y− intercept (0, −6)
f 0 (x) = 3x2 − 7, f 00 (x) = 6x
s
local max at x = −
s
local min at x =
7
3
7
3

s 
 s
s 
s

7  7
∪
, ∞
increasing on −∞, −
3
3
7
decreasing on −
,
3
7
3
inflection point at x = 0
Concave down on (−∞, 0)
Concave up (0, ∞)
(231,166)
y 14
13
12
11
10
9
8
7
6
5
4
3
2
1
-20
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9-8-7-6-5-4-3-2-1 o-1
x
-21 2 3 4 5 6 7 8 9 1011121314151617181920
-3
-4
-5
-6
-7
-8
-9
-10
-11
Graphing...
f(x) = (x-3)(x+1)(x+2) domain = (-∞, ∞)
0.09s
h. f (x) = sin x + cos x
Ans:
f defined everywhere.
x−intercept when x =
3π
+ πn, n an integer.
4
y−intercept (0, 1)
f 0 (x) = cos x − sin x, f 00 (x) = − sin x − cos x
π
local maxima at x = + 2πn, n any integer.
4
5π
local minima at x =
+ 2πn, n any integer.
4
9π
5π
+ 2πn,
+ 2πn , n any integer.
increasing on
4
4
π
5π
decreasing on
+ 2πn,
+ 2πn , n any integer.
4
4
3π
inflection points at x =
+ πn, n any integer.
4
7π
3π
+ 2πn,
+ 2πn , n any integer.
concave up on
4
4
7π
11π
concave down on
+ 2πn,
+ 2πn , n any integer.
4
4
(231,181)
y6
5
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
o
-1
1
2
3
4
5
6
7
x
-2
-3
Graphing...
f(x) = sin(x)+cos(x) domain = (-∞, ∞)
0.11s
i. f (x) = |x2 − 4|
Ans:
f defined everywhere.
x− intercept at x = −2 and x = 2
y− intercept (0, 4)
(
2x
−2x
(
2
−2
0
f (x) =
f 00 (x) =
if x < −2 or 2 < x
if − 2 < x < 2
if x < −2 or 2 < x
if − 2 < x < 2
local maximum at x = 0
local minima at x = −2 and x = 2
increasing on (−2, 0) ∪ (2, ∞)
decreasing on (−∞, −2) ∪ (0, 2)
inflection points at x = −2 and x = 2
concave up on (−∞, −2) ∪ (2, ∞)
concave down on (−2, 2)
(231,175)
y
7
6
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
o
-1 1
2
3
4
5
6
7
8
9
10
x
-2
-3
-4
-5
Graphing...
f(x) = abs(x^2-4) domain = (-∞, ∞)
0.09s
j. f (x) =
x2
x2 + 9
Ans:
f defined everywhere.
x− and y− intercept (0, 0)
f 0 (x) =
18x
18(9 − 3x2 )
00
,
f
(x)
=
(x2 + 9)2
(x2 + 9)3
Horizontal asymptote y = 1
local min at x = 0
increasing (0, ∞)
decreasing (−∞, 0)
√
√
inflection points at x = − 3 and x = 3
√ √ concave up − 3, 3
√ √
3, ∞
concave down −∞, − 3 ∪
(231,194)
y
4
3
2
1
-5
-4
-3
-2
-1
o
1
2
3
4
5x
-1
-2
Graphing...
f(x) = (x^2)/(x^2+9) domain = (-∞, ∞)
0.09s
k. f (x) = √
x
−1
x2
Ans:
Domain of f = (−∞, −1) ∪ (1, ∞)
No x− or y− intercept.
1
3x
f 0 (x) = − 2
, f 00 (x) = 2
3/2
(x − 1)
(x − 1)5/2
Vertical asymptotes at x = −1 and x = 1. Function is undefined on (−1, 1). Horizontal asymptotes y = −1
and y = 1
No critical number. Function is always decreasing inside its domain.
No inflection point, but function changes concavity at the vertical asymptotes.
Concave up on (1, ∞)
Concave down on (−∞, −1)
(231,178)
y
5
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
o
-1
1
2
3
4
5
6
7
x
-2
-3
-4
Graphing...
f(x) = (x)/(sqrt(x^2-1)) domain = (-∞, ∞)
0.12s
√
l. f (x) =
1 − x2
x
Ans:
No y− intercept. x−intercept at x = −1 and x = 1
f 0 (x) = −
x2
√
1
2 − 3x2
00
,
f
(x)
=
x3 (1 − x2 )3/2
1 − x2
f is defined only on [−1, 0) ∪ (0, 1]. f has vertical asymptote at x = 0
No critical number. Function is always decreasing (in its domain).
s
2
Inflection points at x = −
and x =
3
s 


2 
∪ 0,
Concave up −1, −
3
 s

s
2
3
s 
2
3
s

2   2 
Concave down −
,0 ∪
,1
3
3
(231,184)
y
4
3
2
1
-6
-5
-4
-3
-2
-1
o
1
2
3
4
5
6x
-1
-2
-3
Graphing...
f(x) = (sqrt(1-x^2))/(x) domain = (-∞, ∞)
0.08s
m. f (x) =
ln x
x2
Ans:
x− intercept (1, 0)
No y− intercept.
vertical asymptote at x = 0. Function is defined only for x > 0. Horizontal asymptote y = 0
1 − 2 ln x 00
−5 + 6 ln x
,
f
(x)
=
x3
x4
√
local maximum at x = e
√ increasing on 0, e
√
decreasing on
e, ∞
f 0 (x) =
inflection point at x = e5/6
concave up e5/6 , ∞
concave down 0, e5/6
(214,103)
y
1
-3
-2
-1
o
1
2
3
4 x
-1
-2
-3
Graphing...
f(x) = (ln(x))/(x^2) domain = (-∞, ∞)
0.09s
n. f (x) = x + sin x
Ans:
f defined everywhere.
y−intercept (0, 0)
f 0 (x) = 1 + cos x, f 00 (x) = − sin x
Critical numbers at x = π(2n + 1), n any integer. These numbers are neither max or min. All horizontal
tangents.
Function increasing on (π(2n + 1), π(2n + 3)), n any integer.
Inflection points at x = πn, n any integer.
Concave up on (π + 2πn, 2π(n + 1)), n any integer.
Concave down on (2πn, π + 2πn), n any integer.
(230,145)
y 12
11
10
9
8
7
6
5
4
3
2
1
-20
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-16
-15
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-12
-11
-10
-9-8-7-6-5-4-3-2-1 o-1
x
-21 2 3 4 5 6 7 8 9 1011121314151617181920
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
Graphing...
f(x) = x+sin(x) domain = (-∞, ∞)
0.09s
ex
o. f (x) =
x
Ans:
No x− or y− intercept.
f defined everywhere except x = 0.
Vertical asymptote at x = 0. Horizontal asymptote y = 0.
local minimum at x = 1
increasing on (1, ∞)
decreasing on (−∞, 0) ∪ (0, 1)
No inflection point but function changes concavity at vertical asymptote x = 0
concave up (0, ∞)
concave down (−∞, 0)
(219,138)
y
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
o
-1
1
2
3
4
5
6
7
8x
-2
-3
-4
-5
Graphing...
f(x) = (e^x)/(x) domain = (-∞,∞)
0.09s
p. f (x) = √
x
+1
x2
Ans:
f defined everywhere.
x− and y− intercept (0, 0)
f 0 (x) =
(x2
1
+ 1)3/2
f 00 (x) =
(x2
−3x
+ 1)5/2
Horizontal Asymptote: y = 1 (as x → ∞) and y = −1 (as x → −∞)
Always increasing, no local max or min.
Concave up on (−∞, 0), concave down (0, ∞).
Inflection point at x = 0
(231,154)
y
4
3
2
1
-6
-5
-4
-3
-2
-1
o
1
2
3
4
5
6
x
-1
-2
-3
-4
Graphing...
f(x) = x/(sqrt(x^2+1)) domain = (-∞, ∞)
0.12s
q. f (x) = x − 3x1/3
Ans:
f defined everywhere.
√
√
x−intercepts x = 0, x = −3 3, x = 3 3
y−intercept (0, 0)
2
f 00 (x) = x−5/3
3
Increasing on (−∞, −1) ∪ (1, ∞)
f 0 (x) = 1 − x−2/3
Decreasing on (−1, 0) ∪ (0, 1)
Local max at x = −1. Local min at x = 1
x = 0 is a critical number, neither max nor min. It is a vertical tangent.
Concave up on (0, ∞) Concave down on (−∞, 0)
Inflection Point at x = 0
(231,154)
y
4
3
2
1
-6
-5
-4
-3
-2
-1
o
1
2
3
4
5
6
x
-1
-2
-3
-4
Graphing...
f(x) = x-3x^(1/3) domain = (-∞, ∞)
0.1s
r. f (x) =
1
(1 + ex )2
Ans:
f defined everywhere. No x− or y− intercept.
f 0 (x) =
−2ex
(1 + ex )3
f 00 (x) =
−2ex (1 − 2ex )
(1 + ex )4
Horizontal asymptote at y = 0 (as x → ∞) and y = 1 (as x → −∞ )
Always decreasing. No local max or min.
1
1
Concave up on ln
, ∞ Concave down on −∞, ln
2
2
1
Inflection Point at x = ln
2
(258,163)
y
2
1
-3
-2
-1
o
1
2
x
-1
-2
Graphing...
f(x) = 1/(1+e^x)^2 domain = (-∞,∞)
0.12s
s. f (x) = ln(x2 − 3x + 2)
Ans:
Domain of f : (−∞, 1) ∪ (2, ∞)
y−intercept (0, ln 2)
√
√
3− 5
3+ 5
x−intercept at x =
,x=
2
2
−2x2 + 6x − 5
2x − 3
f 00 (x) = 2
f 0 (x) = 2
x − 3x + 2
(x − 3x + 2)2
Vertical asymptotes at x = 1 and x = 2
Increasing on (2, ∞)
Decreasing on (−∞, 1)
No local max or min.
(242,208)
Concave down on whole domain of (−∞, 1) ∪ (2,
y ∞)
7
No inflection point.
6
5
4
3
2
1
-9
-8
-7
-6
-5
-4
-3
-2
-1
o
-1
1
2
3
4
5
6
7
8
9
10 11 12 x13
-2
-3
-4
-5
-6
-7
-8
Graphing...
f(x) = ln(x^2-3x+2) domain = (-∞, ∞)
0.11s
t. f (x) =
x
ex2
Ans:
f defined everywhere. x− and y−intercept (0, 0)
1 − 2x2
2x(2x2 − 3)
00
f
(x)
=
ex2
ex2
Horizontal asymptote y = 0
√ !
√
√ √ !
!
2 2
2
2
,
Decreasing on −∞, −
∪
,∞
Increasing on −
2 2
2
2
√
√
2
2
Local max at x =
. Local min at x = −
.
2
2
f 0 (x) =
s

 s


s 

3
3 
3 
Concave up on 
, ∞ ∪ −
, 0 Concave down on −∞, −
∪ 0,
2
2
2
s
3
Inflection points at x = 0 and x = −
and x =
2
(279,151)
s
s 
3
2
3
2
y
1
-3
-2
-1
o
1
2
3
x
-1
-2
-3
Graphing...
f(x) = x/(e^(x^2)) domain = (-∞, ∞)
0.12s
u. f (x) =
x2
x
−9
Ans:
Domain of f is all ready numbers except x = −3 or x = 3
x− and y− intercept (0, 0)
f 0 (x) = −
x2 + 9
(x2 − 9)2
f 00 (x) = −
2x(x2 + 27)
(x2 − 9)3
Horizontal asymptote y = 0
Vertical asymptote x = −3 and x = 3
Always decreasing within its domain
No local max or local min.
Concave up on (−3, 0) ∪ (3, ∞) Concave down on (−∞, −3) ∪ (0, 3)
Inflection point at x = 0
(294,216)
y
11
10
9
8
7
6
5
4
3
2
1
-16-15-14-13-12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 o-11 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16x17
-2
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-8
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-10
-11
-12
Graphing...
f(x) = x/(x^2-9) domain = (-∞, ∞)
0.11
v. f (x) =
x2
1
−9
Ans:
Domain of f : All real numbers except x = −3 or x = 3
No x− or y− intercept.
f 0 (x) =
−2x
2
(x − 9)2
f 00 (x) =
6(x2 + 3)
(x2 − 9)3
Horizontal asymptote y = 0
Vertical asymptote x = −3 and x = 3
Increasing on (−∞, −3) ∪ (−3, 0)
Decreasing on (0, 3) ∪ (3, ∞)
Local max at x = 0
Concave up on (−∞, −3) ∪ (3, ∞) Concave down on (−3, 3)
No inflection point.
(277,161)
y
5
4
3
2
1
-9
-8
-7
-6
-5
-4
-3
-2
-1
o
-1
1
2
3
4
5
6
7
8
9
10 x
11
-2
-3
-4
-5
-6
-7
-8
-9
Graphing...
f(x) = 1/(x^2-9) domain = (-∞,∞)
0.12
w. f (x) =
x2
x
+9
Ans:
f defined everywhere. x− and y− intercept (0, 0)
f 0 (x) =
−x2 + 9
(x2 + 9)2
f 00 (x) =
2x(x2 − 27)
(x2 + 9)3
Horizontal asymptote y = 0
Increasing on (−3, 3)
Decreasing on (−∞, −3) ∪ (3, ∞)
Local min at x = −3. Local max at x = 3
√
√
Concave up on (−3 3, 0) ∪ (3 3, ∞)
√
√
Concave down on (−∞, −3 3) ∪ (0, 3 3)
√
√
Inflection points at x = 0, x = −3 3, and x = 3 3
(278,225)
y5
4
3
2
1
-6
-5
-4
-3
-2
-1
o
-1
-2
-3
-4
-5
Graphing...
f(x) = x/(x^2+9) domain = (-∞,∞)
1
2
3
4
5
6
7
x. f (x) =
x2
x+8
Ans:
f defined everywhere except x = −8
x− and y− intercept (0, 0)
f 0 (x) = 1 −
64
x(x + 16)
=
2
(x + 8)
(x + 8)2
f 00 (x) =
128
(x + 8)3
Vertical asymptote x = −8
Slant asymptote y = x − 8
Increasing on (−∞, −16) ∪ (0, ∞)
Decreasing on (−16, −8) ∪ (−8, 0)
Local min at x = 0. Local max at x = −16
Concave up on (−8, ∞)
Concave down on (−∞, −8)
No inflection point.
(294,138)
y 28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
-1
-2
o
-65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 -3
5
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
-21
-22
-23
-24
-25
-26
-27
-28
-29
-30
-31
-32
-33
-34
-35
-36
-37
-38
-39
-40
-41
-42
-43
-44
-45
-46
-47
-48
-49
-50
-51
-52
-53
-54
-55
-56
-57
-58
-59
-60
-61
-62
-63
-64
-65
-66
-67
-68
-69
-70
-71
-72
-73
-74
-75
-76
Graphing...
f(x) = x^2/(x+8) domain = (-∞, ∞)
10 15 20 25 30 35 40 45 50 55 60 65 70
y. f (x) =
1
1 − x2
Ans:
f defined everywhere except x = −1 or x = 1
No x− or y− intercept.
f 0 (x) =
2x
(1 − x2 )2
f 00 (x) =
6x2 + 2
(1 − x2 )3
Horizontal asymptote y = 0
Vertical asymptote x = −1 and x = 1
Increasing on (0, 1) ∪ (1, ∞)
Decreasing on (−∞, −1) ∪ (−1, 0)
Local min at x = 0
Concave up on (−1, 1)
Concave down on (−∞, −1) ∪ (1, ∞)
No inflection point.
(223,149)
y
3
2
1
-5
o
-1
-2
-3
-4
-5
-6
-7
-8
-9
Graphing...
f(x) = 1/(1-x^2) domain = (-∞, ∞)
5
10
z. f (x) = x4 − 3x3 + 3x2 − x
Ans:
f defined everywhere.
x− intercept at x = 0 and x = 1
y− intercept (0, 0)
f 0 (x) = 4x3 − 9x2 + 6x − 1 = (x − 1)2 (4x − 1)
Increasing on
f 00 (x) = 12x2 − 18x + 6 = 6(2x − 1)(x − 1)
1
,∞
4
Decreasing on −∞,
1
4
1
4
Critical number at x = 1 is neither a max or min. It is a horizontal tangent.
1
Concave up on −∞,
∪ (1, ∞)
2
1
,1
Concave down on
2
1
Inflection points at x = and x = 1
2
Local min at x = −
y
2
1
o
-1
x
aa. f (x) = x + ln(x2 + 1)
Ans:
Domain of f is all real numbers.
x− and y− intercept (0, 0)
f 0 (x) = 1 +
2x
(x + 1)2
=
x2 + 1
x2 + 1
f 00 (x) =
2(1 − x2 )
(x2 + 1)2
No asymptote of any kind.
Increasing on (−∞, −1) ∪ (−1, ∞)
Critical number at x = −1 is neither max nor min. It is horizontal tangent.
Concave up on (−1, 1)
Concave down on (−∞, −1) ∪ (1, ∞)
Inflection points at x = −1 and x = 1
(336,255)
y
7
6
5
4
3
2
1
-5
o
-1
-2
-3
-4
-5
-6
Graphing...
f(x) = x+ln(x^2+1) domain = (-∞, ∞)
5