Section 3.1 Functions
#1 - 6: Which of the relations define y as a function of x?
Hint: a relation is considered a function if each x is
assigned to one y. That is a relation is not a function
when there are two distinct points that have the same xcoordinate but different y-coordinates.
1) { (1,2) (3,2) (4,2) (5,2)}
2) { (6,1) (7,1) (8,1)}
3) { (1,2) (3,4) (5,6) (7,8) (9,10)}
4) { (1,2) (4,5)}
5) { (3,1) (4,5) (3,6) }
6) { (3,7) (1,5) (1,2) }
#7 - 12: Use the vertical line test to determine whether
the relation defines y as a function of x.
We say the graph of a relation is the graph of a function
provided the graph passes the vertical line test.
Specifically if it is possible to draw at least one vertical
line that touches the graph in more than one place we
say y is NOT a function of x (or just - the graph is not a
function)
If it is not possible to draw a vertical line to touch the
graph in more than one place we say y IS a function of x
(or just – the graph is a function)
7)
8)
y
y
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
9)
10)
y
y
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-6
6
-5
-4
-3
-2
-1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
1
2
3
4
5
6
1
2
3
4
5
6
-6
-6
11)
12)
y
y
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-6
-5
-4
-3
-2
-1
6
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
Section 3.1 functions
#13 – 26: Determine whether the equation defines y as a
function of x. Hint, solve the equation for y and sketch a
graph using your calculator, then solve like problems #712.
13) y = x2
14) y = x2 +4
15) 𝑦 =
16) 𝑦 =
3
𝑥
5
𝑥
17) 𝑦 = √𝑥 + 2
18) 𝑦 = √𝑥 − 2
3
19) 𝑦 = √𝑥 − 2
3
20) 𝑦 = √𝑥 − 5
21) y2 +x2 = 9
22) (x-2)2 + y2 = 16
23) x = y2
24) x + 2 = y2
25) 2x2 + 4y2 = 16
26) 5x2 + 3y2 = 25
#27 – 45: Determine the domain and range of each
function, write your answer in interval notation when
appropriate.
Hint when a function is defined by a finite set of points
the domain of the function consists of the set of all the xcoordinates of the points and the range consists of the
set of all the y-coordinates of the points.
27) f = { 1,2) (3,4) (5,6)}
28) g = {(3,2) (5,7) (9,1) (8,1)}
29) g = {(0,4) (1,5) (2,6) (3,8) (4,1)
30) f = { (0,3) (-1,5)}
The strategy for finding the domain and range when
given a graph of a function is depends on whether the
graph represents a finite number of points or infinitely
many points.
The graphs for problems 31 and 32 are graphs of
functions that have a finite number of points. We find
their domain using the same strategy a problems 27 – 30.
Specifically the domain is the set of all x-coordinates and
the range it the set of all the y-coordinates.
31)
32)
y
14
13
12
(1,12)
11
10
9
8
7
6
5
4
3
2
1
-6
-5
-4
(-5,0)
-3
-2
-1
-2
-3
-4
(-3,-4)
x
1
-1
-5
-6
-7
2
3
4
5
6
The graphs for the second problem 32 through 41 are
graphs of functions that have an infinite number of
points.
There domain and range need to be written in interval
notation.
Domain = [x-coordinate far left point, x-coord. far right point]
Range = [y-coordinate bottom point, y-coordinate top point]
(The brackets will be square [ ] if the coordinate represented is
a point on the graph)
The brackets will be round brackets ( ) if the coordinate does
not represent a point on the graph. For our purposes the xcoordinates of asymptotes and all infinities will be shown with
round parenthesis when we write the domain and range.
32)
33)
14
y
14
12
12
(1,12)
10
10
(0,10)
8
8
6
6
4
2
-8
-6
-4
-2
(5,5)
(3,1)
2
-2
-4
-6
y
4
6
4
2
x
8
x
-8
-6
-4
-2
(-5,0)
(-3,-4)
2
-2
-4
-6
-8
-8
-10
-10
-12
-12
-14
-14
4
6
8
34)
35)
y
y
6
6
5
5
4
4
(2,4)
3
3
2
2
(-1,2)
(0,0)
1
1
x
-6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-5
-4
-3
6
(0,0)
(-1,-2)
-2
-1
x
1
2
3
4
5
-1
-1
-2
-2
-3
-3
-4
(2,-4)
-4
-5
-5
-6
-6
Extend the graphs of 36-45
before finding domain and
range.
36)
37)
y
y
6
6
5
5
4
4
3
3
2
2
(2,1)
1
1
(2,1)
-6
-5
-4
-3
-2
-1
1
2
x
3
4
5
x
6
-6
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
6
38)
39)
y
y
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-6
6
-5
-4
-3
-2
-1
2
3
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
40)
41)
y
y
6
6
5
5
4
4
3
3
2
2
(3,1)
(3,1)
1
1
x
-6
4
(0,0)
(2,0)
-1
1
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
5
6
42)
43)
44)
45)
#46 – 67 Use algebra to find the domain of each
function. Write your answer in interval notation.
Type of function
Square root
Fraction
Polynomial
Algebra to find the domain
Set argument ≥ 0 that is
set what is under the
square root ≥ 0. This gives
the domain in an inequality
notation. Change this to
interval notation.
Set denominator = 0, use
number line to find
domain’s interval notation
No Algebra domain
(−∞, ∞)
46) 𝑓(𝑥 ) = √𝑥 − 2
47) 𝑓(𝑥 ) = √𝑥 − 3
48) 𝑔(𝑥 ) = √3𝑥 + 12
49) 𝑔(𝑥 ) = √2𝑥 + 10
50) 𝑓(𝑥 ) =
51) 𝑓(𝑥 ) =
𝑥+2
𝑥−3
𝑥−6
𝑥−7
52) 𝑓(𝑥 ) =
53) 𝑔(𝑥 ) =
2
𝑥 2 +6𝑥−7
5
𝑥 2 −5𝑥+6
54) 𝑓(𝑥 ) =
55) 𝑓(𝑥 ) =
2
√𝑥−3
3
√𝑥−5
56) 𝑔(𝑥 ) = √5 − 𝑥
57) 𝑘(𝑥 ) = √6 − 𝑥
58) 𝑔(𝑥 ) =
59) 𝑓(𝑥 ) =
4
√6−𝑥
2
√3−𝑥
60) f(x) = 3x + 6
61) g(x) = 2x - 10
62) f(x) = x2 + 4
63) g(x) = x2 + 5
64) h(x) = (x-3)2 + 1
65) f(x) = (x+2)2 – 5
66) f(x) = x3 -6x2 + 7
67) g(x) = x3 – 4x2 + 2x – 3
Section 3.2 operations with functions
#1 - 18: Consider the functions defined by and find the
requested function values.
2
𝑘(𝑥 ) =
𝑥+3
f(x) = 3x + 4
g(x) = x2 + 5x + 6
h(x) = 4
1) f(3)
2) f(-2)
3) g(1)
4) g(0)
2
𝑥+3
f(x) = 3x + 4
g(x) = x2 + 5x + 6
h(x) = 4
𝑘(𝑥 ) =
5) h(2)
6) h(3)
7) k(-5)
8) k(-6)
2
𝑘(𝑥 ) =
𝑥+3
f(x) = 3x + 4
g(x) = x2 + 5x + 6
h(x) = 4
9) f(b)
10) f(c)
11) f(b+1)
12) f(b-2)
2
𝑘(𝑥 ) =
𝑥+3
f(x) = 3x + 4
g(x) = x2 + 5x + 6
h(x) = 4
13) g(2a)
14) g(3a)
15) g(x-2)
16) g(x+1)
2
𝑥+3
f(x) = 3x + 4
g(x) = x2 + 5x + 6
h(x) = 4
𝑘(𝑥 ) =
17) k(a)
18) k(a-2)
#19 - 27: Let f(x) = 2x + 3 and g(x) = 2x2 + 5x + 3. Find
each function.
19) (f + g)(x)
20) (g – f)(x)
#19 - 27: Let f(x) = 2x + 3 and g(x) = 2x2 + 5x + 3. Find
each function.
21) (f/g)(x)
22) (𝑔 ∙ 𝑓 )(𝑥)
#19 - 27: Let f(x) = 2x + 3 and g(x) = 2x2 + 5x + 3. Find
each function.
23) ( g/f)(x)
24) (𝑓 ∘ 𝑔)(𝑥)
#19 - 27: Let f(x) = 2x + 3 and g(x) = 2x2 + 5x + 3. Find
each function.
25) (𝑔 ∘ 𝑓 )(𝑥)
26) (g+f)(x)
27) ( f – g)(x)
#28 - 36: Let f(x) = 2x2 – 5x – 3 and g(x) = x-3. Find each
function.
28) (f + g)(x)
29) (g – f)(x)
#28 - 36: Let f(x) = 2x2 – 5x – 3 and g(x) = x-3. Find each
function.
30) (f/g)(x)
31) (𝑔 ∙ 𝑓 )(𝑥)
#28 - 36: Let f(x) = 2x2 – 5x – 3 and g(x) = x-3. Find each
function.
32) (g/f)(x)
33) (𝑓 ∘ 𝑔)(𝑥)
#28 - 36: Let f(x) = 2x2 – 5x – 3 and g(x) = x-3. Find each
function.
34) (𝑔 ∘ 𝑓 )(𝑥)
35) (g+f)(x)
36) ( f – g)(x)
Section 3.2 operations with functions
#37 - 48: Let h(x) = x2 + 2x+ 1 and k(x) = 2x - 5. Find each
of the following.
37) (h+k)(3)
38) (hk)(-1)
#37 - 48: Let h(x) = x2 + 2x+ 1 and k(x) = 2x - 5. Find each
of the following.
39) (h/k)(5)
40) (k– h)(0)
#37 - 48: Let h(x) = x2 + 2x+ 1 and k(x) = 2x - 5. Find each
of the following.
41) (h– k)(7)
42) (kh)(4)
#37 - 48: Let h(x) = x2 + 2x+ 1 and k(x) = 2x - 5. Find each
of the following.
43) (ℎ ∘ 𝑘)(4)
44) (ℎ ∘ 𝑘)(0)
#37 - 48: Let h(x) = x2 + 2x+ 1 and k(x) = 2x - 5. Find each
of the following.
45) (𝑘 ∘ ℎ)(3)
46) (ℎ ∘ 𝑘)(−2)
#37 - 48: Let h(x) = x2 + 2x+ 1 and k(x) = 2x - 5. Find each
of the following.
47) (𝑘 ∘ ℎ)(1)
48) (𝑘 ∘ ℎ)(−6)
#49 – 57: Let s(x) = x2 +5x – 3 and t(x) = 2x - 7. Find each
of the following.
49) (s/t)(3)
50) (s– t)(4)
#49 – 57: Let s(x) = x2 +5x – 3 and t(x) = 2x - 7. Find each
of the following.
51) (t+s)(6)
#49 – 57: Let s(x) = x2 +5x – 3 and t(x) = 2x - 7. Find each
of the following.
52) (𝑠 ∘ 𝑡)(4)
53) (𝑠 ∘ 𝑡)(0)
#49 – 57: Let s(x) = x2 +5x – 3 and t(x) = 2x - 7. Find each
of the following
54) (𝑠 ∘ 𝑡)(3)
55) (𝑠 ∘ 𝑡)(−2)
#49 – 57: Let s(x) = x2 +5x – 3 and t(x) = 2x - 7. Find each
of the following
56) (𝑡 ∘ 𝑠)(1)
57) (𝑡 ∘ 𝑠)(−6)
#58-67: Find the difference quotient; that is find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
58) f(x) = 2x + 6
59) f(x) = 3x – 7
#58-67: Find the difference quotient; that is find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
60) f(x) = 5x + 4
61) f(x) = 9x - 5
#58-67: Find the difference quotient; that is find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
62) f(x) = x2 – 5
63) f(x) = x2 + 1
#58-67: Find the difference quotient; that is find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
64) f(x) = x2 + 3x+ 5
65) f(x) = x2 + 5x – 3
#58-67: Find the difference quotient; that is find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
66) f(x) = x2 – 2x + 4
67) f(x) = x2 – 5x + 8
Section 3.3 the graphs of functions
1) We will refer to the function in the graph as f.
a) find the x-intercepts
b) find the y-intercept
c) for what values of x is f(x) = 5
d) find f(5)
e) what is the domain of f
f) what is the range of f
2) We will refer to the function in the graph as g.
y
14
13
12
11
10
(6,7)
9
8
7
6
5
4
3
(-1,0)
(5,0)
2
1
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
-1
-2
(0,-5)
-3
-4
-5
-6
-7
-8
(2,-9)
-9
a) find the x-intercepts
b) find the y-intercept
c) for what values of x is g(x) = 7
d) find g(0)
e) what is the domain of g
f) what is the range of g
4
5
6
x
7
8
9
3) We will refer to the function in the graph as h.
y
14
13
12
11
10
9
(-1,9)
(0,8)8
7
6
(-3,5)
(1,5)
5
4
3
2
1
-9
-8
-7
-6
(-4,0)
-5
-4
-3
-2
(2,0)
1
2
-1
x
3
-1
-2
-3
-4
-5
-6
(-5,-7)
-7
-8
-9
a) find the x-intercepts
b) find the y-intercept
c) for what values of x is h(x) = 0
d) find h(0)
e) what is the domain of h
f) what is the range of h
4
5
6
7
8
9
4) We will refer to the function in the graph as f.
a) find the x-intercepts
b) find the y-intercept
c) for what values of x is f(x) = 9
d) find f(5)
e) what is the domain of f
f) what is the range of f
#5 – 12, we will call each function graphed in problems 512 f(x). For what values of x is
a) f(x) ≥ 0 b) f(x) ≤0
5)
6)
#5 – 12, we will call each function graphed in problems 512 f(x). For what values of x is
a) f(x) ≥ 0 b) f(x) ≤0
7).
8)
#5 – 12, we will call each function graphed in problems 512 f(x). For what values of x is
a) f(x) ≥ 0 b) f(x) ≤0
9)
10)
#5 – 12, we will call each function graphed in problems 512 f(x). For what values of x is
a) f(x) ≥ 0 b) f(x) ≤0
11)
12)
Section 3.3 graphs of functions
#13 – 23: Find the following:
a) the interval(s) where the function graphed is
increasing
b) the interval(s) where the function graphed is
decreasing
c) The values of x (if any) where the function has a local
maximum
d) The local maximum value (if any)
e) The values of x (if any) where the function has a local
minimum
f) The local minimum values (if any)
13)
14)
#13 – 23: Find the following:
a) the interval(s) where the function graphed is
increasing
b) the interval(s) where the function graphed is
decreasing
c) The values of x (if any) where the function has a local
maximum
d) The local maximum value (if any)
e) The values of x (if any) where the function has a local
minimum
f) The local minimum values (if any)
15)
16)
#13 – 23: Find the following:
a) the interval(s) where the function graphed is
increasing
b) the interval(s) where the function graphed is
decreasing
c) The values of x (if any) where the function has a local
maximum
d) The local maximum value (if any)
e) The values of x (if any) where the function has a local
minimum
f) The local minimum values (if any)
17)
#13 – 23: Find the following:
a) the interval(s) where the function graphed is
increasing
b) the interval(s) where the function graphed is
decreasing
c) The values of x (if any) where the function has a local
maximum
d) The local maximum value (if any)
e) The values of x (if any) where the function has a local
minimum
f) The local minimum values (if any)
18)
19)
#13 – 23: Find the following:
a) the interval(s) where the function graphed is
increasing
b) the interval(s) where the function graphed is
decreasing
c) The values of x (if any) where the function has a local
maximum
d) The local maximum value (if any)
e) The values of x (if any) where the function has a local
minimum
f) The local minimum values (if any)
20)
21)
#13 – 23: Find the following:
a) the interval(s) where the function graphed is
increasing
b) the interval(s) where the function graphed is
decreasing
c) The values of x (if any) where the function has a local
maximum
d) The local maximum value (if any)
e) The values of x (if any) where the function has a local
minimum
f) The local minimum values (if any)
22)
23)
24) Find the average rate of change of f(x) = (x-2)2 -4
a) from 1 to 2
b) from 3 to 5
25) Find the average rate of change of f(x) = (x-3)2 – 2
a) from 1 to 3
b) from 4 to 5
26) find the average rate of change of f(x) = x3 – 2x + 1
a) from -3 to -2
b) from -1 to 1
27) Find the average rate of change of f(x) = x3 – 3x2 + 5
a) from -3 to -2
b) from 4 to 6
28) The number of people P(t) (in hundreds) infected t
days after an epidemic begins is approximated by
10ln(0.19𝑡 + 1)
(
)
𝑃 𝑡 =
0.19𝑡 + 1
The graph modeling the first 40 days of the epidemic is
depicted below. The x-coordinate of each point
represents the number of days since the epidemic began
and the y-coordinate the number of people (in hundreds)
infected.
a) Find the interval(s) where the graph is increasing and
decreasing. Interpret the result.
b) What was the maximum number of people infected?
When did it occur?
29) The function A(x) = 0.003631x3 -0.03746x2 + 0.1012x
+ 0.009
approximates blood alcohol concentration in a 170-lb
woman x hours after drinking 2 ounces of alcohol on an
empty stomach. The graph of this function for the first 5
hours is drawn below. Where the x-coordinate of each
point represents how long it has been since the woman
had her drink and the y-coordinate represents her blood
alcohol concentration.
a) Find the intervals where the graph is increasing and
decreasing. Interpret the result.
b) What was the maximum blood alcohol concentration?
When did it occur?
30) The percent of concentration of a drug in the
bloodstream x hours after a drug is administered is given
by the function:
4𝑥
3𝑥 2 + 27
The graph for the first 10 hours is drawn below.
𝐾(𝑥 ) =
a) Find the interval(s) where the graph is increasing and
decreasing. Interpret the result.
b) What was the maximum concentration? When did
this occur?
Section 3.4: Piecewise-defined functions
#1-6: Find the indicated value for each function.
1)
𝑓 (𝑥 ) = {
a) f(-5)
2) 𝑓 (𝑥 ) = {
a) f(0)
b) f(5)
c) f(6)
3𝑥, 𝑖𝑓 𝑥 < 0
2𝑥 + 1, 𝑖𝑓 𝑥 ≥ 0
b) f(0)
c) f(2)
𝑥 − 5, 𝑖𝑓 𝑥 ≤ 5
2𝑥 − 4, 𝑖𝑓 𝑥 > 5
𝑥 − 5, 𝑖𝑓 𝑥 < −1
𝑥, 𝑖𝑓 − 1 ≤ 𝑥 ≤ 2
3) 𝑔(𝑥 ) = {
𝑥 + 2, 𝑖𝑓 𝑥 > 2
a) g(-1)
b) g(2)
2𝑥, 𝑖𝑓 𝑥 ≤ 0
4) g(x) = {𝑥, 𝑖𝑓 0 < 𝑥 ≤ 3
−5𝑥 𝑖𝑓 𝑥 > 3
a) g(0)
b) g(3)
c) g(-2)
c) g(0)
𝑥 2 − 10, 𝑖𝑓 𝑥 < −10
5) 𝑘(𝑥 ) = {𝑥 2 , 𝑖𝑓 − 10 ≤ 𝑥 ≤ 10
𝑥 2 + 10. 𝑖𝑓 𝑥 > 10
a) k(-10)
b) k(11)
c) k(0)
2𝑥 2 − 3, 𝑖𝑓 𝑥 < 2
6) 𝑘(𝑥 ) = { 𝑥 2 , 𝑖𝑓 2 ≤ 𝑥 ≤ 4
5𝑥 − 7 𝑖𝑓 𝑥 > 4
a) k(2)
b) k(4)
c) k(5)
#7-12: sketch a graph of each function.
7)
𝑓 (𝑥 ) = {
8) 𝑓 (𝑥 ) = {
3𝑥, 𝑖𝑓 𝑥 < 0
2𝑥 + 1, 𝑖𝑓 𝑥 ≥ 0
𝑥 − 5, 𝑖𝑓 𝑥 ≤ 5
2𝑥 − 4, 𝑖𝑓 𝑥 > 5
𝑥 − 5, 𝑖𝑓 𝑥 < −1
𝑥, 𝑖𝑓 − 1 ≤ 𝑥 ≤ 2
9) 𝑔(𝑥 ) = {
𝑥 + 2, 𝑖𝑓 𝑥 > 2
2𝑥, 𝑖𝑓 𝑥 ≤ 0
10) g(x) = {𝑥 + 1, 𝑖𝑓 0 < 𝑥 ≤ 3
−5𝑥 𝑖𝑓 𝑥 > 3
𝑥 2 − 2, 𝑖𝑓 𝑥 < −1
11) 𝑘(𝑥 ) = {𝑥 2 , 𝑖𝑓 − 1 ≤ 𝑥 ≤ 1
𝑥 2 + 2. 𝑖𝑓 𝑥 > 1
2𝑥 2 − 3, 𝑖𝑓 𝑥 < 2
12) 𝑘(𝑥 ) = { 𝑥 2 , 𝑖𝑓 2 ≤ 𝑥 ≤ 4
5𝑥 − 7 𝑖𝑓 𝑥 > 4
13) In the 1995 tax form a tax rate schedule is given for
people whose filing status is single. Part of the table is
shown below:
If the taxable
But not then the tax
income is
over-is...
over...
$0 $23,350
15%
$3,502.50 +
$23,350 $56,550
28%
$12,798.50 +
$56,550 $117,950
31%
of the
amount
over-$0
$23,350
$56,550
a. Write the defining rule for a piecewise function
T(x) giving the tax owed by a person whose
taxable income is x, where x is less than
$117,950.
b. Evaluate the function to find the tax owed by a
single person whose taxable income in 1995 was
$31,950.
14) In the 2005 tax form a tax rate schedule is given for
people whose filing status is single. Part of the table is
shown below:
If the taxable
But not then the
income is
over-tax is...
over...
$0 $28,000
10%
$2800 +
$28,000 $60,000
20%
9200 +
$60,000 $200,000
25%
of the
amount
over-$0
$28,000
$60,000
a) Write the defining rule for a piecewise
function T(x) giving the tax owed by a person
whose taxable income is x, where x is less than
$200,000.
b) Evaluate the function to find the tax owed by
a single person whose taxable income in 2005
was $50,000.
15) Assume you work at a company where you are paid
hourly.
You are paid $7.80 per hour for regular time (less than or
equal to thirty-five hours)
and time and a half for overtime hours up to forty-five
hours in one week.
If you are asked to work forty-five or more hours in one
week you are paid double-time.
a) Write this information in the form of a piecewisedefined function
b) Use your formula to compute the pay for working 50
hours in a week.
16) Assume you work at a company where you are paid
hourly.
You are paid $10.00 per hour for regular time (less than
or equal to forty hours)
and time and a half for overtime hours up to fifty hours
in one week.
If you are asked to work fifty or more hours in one week
you are paid double-time.
a) Write this information in the form of a piecewisedefined function
b) Use your formula to compute the pay for working 50
hours in a week.
Section 3.5 Transformations of functions:
Here is a summary of the rules presented in this section.
Up and down
Transformation
shifts
y = f(x) + k (k>0) Shift the graph
UP k units
Y = f(x) – k (k>0) Shift the graph
DOWN k units
Left and
right shifts
y = f(x+h)
(h >0)
y = f(x-h)
(h>0)
Table
Use same “x”
values
Use same “x”
values
Transformation
Table
Shift graph LEFT
h units
Shift graph RIGHT
h units
Subtract “h” from
each “x” value
Add “h” to each “x”
value
Reflections Transformation
y= -f(x)
REFLECTS graph about
x-axis
y= f(-x)
REFLECTS graph about
y-axis
Table
Use same “x”
values
Change the sign of
each “x” value
Compressing and
stretching
y = af(x) (a >0)
Compressing and
stretching
y = f(ax) (a >0)
Transformation
Table
STRETCHES the
graph when
a>1
COMPRESSES
graph when
0<a<1
Use same
values of x
Use same
values of x
Transformation
Table
STRETCHES the
graph when
a>1
Divide
numbers in
x-column by
“a”
Multiply
numbers in
the x-column
by reciprocal
of “a”
COMPRESSES
graph when
0<a<1
Section 3.5 Transformations of functions
#1-6: Complete the following horizontal transformation
problems.
1) Let f(x) = |𝑥 |
a) Sketch a graph of f(x)
b) Find f(x-3)
c) Make a table of values and sketch a graph of f(x-3) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of
f(x-3) from the graph of f(x)
e) Find f(x+2)
f) Make a table of values and sketch a graph of f(x+2) on
the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x+2) from the graph of f(x)
2) Let f(x) = |𝑥 |
a) Sketch a graph of f(x)
b) Find f(x-4)
c) Make a table of values and sketch a graph of f(x-4)
d) Describe the transformation to obtain the graph of
f(x-4) from the graph of f(x)
e) Find f(x+3)
f) Make a table of values and sketch a graph of f(x+3) on
the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x+3) from the graph of f(x)
3) Let f(x) = 𝑥 2
a) Sketch a graph of f(x)
b) Find f(x-2)
c) Make a table of values and sketch a graph of f(x-2)
d) Describe the transformation to obtain the graph of
f(x-2) from the graph of f(x)
e) Find f(x+1)
f) Make a table of values and sketch a graph of f(x+1) on
the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x+1) from the graph of f(x)
Section 3.5 Transformations of functions:
4) Let f(x) = 𝑥 2
a) Sketch a graph of f(x)
b) Find f(x-5)
c) Make a table of values and sketch a graph of f(x-5)
d) Describe the transformation to obtain the graph of
f(x-5) from the graph of f(x)
e) Find f(x+3)
f) Make a table of values and sketch a graph of f(x+3) on
the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x+3) from the graph of f(x)
5) Let f(x) = √𝑥 + 2
a) Sketch a graph of f(x)
b) Find f(x-3)
c) Make a table of values and sketch a graph of f(x-3)
d) Describe the transformation to obtain the graph of
f(x-3) from the graph of f(x)
e) Find f(x+1)
f) Make a table of values and sketch a graph of f(x+1) on
the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x+1) from the graph of f(x)
6) Let f(x) = √𝑥 + 4
a) Sketch a graph of f(x)
b) Find f(x-5)
c) Make a table of values and sketch a graph of f(x-5)
d) Describe the transformation to obtain the graph of
f(x-5) from the graph of f(x)
e) Find f(x+2)
f) Make a table of values and sketch a graph of f(x+2) on
the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x+2) from the graph of f(x)
#7-12: Complete the following vertical transformation
problems.
7) Let f(x) = |𝑥 + 3|
a) Sketch a graph of f(x)
b) Find f(x) - 3
c) Make a table of values and sketch a graph of f(x) – 3
on the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of
f(x) – 3 from the graph of f(x)
e) Find f(x) + 2
f) Make a table of values and sketch a graph of f(x) + 2
on the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x+2) from the graph of f(x)
8) Let f(x) = |𝑥 + 1|
a) Sketch a graph of f(x)
b) Find f(x) – 4
c) Make a table of values and sketch a graph of f(x) – 4
d) Describe the transformation to obtain the graph of
f(x) – 4 from the graph of f(x)
e) Find f(x) + 3
f) Make a table of values and sketch a graph of f(x) + 3
on the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x) + 3 from the graph of f(x)
9) Let f(x) = (𝑥 − 3)2
a) Sketch a graph of f(x)
b) Find f(x) – 2
c) Make a table of values and sketch a graph of f(x) – 2
d) Describe the transformation to obtain the graph of
f(x) – 2 from the graph of f(x)
e) Find f(x) + 1
f) Make a table of values and sketch a graph of f(x) + 1
on the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x) + 1 from the graph of f(x)
10) Let f(x) = (𝑥 − 1)2
a) Sketch a graph of f(x)
b) Find f(x) – 5
c) Make a table of values and sketch a graph of f(x) – 5
d) Describe the transformation to obtain the graph of
f(x) – 5 from the graph of f(x)
e) Find f(x) + 3
f) Make a table of values and sketch a graph of f(x) + 3
on the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x) + 3 from the graph of f(x)
11) Let f(x) = √𝑥 − 4
a) Sketch a graph of f(x)
b) Find f(x) – 3
c) Make a table of values and sketch a graph of f(x) – 3
d) Describe the transformation to obtain the graph of
f(x) – 3 from the graph of f(x)
e) Find f(x) + 1
f) Make a table of values and sketch a graph of f(x) + 1
on the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x) + 1 from the graph of f(x)
12) Let f(x) = √𝑥 + 2
a) Sketch a graph of f(x)
b) Find f(x) – 5
c) Make a table of values and sketch a graph of f(x) – 5
d) Describe the transformation to obtain the graph of
f(x) – 5 from the graph of f(x)
e) Find f(x) + 2
f) Make a table of values and sketch a graph of f(x) + 2
on the same graph you have already created
g) Describe the transformation to obtain the graph of
f(x) + 2 from the graph of f(x)
#13 – 16: Complete the following reflection
transformation problems
13) Let f(x) = |𝑥 + 2|
a) Sketch a graph of f(x)
b) Find f(-x)
c) Make a table of values and sketch a graph of f(-x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of f(x) from the graph of f(x)
e) Find -f(x)
f) Make a table of values and sketch a graph of -f(x) on
the same graph you have already created
g) Describe the transformation to obtain the graph of f(x) from the graph of f(x)
14) Let f(x) = |𝑥 + 3|
a) Sketch a graph of f(x)
b) Find f(-x)
c) Make a table of values and sketch a graph of f(-x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of f(x) from the graph of f(x)
e) Find -f(x)
f) Make a table of values and sketch a graph of -f(x) on
the same graph you have already created
g) Describe the transformation to obtain the graph of f(x) from the graph of f(x)
15) Let f(x) = (𝑥 − 1)2
a) Sketch a graph of f(x)
b) Find f(-x)
c) Make a table of values and sketch a graph of f(-x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of f(x) from the graph of f(x)
e) Find -f(x)
f) Make a table of values and sketch a graph of -f(x) on
the same graph you have already created
g) Describe the transformation to obtain the graph of f(x) from the graph of f(x)
16) Let f(x) = (𝑥 − 3)2
a) Sketch a graph of f(x)
b) Find f(-x)
c) Make a table of values and sketch a graph of f(-x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of f(x) from the graph of f(x)
e) Find -f(x)
f) Make a table of values and sketch a graph of -f(x) on
the same graph you have already created
g) Describe the transformation to obtain the graph of f(x) from the graph of f(x)
#17 – 20: Complete the following compressing and
stretching problems
17) Let f(x) = |𝑥 |
a) Sketch a graph of f(x)
b) Find f(2x)
c) Make a table of values and sketch a graph of f(2x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of
f(2x) from the graph of f(x)
1
e) Find 𝑓 ( 𝑥)
2
1
f) Make a table of values and sketch a graph of 𝑓 ( 𝑥)on
2
the same graph you have already created
g) Describe the transformation to obtain the graph of
1
𝑓 ( 𝑥)from the graph of f(x)
2
18) Let f(x) = |𝑥 |
a) Sketch a graph of f(x)
b) Find f(3x)
c) Make a table of values and sketch a graph of f(3x)
d) Describe the transformation to obtain the graph of
𝑓(3𝑥) from the graph of f(x)
1
e) Find 𝑓 ( 𝑥)
3
1
f) Make a table of values and sketch a graph of 𝑓 ( 𝑥)
3
on the same graph you have already created
g) Describe the transformation to obtain the graph of
1
𝑓 ( 𝑥) from the graph of f(x)
3
19) Let f(x) = 𝑥 2
a) Sketch a graph of f(x)
b) Find f(4x)
c) Make a table of values and sketch a graph of f(4x)
d) Describe the transformation to obtain the graph of
f(4x) from the graph of f(x)
1
e) Find 𝑓 ( 𝑥)
4
1
f) Make a table of values and sketch a graph of 𝑓 ( 𝑥))
4
on the same graph you have already created
g) Describe the transformation to obtain the graph of
1
𝑓 ( 𝑥) from the graph of f(x)
4
20) Let f(x) = 𝑥 2
a) Sketch a graph of f(x)
b) Find f(2x)
c) Make a table of values and sketch a graph of f(2x)
d) Describe the transformation to obtain the graph of
f(2x) from the graph of f(x)
1
e) Find 𝑓 ( 𝑥)
2
1
f) Make a table of values and sketch a graph of 𝑓 ( 𝑥)on
2
the same graph you have already created
g) Describe the transformation to obtain the graph of
1
𝑓 ( 𝑥) from the graph of f(x)
2
#21 – 25: Complete the following compressing and
stretching problems
21) Let f(x) = |𝑥 |
a) Sketch a graph of f(x)
b) Find 3f(x)
c) Make a table of values and sketch a graph of 3f(x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of
3f(x) from the graph of f(x)
1
e) Find 𝑓(𝑥)
3
1
f) Make a table of values and sketch a graph of 𝑓(𝑥) on
3
the same graph you have already created
g) Describe the transformation to obtain the graph of
1
3
𝑓(𝑥)from the graph of f(x)
22) Let f(x) = |𝑥 |
a) Sketch a graph of f(x)
b) Find 2f(x)
c) Make a table of values and sketch a graph of 2f(x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of
2f(x) from the graph of f(x)
1
e) Find 𝑓(𝑥)
2
1
f) Make a table of values and sketch a graph of 𝑓(𝑥) on
2
the same graph you have already created
g) Describe the transformation to obtain the graph of
1
2
𝑓(𝑥)from the graph of f(x)
23) Let f(x) = 𝑥 2
a) Sketch a graph of f(x)
b) Find 3f(x)
c) Make a table of values and sketch a graph of 3f(x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of
3f(x) from the graph of f(x)
1
e) Find 𝑓(𝑥)
3
1
f) Make a table of values and sketch a graph of 𝑓(𝑥) on
3
the same graph you have already created
g) Describe the transformation to obtain the graph of
1
3
𝑓(𝑥)from the graph of f(x)
24) Let f(x) = 𝑥 2
a) Sketch a graph of f(x)
b) Find 2f(x)
c) Make a table of values and sketch a graph of 2f(x) on
the same graph as the graph of f(x)
d) Describe the transformation to obtain the graph of
2f(x) from the graph of f(x)
1
e) Find 𝑓(𝑥)
2
1
f) Make a table of values and sketch a graph of 𝑓(𝑥) on
2
the same graph you have already created
g) Describe the transformation to obtain the graph of
1
2
𝑓(𝑥)from the graph of f(x)
#25 – 50, write the function whose graph is the graph of
f(x) = x2, but is
25) Shifted to the left 2 units
26) Shifted to the left 3 units
27) Shifted to the right 5 units
28) Shifted to the right 4 units
29) Reflected over the x-axis
30) Reflected over the y-axis
31) Shifted up 2 units
32) Shifted up 3 units
33) Shifted down 4 units
34) Shifted down 6 units
35) Shifted to the right 2 units and down 3 units
36) Shifted to the right 3 units and down 4 units
37) Shifted to the left 2 units and down 3 units
38) Shifted to the left 3 units and down 4 units
39) Shifted to the right 2 units and up 3 units
40) Shifted to the right 3 units and up 4 units
41) Shifted to the left 2 units and up 3 units
42) Shifted to the left 3 units and up 4 units
43) reflected over x-axis and up two units
44) reflected over x-axis and up 3 units
45) reflected over x-axis and down 2 units
46) reflected over x-axis and down 4 units
47) reflected over x-axis and right 3 units
48) reflected over x-axis and right 2 units
49) reflected over x-axis and left 2 units
50) reflected over x-axis and left 4 units
#51 – 56: Use the graph of f(x) below to sketch a graph and describe the transformation.
51)
51a) f(x – 2)
51b) f(x + 1)
51c) f(x) – 2
51d) f(x) + 1
51e) f(x-2) + 1
51f) f(x + 1) – 2
51g) f(-x)
51h) –f(x)
Section 3.5 Transformation of functions
52)
52a) f(x – 3)
52b) f(x + 4)
52c) f(x) – 3
52d) f(x) + 4
52e) f(x-3) + 4
52f) f(x + 4) – 3
52g) f(-x)
52h) –f(x)
53)
53a) f(x – 1)
53b) f(x + 2)
53c) f(x) – 1
53d) f(x) + 2
53e) f(x-1) + 2
53f) f(x + 2) – 1
53g) f(-x)
53h) –f(x)
54)
54a) f(x – 5)
54b) f(x + 4)
54c) f(x) – 5
54d) f(x) + 4
54e) f(x-5) + 4
54f) f(x + 4) – 5
54g) f(-x)
54h) –f(x)
55)
55a) f(x – 2)
55b) f(x + 2)
55c) f(x) – 3
55d) f(x) + 3
55e) f(x-2) + 3
55f) f(x + 2) – 3
55g) f(-x)
55h) –f(x)
56)
56a) f(x – 1)
56b) f(x + 2)
56c) f(x) – 1
56d) f(x) + 2
56e) f(x-1) + 2
56f) f(x + 2) – 1
56g) f(-x)
56h) –f(x)
Section 3.6: mathematical modeling – building functions
1) A campground owner has 800 meters of fencing. He
wants to enclose a rectangular field. Let W represent the
width of the field. Follow these steps to find the
dimensions of the field that yields the largest area.
a) Write an equation for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area.
2) A campground owner has 1000 meters of fencing. He
wants to enclose a rectangular field bordering a river,
with no fencing needed along the river. Let W represent
the width of the field. Follow these steps to find the
dimensions of the field that yields the largest area.
a) Write an equation for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area.
3) A campground owner has 1400 meters of fencing. He
wants to enclose a rectangular field bordering a river,
with no fencing needed along the river, and let W
represent the width of the field. Follow these steps to
find the dimensions of the field that yields the largest
area.
a) Write an equation for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area.
4) A campground owner has 1000 meters of fencing. He
wants to enclose a rectangular field bordering a river,
with no fencing needed along the river, and let W
represent the width of the field.
a) Write an equation for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area.
5) A fence must be built to enclose a rectangular area of
20,000 square feet. Fencing material costs $2.50 per
foot for the two sides facing north and south (call these
sides the length, and $3.20 per foot for the other two
sides (call these sides the length). Follow these steps to
find the cost of the least expensive fence.
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost
d) Find the value of L leading to the minimum cost
e) Find the minimum cost.
6) A fence must be built to enclose a rectangular area of
20,000 square feet. Fencing material costs $2.00 per
foot for the two sides facing north and south (call these
sides the length, and $4.00 per foot for the other two
sides (call these sides the length). Follow these steps to
find the cost of the least expensive fence.
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost
d) Find the value of L leading to the minimum cost
e) Find the minimum cost.
7) A fence must be built in a large field to enclose a
rectangular area of 25,600 square meters. One side of
the area is bounded by an existing fence; no fence is
needed there. Material for the fence costs $3.00 per
meter for the two ends, and $1.50 per meter for the side
opposite the existing fence. Find the cost of the least
expensive fence.
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost
d) Find the value of L leading to the minimum cost
e) Find the minimum cost.
8) A fence must be built in a large field to enclose a
rectangular area of 10,000 square meters. One side of
the area is bounded by an existing fence; no fence is
needed there. Material for the fence costs $5.00 per
meter for the two ends, and $2.00 per meter for the side
opposite the existing fence. Find the cost of the least
expensive fence.
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost
(round to 2 decimals)
d) Find the value of L leading to the minimum cost
(round to 2 decimals)
e) Find the minimum cost.
9) An open box with a square base is to be made from a
square piece of cardboard 10 inches on a side by cutting
out a square ( x inches by x inches) from each corner and
turning up the sides. (round to 2 decimals if needed)
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Graph the volume function using your graphing
calculator and find the value of x that makes V the
largest.
10) An open box with a square base is to be made from a
square piece of cardboard 12 inches on a side by cutting
out a square ( x inches by x inches) from each corner and
turning up the sides.
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Graph the volume function using your graphing
calculator and find the value of x that makes V the
largest.
11) An open box is to be made by cutting a square
corner of a 20 inch by 20 inch piece of metal then folding
up the sides. What size square should be cut from each
corner to maximize volume? (round to 2 decimals if
needed)
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Graph the volume function using your graphing
calculator and find the value of x that makes V the
largest. (round to 2 decimal places if needed)
12) An open box is to be made by cutting a square
corner of a 30 inch by 30 inch piece of metal then folding
up the sides. What size square should be cut from each
corner to maximize volume?
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Graph the volume function using your graphing
calculator and find the value of x that makes V the
largest. (round to 2 decimal places if needed)