Week 6: Functions, Equations, and Graphs
Function: shows the relationship between values from one set of numbers to another set.
Input: the first set of numbers.
- represented by “x” in an ordered pair
- is the independent variable
Output: the second set of numbers
- represented by “y” in an ordered pair
- is the dependent variable (it will change according to the value of the input. )
Example: Identify the dependent and independent variables in this situation. Then write an equation
to represent the relationship.
Part A: Sam is trying to figure out his total cost for the bus rides he has taken while on vacation.
Each bus ride costs $2.50.
Rides: independent variable (input)
Cost: dependent variable (output)
Equation: cost = $2.50 x the number of rides
c = 2.50r
Part B: How much will it cost for 4 bus rides?
C = 2.50(4)
C = $10.00
It will cost $10 for 4 bus rides.
You Try:
1) Janna is buying a laptop and a flash drive. The total cost, c, will include the price, p, of the laptop,
plus $12.50 for the flash drive.
Independent Variable:
Dependent Variable:
Equation: __________________________________
Find the total cost if the price of the laptop is $375.00
2) Kenneth earns $9 per hour mowing lawns. The total amount earned, t, will depend on the number
of hours, h, he works. Which equation represents this situation?
a. t = 9h
c. t = 9 + h
b. h = 9t
d. h = 9 + t
3) Hakeem wants to rent a bike. Bike rental costs $1.25 per hour plus a one-time fee of $3. Which
equation represents the relationship between the number of hours, h, Hakeem rents the bike and
the total cost, c?
a. h=1.25c + 3
c. h = 3c + 1.25
b. c = 1.25h + 3
d. c = 3h + 1.25
4) Neema purchases a gallon of milk for $3 and some tomatoes fro $2 per pound. Which equation
represents the relationship between the number of pounds, p, of tomatoes purchased and the total
amount, a, Neema spends.
a. p = 3a + 2
c. p = 2a +3
b. a = 3p + 2
d. a = 2p + 3
Function Tables:
Example: Bob’s school club washes cars to earn money. They earn $5 for each car they wash. Write an
equation to represent the amount of money the club can earn.
Equation: 5 × input = output
5x = y
 let x represent the input, and y represent the output.
Make a function table to show the amount of money Bob’s school earns washing 5,10, 15, and 20 cars.
Input (x)
Output (y)
5
25
10
50
15
75
20
100
The rule is “multiply by 5”
The input values are 5, 10, 15, 20
Each output is 5 times the corresponding input.
You can also determine a rule and write an equation when given a function table:
Example: Write an equation to show the relationship shown in this table:
Input (x)
30
35
40
45
50
Output (y)
6
7
8
9
10
Find a pattern: Think “What can I do to each ‘x’ value to find its corresponding ‘y’ value?
- Since the y values are less than the x values, you know it must be dividing or subtracting
The pattern each time is to divide x by 5 to get y.
𝒙
The equation is y = x ÷ 5 or y = 𝟓
You Try:
Write an equation to represent the relationship shown in the table. Then find the missing value in
the table:
1)
X
20
40
60
Y
23
43
X
3
4
5
Y
18
24
39
Equation: ________________________________
80
83
2)
Equation: ________________________________
6
Use the equation to complete the table:
3) y = 7x
X
1
2
3
Y
4
4) y = x – 2
X
5
8
11
14
Y
5) A bowling ally charges $1.50 for shoe rental and $2 per game. The table shows the total cost, y, of
bowling x games:
Which equation could be used to find the total cost, y of bowling x
Number of Cost, y
games?
Games, x
a.
y = 2x + 1.5
1
3.50
b.
y = 1.5x + 2
2.
5.50
c.
y=x+2
3
?
d.
y = x + 1.5
4
9.50
e.
6) What is the total cost of bowling 3 games?
a. $7.50
b. $6.50
c. $50.00
d. $4.50
7) The table below shows the price of student notebooks at a school store.
Each notebook is the same price.
Number of
Notebooks
2
4
6
8
9
Price in $
6
12
18
24
?
According to the relationship shown in the table, what would be
the price, in dollars, of 9 notebooks?
a.
25
b.
27
c.
30
d.
42
Graphs
A straight line on a coordinate plane can represent a linear function.
Remember:
- The horizontal axis is the x-axis (going across)
- The vertical axis is the y-axis (going up and down)
- Ordered pairs are always written in the form (x,y)  (input, output)
Example 1: Graph the function y = 3x on a coordinate plane.
Step 1:Make a table of values for this function. You can choose any values for x, and then determine the
corresponding y-value. It is usually best to pick x = 0,1,2,3…
X
0
1
2
3
Y
0
3
6
9
The values from this table can be
written as the ordered pairs:
(0,0), (1,3), (2,6), (3,6)
Step 2: Draw a coordinate plane.
Graph each point from the table on
the plane. Then connect the points with a straight line
going in each direction.
**Remember, the points on the linear function can
have positive and negative coordinates too. In this
example, the point (-2,-6) is on the line because
if x = -2, y = 3(-2) = -6
** In some situations, it does not make sense to have
negative values for x and y. In these cases, the graphs should appear only in quadrant one.
(Example, how far a car travels, cost for an item…)
Example 2: Graph the function y = 4x + 1
Step 1: Make a table of values
For this function, I will pick some negative values for x. I want to pick smaller numbers, because we are
going to multiply by 4 and then add 1. This will keep the
y value lower and easier to graph.
X
-2
-1
0
1
Y
-7
-3
1
5
Step 2: Graph the ordered
pairs from the table. Connect
the points with a straight line.
(-2,-7), (-1,-3), (0,1), (1,5)
You Try:
1) Draw the graph of :
y=x–2.
x
y
2) Determine the relationship between x and y from the table below. Write an equation to
represent this relationship. Then graph the equation on the coordinate plane:
x
2
3
4
5
6
y
1
1.5
2
2.5
3
Equation:
y=
More Practice:
1) There are 12 boys in a math class. The total number of students s depends on the number of girls
in the class g. Which equation represents this situation?
A)
B)
C)
D)
g = 12s
s = 12g
g = 12 + s
s = 12 + g
2) A store received a shipment of soup cans. The clerk put an equal number of cans on each of 4
shelves. Which equation represents the relationship between the total number of cans t and the
number of cans on each shelf n?
A)
B)
C)
D)
n=t÷4
n=t–4
n = 4t
n=t+4
3) Marcus sold brownies at a bake sale. He sold d dollars worth of brownies. He spent $5.50 on
materials, so his total profit p can be found by subtracting $5.50 from his earnings. Which
equation represents this situation?
A)
B)
C)
D)
p = d + 5.50
p = d – 5.50
p = 5.50d
p = d ÷ 5.50
4) A balloon rises at the rate of 12 feet, f, per second, s. Which equation represents this scenario?
a. f = 12s
c. f = 12 - s
b. f = 12 + s
d. f = s/12
5) Which table represents the scenario in the previous problem?
a.
c.
f
2
4
6
8
s
s
12 24 36 48
f
b.
12
1
24
2
36
3
48
4
s
f
1
12
2
36
3
24
4
48
t
d
1
20
2
40
3
60
4
80
d
t
1
20
2
40
3
60
4
80
d.
s
f
1
12
2
24
3
36
4
48
6) Which table represents the equation t = 20d?
a.
t
2
4
6
8
d
20 40 60 80
b.
c.
d.
d
t
2
20
4
40
6
60
8
80
7) It takes Phillip 4.5 minutes to run a mile. Let m represent the number of minutes and d represent
the number miles ran. What is the missing value in the table?
m
d
4.5 9
1
2
3
18
4
a. 3
c. 13
b. 13.5
d. 17.5
8) Which equation represents the function shown in the table?
x
y
8
4
10
6
12
8
14
10
a. y = 2x
c. y = x - 4
b. y = x/2
d. y = x +4
9) Which scenario can the graph represent?
a. Melissa earns $12 per hour
b. The temperature increased by 2 per hour
c. Zac ran a mile every 15 minutes
d. Maria folds 10 napkin every minute
10) Mel wants to graph the function represented by the table. Which ordered pair is a
point on the graph of the function?
x
y
1
8
2
16
3
24
4
32
a. (1,2)
c. (8,16)
b. (3,24)
d. (32,4)
11) A bookstore charges $4 for all children’s books. Michael graphs the function that gives
the total cost, y, in dollars to buy x number of books. Which ordered pair is a point on the
graph of the function?
a. (2,8)
c. (4,12)
b. (5,2)
d. (9,18)
12) If x < c and x = 17, which of the following could be values of c?
A. (15, 16, 17)
B. (17, 18, 19)
C. (13, 20, 22)
D. (19, 24, 27)
13)
Darryl has 5 hotdogs at home. He goes to the store and purchases more. Each pack has 8
hot dogs. Which equation displays the number of hot dogs Darryl has? Explain.
A. y = 5x + 8
B. y = 5x – 8
C. y = 8x + 5
D. y = 8x – 5
14)Which equation describes the table?
x
1
2
3
4
y
1.23
2.46
3.69
4.92
A. y = 2x
B. y = 1.23x
15)Which equation describes the table?
x
y
A. y = 5x
C. y = 2x + 5
0
5
B. y = 5x + 2
D. x = 3y + 5
2
9
3
11
7
19
16)What number should replace the question mark?
A. 6
Feet
3
6
9
15
24
Yards
1
2
3
5
?
B. 7
C. 8
D. 9
E. 12
C. y = x + 1.23
D. y = x/1.23
17)Miles works at the Apple Store for 6 hours and makes $12.25 per hour plus a sales bonus of
$18.65. How much did he make in total?
A. y = $92.15
B. y = $86.35
C. y = $78.75
D. y = $96.35
18)Which statement is true?
A. x = output, y = input
C. x = input, y = output
B. x = constant, y = coefficient
D. x = coefficient, y = constant
The graph at the right shows the cost of buying comic books from an online retailer. The shipping charge
is always the same no matter how many comic books are purchased.
Use the graph.
19)Fill in this table of values using the coordinates on the graph:
X
Y
0
1
2
3
4
16
14
12
10
20) How much does each book cost?
a. $1
b. $2
c. $3
d. $5
(2,8)
8
(1,5)
6
21)What is the shipping cost?
a. $1
b. $3
c. $2
d. $5
4
22) Which equation represents the total cost in dollars c for n
comic books?
a. c = 3n + 2
b. c = 3n
c. c = 2n + 3
d. c = 2n
2
0
0
5
23)The linear equation y=5x represents the cost y in dollars of x
pounds of dog food. Which ordered pair lies on the graph of this equation?
a. (3,15)
c. (2,3)
b. (3,10)
d. (2,12)
24)Dee is driving at an average speed of 50 miles per hour. Write an equation for the relationship that
gives the distance y, in miles that Dee drives in x hours
Equation: _____________________________
How many miles will Dee drive in 4 hours?
How many miles will Dee drive in 3½ hours?
Simplify. Follow the Order of Operations.
1. 54 ÷ 9 − 3  2
a.
b.
c.
d.
0
b. 1
c. 3
d. 6
2. 10 + 32  2 – 8
a.
b.
c.
d.
30
b. 24
c. 20
d. 14
3. Evaluate p + (n − 1)  3 for n = 7 and p = 9
a.
b.
c.
d.
51
45
c. 31
d. 27
4. Evaluate 18 − z  (i2 −8) for i = 4 and z = 2
a.
b.
c.
d.
0
2
18
128
5. Evaluate e2 + 4w ÷ 2 − 1 for e = 6 and w = 3.
a.
b.
c.
d.
41
23
20
17