non proportional equations and graphs

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Grade 8 Mathematics

Teacher Facing
February 12, 2025

Unit 13: Proportional and Nonproportional Relationships

Lesson 6: Graphs and Equations of Linear Relationships

Do Now

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●Plot the following points on the
graph: (–2 , 3); (0, 1); (2, –1); (4, –3).

●Connect the points with a line.

●Does the graph represent a
proportional relationship? Explain.

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LO: SWBAT represent linear non-proportional
situations with graphs and equations in the form
of y = mx + b, where b ≠ 0.

DOL: Given a set of problems, students will
represent linear non-proportional situations with
graphs and equations in the form of y = mx + b,
where b ≠ 0 in at least 4 of 5 questions.

MATH.8.5B Represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, b ≠ 0 .

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Run = Change in x

Linear Function
y = mx + b
m = slope
b = y-intercept

Slope =

Rise = Change in y

Academic Vocabulary

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Academic Vocabulary

Linear Equation

y = mx + b

rate of
change

initial
value

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Academic Vocabulary

Slope/Rate of Change

(x1 y1), (x2, y2)

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In the last unit, we

explored proportional

relationships and

rate of change.
In what form are

equations of
proportional

relationships written?

What does the
“k” represent?

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Today, we will explore
non-proportional linear

relationships.

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Sarah runs a business selling homemade candles. She charges $5
per candle. If x represents the number of candles sold and y
represents the total earnings, what equation represents this
relationship?

Sarah runs a business selling homemade candles. She charges
$5 per candle, but there is also a one-time setup fee of $10 for
custom orders. If x represents the number of candles sold and
y represents the amount of money earned, what equation
represents this relationship?

How are the verbal descriptions
and equations similar? How are

they different?

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Sarah runs a business selling homemade
candles. She charges $5 per candle, but
there is also a one-time setup fee of $10
for custom orders. If x represents the
number of candles sold and y represents
the amount of money earned.

Bottles

Purchased 0

1

2

3

4

Total Cost

($)

What are the

characteristics of
the graph we just

created?

Where do we see
components of a
linear relationship

on the graph?

number of candles

total cost ($)

x

y

y – intercept

slope – intercept form

Slope

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Time (Seconds)

Altitude (meters)

(5, 640)

(15, 620)

(35, 580)

(45, 560)

What is the rate of the balloon's descent?

Why is the rate of
change negative?

Write an equation to represent the altitude in
meters, y, over time, x, in seconds.

A hot air balloon starts at 650 meters in the air makes its descent at a constant rate.
The graph shows the altitude in meters of the ballon over time in seconds.

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Ella runs an ice cream stand where she pays a fixed fee for her stall, in
addition to extra costs for each flavor of ice cream she buys. The graph
represents the relationship between y, her total operating costs, and x,
the number of flavors of ice cream she purchases.

Write an equation to represent the relationship.

Step 1: Identify the initial value.

ice cream flavors

total operating cost ($)

Step 2: Determine the slope.

Step 3: Write equation in the

form y = mx + b.

x

y

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Aiden is draining a pool for cleaning. There’s a fixed amount of water at
the start, and the water level decreases as he pumps it out. The graph
shows the relationship between y, the water left in the pool, and x, the
number of hours he pumps the water.

Which equation best represents the
relationship?

A. y = 500x + 1,000

B. y = –1,000x + 5,000

C. y = –500x + 5,000

D. y = 1,000x – 5,000

time (hours)

gallons of water

x

y

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Graph the relationship
between x and y.
y = 3x – 2

x

y

–2

–1

0

1

2

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Which equation best
represents the relationship?

A. y = 2x + 1

B. y = –2x – 1

C. y = –2x + 1

D. y = 2x – 1

x

y

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Chelsea opens a savings account and starts with
a set amount of money. However, each month,
she withdraws money to pay for her summer
trips. The graph shows the relationship between
y, the total amount of money left in her account,
and x, the number of months since she began
making withdrawals.

Which statements are true about Chelsea’s
account? Select TWO correct answers.

A.The initial amount in the account was $500.

B.The initial amount in the account was $750.

C.Chelsea withdrew $750 per month.

D.Chelsea deposited $25 a month.

E.Chelsea withdrew $25 a month.

months

account balance ($)

x

y

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The graph shows the relationship
between y, the total cost of renting a
kayak, and x, the number of hours
the kayak is rented.

●How much would it cost to rent
the kayak for 7.2 hours?

●How many hours is the kayak
rented for if the total cost of the
rental is $47?

time (hours)

total cost ($)

x

y

Hint: Write an

equation in the form

y = mx + b.

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Let’s practice!

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Gemma has $5 in her bank account and adds $30 dollars every month.
Based on this information, which representation best shows this relationship
between the amount in Gemma’s bank account, y, and the number of months
that have passed, x?

Months

Months

Amount in Bank Account

Months

Amount in Bank Account

Amount in Bank Account

A.

B.

C.

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Gizelle has a large container filled with water. She is draining the
tank at a constant rate. What value represents the rate of change?

–10 cups per minute

B.

–1 cup per minute

C.

–5 cups per minute

D.

–6 cups per minute

Water Capacity

Minutes

Cups of Water

How many cups of water did

Gizelle start with?

x

y

A

B

C

Which graph has a rate of change of 2?

x

y

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Lila is cleaning her fish tank. She starts with a full tank of water, and as she uses
a siphon to drain the water, the level gradually decreases. The graph represents
the relationship between y, the amount of water left in the tank, and x, the
number of hours she spends siphoning the water out.

Which equation best represents the
relationship?

A. y = 20x + 50

B. y = –10x + 50

C. y = –50x + 10

D. y = –20x – 50

time (hours)

gallons of water

x

y

A. y = 6x + 1

B. y = 3x + 1

C. y = 2.5x + 1

D. y = 5x + 1

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Which equation best
represents the
relationship between
x and y in the graph?

x

y

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Jesse had a collection of 24 comic books. He sold 6 comic books an
hour at a garage sale until his entire collection was sold. Which
graph best represents the situation?

A.

B.

C.

D.

time (hours)

comic books

time (hours)

time (hours)

time (hours)

comic books

comic books

comic books

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Jared borrowed money from
his sister. He pays her back
the same amount every week
until he has paid her back.

● According to the graph,

how much money did Jared
borrow from his sister?

● How much money did he

give her every week?

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Bailey goes to the movies
with her siblings. She
purchases a large popcorn
and each of their movie
tickets. The graph
represents the relationship
between y, the total cost
and x number of tickets.

Write an equation that
represents the relationship
between x and y.

tickets

total cost ($)

x

y