3B Rate of Change

​Rate of change (SLOPE): a RATE that describes how one quantity CHANGES in relation to ANOTHER quantity

​​y = mx + b
OR
f(x) = mx + b
where m represents slope

​

​3B: calculate the rate of change of a linear function represented as a table, graph, or in real-world (word) problems

​Printer ink costs $18.99 per cartridge. The company that sells the ink charges a flat rate shipping fee of $7.95, no matter the number of cartridges purchased in a single order.

m = 18.99 since the cost is $18.99 PER cartridge

​GRAPH

Drag and Drop

The water level in a tank is changing at a constant rate. The graph shows the relationship between time in hours, x, and the water level in meters, y.



Complete the statement that describes the rate of change of the water level with respect to time.



The water level ​

Placeholder

at a rate of ​

Placeholder

m/h.

Drag these tiles and drop them in the correct blank above

decreases

increases

0.25

1

4

Multiple Choice

The graph of a line is shown on the grid. The coordinates of both points indicated on the graph of the line are integers.

What is the rate of change of y with respect to x for this line?

• $$\frac{5}{6}$$

• $$-\frac{6}{5}$$

• $$\frac{2}{3}$$

• $$-\frac{5}{6}$$

Multiple Choice

The graph of a linear function is shown on the grid.

What is the rate of change of y with respect to x for this function?

• $$\frac{7}{9}$$

• $$-\frac{7}{9}$$

• $$\frac{3}{4}$$

• $$-\frac{3}{4}$$

Math Response

The graph of a linear function is shown on the grid.

What is the rate of change of y with respect to x for this function?

Type answer here

Correct Answer: -0.2

Deg^°

Rad

Dropdown

A savings account balance can be modeled by the graph of the linear function shown on the grid.

The rate of change of the balance with respect to the number of deposits is ​

Select answer

per deposit.

TABLE

​x	y​
​-4	​10
-2​	7​
6​	-5​

​x	f(x)
​-3	-​130
2​	0​
5​	78
11	234

Math Response

A construction worker built steps up to a home. The table shows the linear relationship between the height off the ground in inches and the number of steps.

What is the rate of change of height off the ground with respect to the number of steps?

Type answer here

Correct Answer: 7

Deg^°

Rad

Dropdown

A worker is packing items in boxes. The table shows the linear relationship between the number of items the worker has packed in boxes after different amounts of time.



Complete the statement to describe the rate of change of the number of items the worker packed in boxes with respect to the number of minutes the worker has been packing items in boxes.



The worker packed ​

Select answer

items in boxes per minute.

Math Response

The table shows the linear relationship between the distance in feet below sea level and the time in seconds traveled by a submarine.

What is the rate of change of the distance in feet below sea level with respect to time that the submarine traveled?

Type answer here

Correct Answer: 8

Deg^°

Rad

Multiple Choice

The values in the table represent a linear relationship between x and y.

What is the rate of change of y with respect to x ?

• 10

• 17

• -10

• -17

Dropdown

A contractor's total earnings from a job include a fixed amount plus an amount based on the number of hours worked. The values in the table represent the linear relationship between the number of hours worked and the contractor's total earnings in dollars.



The rate of change of the contractor's total earnings in dollars with respect to the number of hours worked is ​

Select answer

per hour worked.

Multiple Choice

The table shows the linear relationship between the average height in feet of trees on a tree farm and the number of years since the trees were planted.

What is the rate of change of the average height in feet of the trees on the farm with respect to the number of years since the trees were planted?

• 14 ft/yr

• 7 ft/yr

• 3 ft/yr

• 10 ft/yr

​The cost of dance lessons offered at Wayside are determined by the following:

• A $30 registration fee plus $45 per month

WORDS

​EACH, PER, FOR EVERY, RATE, CONSTANT RATE, RECURRING

​Since the cost is $45 PER month,
45 is the rate of change (slope).

Since $12.95 is a RECURRING monthly fee
(meaning $12.95 PER month),
12.95 is the rate of change (slope).

There is a recurring monthly fee of $12.95 to join the streaming service UpStream. You can also purchase movies for an additional $3.99.

Math Response

The function y = 3.75 + 1.5(x -1) can be used to determine the cost in dollars for a taxi ride of x miles.

What is the rate of change of the cost in dollars with respect to the number of miles?

$_____ per mile

Type answer here

Correct Answer: 1.5

Deg^°

Rad

Math Response

Printer ink costs $18.99 per cartridge. The company that sells the ink charges a flat rate shipping fee of $7.95, no matter the number of cartridges purchased in a single order.

What is the rate of change of the cost in dollars with respect to the number of ink cartridges purchased?

Type answer here

Correct Answer: 18.99

Deg^°

Rad

Math Response

The cost of dance lessons offered at Wayside are determined by the following:

• A $30 registration fee plus $45 per month.

What is the rate of change of the cost of dance lessons at Wayside with respect to the number of months of participation?

Type answer here

Correct Answer: 45

Deg^°

Rad