5.2 Graphing and Analyzing Relations

Leanne Thompson

5.2 Graphing and Analyzing Relations

A relation is the relationship between a set of values. In the previous lesson, we expressed relations in the form of an equation, ordered pair, table of values, and on a graph. All of these forms showed us the relationship between x and y in a different way. In these relations from the previous lesson, we consider x to be the input or the independent variable and y to be the output or the dependent variable. In this lesson, we will continue to graph relations, then using the graph, we will analyze them.

Practice 1

Consider the relation $y=3x-4$ .
a) Fill out the table of values below.
b) Plot the ordered pairs on the coordinate plane. Connect the points with a line. Draw arrows at both ends of the line to indicate that it continues in both directions.

x	y	*(x, y)*
–2
–1
0
1

Blank coordinate plane. Both axes have a minimum of -12 and maximum of 12 and a scale of 2.

c) What is the value of y when x = 4? Use the extended line on your graph from part b) to help you figure this out.

d) What is the value of x when y = 11? Use the extended line on your graph from part b) to help you figure this out.

e) Algebraically verify the answers to parts c) and d) using the equation of the line.

Practice 2

Consider the equation $y=\frac{1}{2}x+1$ .
a) Fill out the table of values below.
b) Plot the ordered pairs on the coordinate plane. Connect the points with a line. Draw arrows at both ends of the line to indicate that it continues in both directions.

x

y

(x, y)

–10

–6

–2

6

c) What is the value of y when x = 4? Use the extended line on your graph from part b) to help you figure this out

d) What is the value of x when y = 6? Use the extended line on your graph from part b) to help you figure this out.

e) Algebraically verify the answers to parts c) and d) using the equation of the line.

Putting data onto a graph is beneficial because it allows you to see relations visually. Seeing real-world data in this way can be useful because it allows us to identify patterns and make predictions based on these patterns. We will now graph and analyze relations that represent real-life applications. Some examples of relations that could be graphed are the following:

The temperature and the number of plants being planted
The population of rabbits in Canada over time
How frequently a person shops for groceries and how much food they waste

List two more examples of relations.

Practice 3

Maleska recorded her child’s height and weight every year. His height, in inches, and weight, in pounds, are listed in the table below and are shown as ordered pairs in the third column.
a) Label each axis and give the graph a title.
b) Plot the points on the coordinate plane.
c) In which quadrant does this graph exist? Why does it only exist in one quadrant?

Height (in)	Weight (lb)
28	22
31	27
33	33
37	35
40	41
42	45

Quadrant 1 of a blank coordinate plane. The x-axis has a minimum of 0 and maximum of 50 and a scale of 10. The y-axis has a minimum of 0 and maximum of 50 and a scale of 10.

Practice 4

At the art gallery where he works, Archisma gets paid $200 per week plus 15% of the sales he makes, so the equation $y=200+0.15x$ gives the amount, y, that he earns for selling x dollars of artwork. Calculate the amount Archisma earns for selling $800, $1600, $2000, and $2500.
a) Fill in the table of values.
b) Label each axis and give the graph a title.
c) Plot the points, then draw a line to connect the points.

–

Quadrant 1 of a blank coordinate plane. The x-axis has a minimum of 0 and maximum of 4000 and a scale of 100. The y-axis has a minimum of 0 and maximum of 800 and a scale of 10.

d) Use your graph to find how much Archisma will earn if he sells $3000 worth of artwork.

e) Use your graph to find how much artwork Archisma would need to sell to make $800.

f) Algebraically confirm your answers from parts d) and e).

g) Were the answers you got in part f) consistent with your answers in parts d) and e)? If they were not, why do you think that was the case?

Homework

Consider the equation $y=-x-3$ .
a) Fill out the table of values below.
b) Plot the ordered pairs on the coordinate plane. Connect the points with a line. Draw arrows at both ends of the line to indicate that it continues in both directions.

x y (x, y)

–3

–1

0

2

c) What is the value of y when x = 6? Use the extended line on your graph to help you figure this out.

d) What is the value of x when y = 4? Use the extended line on your graph to help you figure this out.

e) Algebraically verify the answers to parts c) and d) using the equation of the line.
Consider the equation $y=-3x+3$ .
a) Fill out the table of values below.
b) Plot the ordered pairs on the coordinate plane. Connect the points with a line. Draw arrows at both ends of the line to indicate that it continues in both directions.

x y (x, y)

–

–

–

–

c) What is the value of y when x = 3? Use the extended line on your graph to help you figure this out.

d) What is the value of x when y = –3? Use the extended line on your graph to help you figure this out.

e) Algebraically verify the answers to parts c) and d) using the equation of the line.
Carissa recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table below and are shown as ordered pairs in the third column.
a) Label each axis and give the graph a title.
b) Plot the points on the provided coordinate plane.
c) Which quadrant does this graph exist in? Why does it only exist in one quadrant?

Age (yrs) Weight (lb)

0 7

2 11

4 15

6 16

8 19

10 20

12 21
Simone and Jon rented a minivan to go on vacation to the Yukon. It cost them $594 plus $0.32 per mile to rent the mini van, so the linear equation $y=594+0.32x$ gives the cost, y, for driving x miles. Calculate the rental cost for driving 400, 800, 1000, and 1200 miles.
a) Fill in the table of values.
b) Label each axis and give the graph a title.
c) Graph the line, then draw a line to connect the points.

– –

– –

– –

– –

– –

d) Use your graph to find approximately how much Simone and Jon will pay if they drive 200 miles.

e) Use your graph to find approximately how many miles they went if they paid $800 for the minivan.

f) Algebraically confirm your answers from parts d) and e).

g) Were the answers you got in part f) consistent with your answers in parts d) and e)? If they were not, why do you think that was the case?
Alpine Attire adjusts the cost of their Polar Vibe winter jackets as the temperature increases. To help them decide what they should price the jacket at they use the formula $C=400-2t$ , where t is the increase in temperature in degrees and C is the new cost of the jacket in dollars.
a) Fill in the table of values.
b) Label each axis and give the graph a title.
c) Graph the line, then draw a line to connect the points.

Increase in Temperature (°C) New Cost of Jacket ($)

5

10

15

20

d) What is the original cost of the jacket? Explain how you know.

e) How much would the temperature have to increase to get a 5% discount on a jacket?

f) If the jacket costs $338, how much has the temperature increased?

g) How much would the temperature have to increase for the jacket to be free? Why is this unlikely to happen?

Answers

1.

a)

x

y

(x, y)

–3

0

(–3, 0)

–1

–2

(–1, –2)

0

–3

(0, –3)

2

–5

(2, –5)

c) y = –9
d) x = –7
e) y = –9, x = –7

b)

2.

a) Answers may vary.

x	y	*(x, y)*
–2	9	(–2, 9)
–1	6	(–1, 6)
0	3	(0, 3)
1	0	(1, 0)

c) y = –6
d) x = 2
e) y = –6, x = 2

b)

3.

a) and b)

A graph that plots the points (0, 7), (2, 11), (4, 15), (6, 16), (8, 19), (10, 20) and (12, 21).

c) The graph is located in quadrant I because age and weight are always represented with positive numbers.

4.

a)

Distance Driven (mi)	Cost ($)
400	722
800	850
1000	914
1200	978

d) $658
e) 643.75 mi
f) $658, 643.75 mi
g) Answers may vary.

b) and c)

5.

a)

Increase in Temperature (°C)	New Cost of Jacket ($)
5	390
10	380
15	370
20	360

d) $400
e) 10°C
f) 31°C
g) 200°C

b) and c)

Quadrant 1 of a blank coordinate plane. The x-axis has a minimum of 0 and maximum of 35 and a scale of 1. The y-axis has a minimum of 0 and maximum of 500 and a scale of 25. With equation C=400-2t graphed.

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

License

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