Unit 4: Linear and Exponential Functions
Study Guide
39 SWBAT identify functions as relations where each input is paired with exactly one output
For a relation to be a function, every input value (x) must have one and only one output value (y)!
1. Which relations is NOT a function?
Relation #1 {(3, 4), (4, 5), (6, 7), (8, 9)}
Relation #2 {(3, 4), (4, 5), (6, 7), (3, 9)}
Relation #3 {(-3, 4), (4, -5), (0, 0), (8, 9)}
Relation #4 {(8, 11), (34, 5), (6, 17), (8, 11)}
2. Which of the following mapping diagrams shows a function? Explain how you can tell that the
relation you have chosen is a function.
3. For the following relation to be a function, x cannot be what values?
{(-2, 5), (3, 1), (7, -9), (-6, 4), (x, 1)}
4. Is the relation shown in the table to the right a function? Explain why or why not.
x
-1
0
1
2
y
-1
0
1
8
40 SWBAT understand that if f is a function and x is an element of its domain, then f(x) denotes the
output of f corresponding to the input x and that the graph of f is the graph of f(x) = y
41 SWBAT use function notation and evaluate functions for inputs in their domains
5. Which of the following equations is written in function notation?
a. x(f) = 5x
b. y = 3x3 + 11
c. func(x) = ½x - 3
d. h(x) = x2 + 7
6. What is the range of the function f (x) = 2x - 5 given a domain {-5, 0, 2, 11}?
7. What is the following function’s output when the input is {-3, 1, 6, 12, 15}
f (x) = 2 x - 3x +10
8. Evaluate each of the following functions for the four function operations listed below.
Function
Function Operation
æ1 ö
æ1
ö
Find f ç x ÷
Find f ç x + 2 ÷
Find - f (x)
Find f (-x)
f (x) = 8x + 4
è4 ø
è4
ø
æ1 ö
æ1
ö
Find f ç x ÷
Find f ç x +1÷
Find - f (x)
Find f (-x)
f (x) =12x + 6
è6 ø
è6
ø
æ3 ö
æ1
ö
Find f ç x ÷
Find f ç x +10 ÷
Find - f (x)
Find f (-x)
f (x) =12x +8
è4 ø
è4
ø
1
Find f ( 6x )
Find f (8x + 4)
Find - f (x)
Find f (-x)
f (x) = x + 4
2
42 SWBAT identify the six basic functions
There are 6 basic functions.
9. Name each of the six functions below and write the equation for each.
43 SWBAT graph transformations of basic functions including vertical shifts, stretches, shrinks, and
reflections
The table below describes the different types of transformations of the graph of a function.
10. Describe the
transformation(s)
that produced each
of the daughter
functions graphed
below.
a.
Vertical
Shift
Dilations
Algebraic
Add a constant
Subtract a constant
Multiply by a constant
Divide by a positive constant
Reflections
Multiply by a negative constant
b.
Graphical
Shift upward
Shift downward
Stretch
Shrink
Flip over line a reflection:
if – f(x) x-axis is line of reflection
if f(-x) y-axis is line of reflection
c.
Explain how the graph of a function will change when the function is transformed from f(x) to g(x).
1
11. f (x) = x 2 - 3  g(x) = x 2 + 7
13. f (x) = 2x +10  g(x) = x + 3
2
12. f (x) = x - 2  g(x) = - x + 2
3
3
14. f (x) = x  g(x) = (-x)
15. The original equation for a function is f (x) = 2x + 7 . Which equation will shift the graph of the
original equation up 2 units?
a.
f (x) = 2x - 2
b.
f (x) = 2(2x + 7)
c.
f (x) = 2x + 9
3
x - 3 , now change g(x) to g(8x). g(8x)=?
4
a. 6x - 3
b. 8x - 3
c. 6x + 6
d.
f (x) = 2x + 2
16. Assume g(x) =
17.
18.
19.
20.
21.
d. 8x -1
22.
44 SWBAT calculate the y-intercept of a linear function given two points or one point and slope
To calculate the y-intercept of a linear function given two points…
y -y
First, use slope formula 2 1 to find the slope of the line.
x2 - x1
Next, choose one point (x, y) and use your slope m, substitute these values into the formula y = mx + b
and solve for b!
23. What is the formula for slope?
24. Which of the following answer choices shows the points (3, -6) and (1, 4) labeled correctly to plug
into slope formula?
x1
y1
x2 y2
x1
x2
y1 y2
a. (3, -6) and (1, 4)
y1
x1
y2 x2
x2
y1
x1 y2
c. (3, -6) and (1, 4)
b. (3, -6) and (1, 4)
d. (3, -6) and (1, 4)
25. Use slope formula to find the slope of the line that passes through (5, 3) and (1, 1)
26. What equation will allow you to determine the y-intercept of a line given the slope and one point?
27. Find the y-intercept of the line that passes through (5, 3) and has a slope of 1.
28. Find the y-intercept of the line that passes through the point (-6, 1) and has a slope of
29. Write an equation for the line that passes through (3, 2) and (-6, 4)
2
.
3
30. Write an equation for the line that passes through (1, 9) and has a slope of 3.
31. Write an equation for the line that passes through (2, -7) and (5, 0)
32. Write an equation for the line that passes through (3, 0) and has a slope of -
3
4
45 SWBAT classify functions as even, odd or neither given a graph or function rule
SEY OOO
A function is EVEN if and only if the arms of the function point the SAME direction and it is symmetrical
with respect the to Y-AXIS.
To check for symmetry with respect to the Y-AXIS, reflect the original function over the y-axis. If
the function is in the exact same position as the original, it is symmetrical.
A function is ODD if and only if the arms of the functions point OPPOSITE directions and it is symmetrical
with respect to the ORIGIN
To check for symmetry with respect to the ORIGIN, reflect the original function over the y-axis and
the x-axis. If the function is in the exact same position as the original, it is symmetrical.
Tell whether each of the following functions is even, odd or neither. Explain how you know.
33.
34.
35.
36. Which transformations will make it such that a function with arms pointing in the same direction
(i.e. quadratic, absolute value) is NOT even?
37. Which transformations will make it such that a function with arms pointing in opposite directions
(i.e. linear, cubic) is NOT odd?
38. Which of the basic functions will never be even nor odd?
Describe how each of the following functions has been transformed from the parent function. Tell
whether each function is even, odd, or neither.
39. f (x) = -x
40. f (x) = x 2 + 3
41. f (x) = -x + 2
42. f (x) = x 3 -1
35 SWBAT determine if a sequence is arithmetic and find the common difference
36 SWBAT write a closed rule to represent an arithmetic sequence and determine the terms of an
arithmetic sequence given a closed rule
A Closed Rule is an equation that represents the relationship between the term number, n, and the value
of that term an. A(n) = a + (n – 1)d or an = a1 + (n -1)d, where A(n) or an = nth term, a or a1 = 1st term, n =
term number, d = common difference
1st Determine the first term, a, and the common difference, d
2nd Plug these values into the formula shown above
3rd Simplify by distributing d and combining like terms
43. For the sequence -18, -15, -12, -9, -6…, write and simplify a closed rule to represent the sequence
and use this closed rule to determine the 5th, 20th, and 37th terms in the sequence.
44. Which arithmetic sequence matches the closed rule an = 7n – 4?
a. 3, 7, 11, 15, 19,…
c. 4, 7, 10, 13, 16,…
b. 3, 10, 17, 24, 31,…
d. 4, 11, 18, 25, 32,…
45. A sequence is defined by the closed rule an = 12n – 15. What is the 50th term in this sequence?
a. 7
b. 67
c. 585
d. 600
46. The first 4 terms in a sequence and -6, -1, 4, 9… What is the 14th term in the sequence?
a. 44
b. 45
c. 50
d. 59
47. What value is missing in the table?
a. 15
b. 18
c. 24
d. 36
48. What is the 20th term in the sequence 1, 4, 7, 10,…?
a. -12
b. 33
c. 58
d. 61
48 SWBAT determine if a sequence is geometric and find the common ratio.
49 SWBAT write a function rule to represent a geometric sequence and determine the terms of a
geometric sequence given a function rule.
An arithmetic sequence is a sequence where there
is a common difference between consecutive
numbers, a common number is added to each term
to get the next term.
Closed rule for an arithmetic sequence:
A(n) = a + (n – 1)d
where a(n) = nth term
a = 1st term
n = term number
d = common difference
A geometric sequence is a sequence where there
is a common ratio between consecutive numbers,
a common number multiplied by each term to get
the next term.
Function rule for an arithmetic sequence:
A(n) = arn-1
where a(n) = nth term
a = 1st term
n = term number
r = common ratio
Tell which of the following sequences is arithmetic or geometric, then list the first term and the common
difference or common ratio and write a rule to represent each sequence.
49. 3, 12, 48, 192,…
50. 2, 4, 6, 8,…
51. 80, 20, 5,
5
,…
4
52. 2, -6, 18, -54,…
53. 1, -3, -7, -11,…
54. Which equation could you use to find the next term in the pattern 3, 6, 12, 24, 48,…?
b. A(n) = 3·2n
a. A(n) = 3n-1
c. A(n) = 3(2)n-1
d. A(n) = 3n2
50 SWBAT evaluate and graph exponential functions
A function in the form y = a ·b x , where a is a nonzero constant, b is
greater than 0 and not equal to 1, and x is a real number.
Exponential Function
Evaluate each of the following exponential functions. Graph each function on a separate piece of paper.
55.
x
-1
0
1
2
3
56.
f (x) = 3·2 x
f(x)
57.
x
-1
0
1
2
3
f (x) =100·0.5x f(x)
51 SWBAT identify that exponential functions grow by equal factors over equal intervals and write
exponential functions to represent situations where this is true
52 SWBAT identify that linear functions grow by equal differences over equal intervals and write
exponential functions to represent situations where this is true
53 SWBAT determine whether a real world situation can be modeled by a linear or exponential function
and write linear and exponential functions to model and answer questions about real-world
situations
Consider each of the following situations…
If Exponential
a. Tell whether the function models
exponential growth or decay.
b. List the growth or decay factor, b, and tell
what is means in terms of the situation.
c. List the initial value, a, and tell what it
means in terms of the situation.
d. Write a function rule to model the situation.
e. Answer the question given.
If Linear
a. Tell whether the function models linear
growth or decline.
b. List the rate of change, m, and tell what it
means in terms of the situation.
c. List the initial value, b, and tell what it
means in terms of the situation.
d. Write a function rule to model the situation.
e. Answer the question given.
58. Jason deposits $500 into an account earning 10% interest.
59. You are given a $20 iTunes gift card and use the gift card to buy songs for $1.00 each.
60. The population of bald eagles was 621 in 1980 and has decreased 3% every year
61. Caylin is charged a $5 flat fee to ride a taxi plus an additional $1.50 for each mile she travels.
62. Sarah spent $100 to start her Mary Kay business. Since her initial investment, she has earned $250
per month selling Mary Kay.
63. The number of people in the United States who have a master’s degree has increased 3% per year
since 2000 when it was 1,231,590
EXTRA CREDIT
For extra practice modeling real-world situations with Linear and Exponential functions, complete
questions #1-30 on the Linear and Exponential Functions Quest (see Unit 4 Homework or Assessments).
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