Rate of change and initial value worksheet

Grade 8 math, Common Core 8.F.B.4. 10 problems with an answer key and worked steps. Free to print, and you can make new problems any time.

By the StudyWiz teamUpdated October 23, 2026How we write our pages

Print the answer key

NameDate

Rate of change and initial value

Grade 8 math, Common Core 8.F.B.4

Find the rate of change and the initial value, then write y = mx + b.

1.
A line goes through (1, 6) and (2, 9).
2.
A tank holds 33 gallons of water and drains 3 gallons each minute. Let x be the minutes and y the gallons left.
3.
A tank holds 27 gallons of water and drains 3 gallons each minute. Let x be the minutes and y the gallons left.
4.
Use the table.
x 0 2 4 6
y 17 5 −7 −19
5.
Use the table.
x 0 2 4 6
y 13 9 5 1
6.
A line goes through (4, 19) and (5, 21).
7.
Use the table.
x 0 2 4 6
y 25 17 9 1
8.
Use the table.
x 0 2 4 6
y 22 14 6 −2
9.
A line goes through (1, −3) and (5, 5).
10.
Use the table.
x 0 2 4 6
y 40 52 64 76

x	0	2	4	6
y	17	5	−7	−19

x	0	2	4	6
y	13	9	5	1

x	0	2	4	6
y	25	17	9	1

x	0	2	4	6
y	22	14	6	−2

x	0	2	4	6
y	40	52	64	76

Answer key: Rate of change and initial value

Grade 8 math. Each answer has a few worked steps, to help check the method as well as the answer.

1. Rate 3, initial value 3: y = 3x + 3
Rate: (9 − 6) ÷ (2 − 1) = 3b = 6 − 3(1) = 3
2. Rate −3, initial value 33: y = −3x + 33
3. Rate −3, initial value 27: y = −3x + 27
4. Rate −6, initial value 17: y = −6x + 17
y changes by −12 for every 2 in x, so the rate is −6
5. Rate −2, initial value 13: y = −2x + 13
y changes by −4 for every 2 in x, so the rate is −2
6. Rate 2, initial value 11: y = 2x + 11
Rate: (21 − 19) ÷ (5 − 4) = 2b = 19 − 2(4) = 11
7. Rate −4, initial value 25: y = −4x + 25
y changes by −8 for every 2 in x, so the rate is −4
8. Rate −4, initial value 22: y = −4x + 22
y changes by −8 for every 2 in x, so the rate is −4
9. Rate 2, initial value −5: y = 2x − 5
Rate: (5 − (−3)) ÷ (5 − 1) = 2b = −3 − 2(1) = −5
10. Rate 6, initial value 40: y = 6x + 40
y changes by 12 for every 2 in x, so the rate is 6

The answer key prints on its own page. You’re welcome to print and copy these worksheets for your own kids or class.

About rate of change and initial value

Many real situations are linear: a starting amount plus a constant rate, like a $50 fee plus $30 an hour. The rate of change is m, how much y changes for each 1 of x. The initial value is b, the value of y when x is 0. Together they give the equation y = mx + b.

Worked example 1

A line goes through (1, 3) and (3, −7).

Step 1.Rate: (−7 − 3) ÷ (3 − 1) = −5
Step 2.b = 3 − (−5)(1) = 8

Answer: Rate −5, initial value 8: y = −5x + 8

Example 2

A tank holds 32 gallons of water and drains 3 gallons each minute. Let x be the minutes and y the gallons left.

Answer: Rate −3, initial value 32: y = −3x + 32

Example 3

A plumber charges a $30 visit fee plus $39 an hour. Let x be the hours and y the total cost.

Answer: Rate 39, initial value 30: y = 39x + 30

Watch out for

The initial value is what you have before anything happens (x = 0).
Words like “per”, “each” and “every” point to the rate of change.
A decreasing amount, like water draining, has a negative rate of change.

Practice rate of change and initial value online

On StudyWiz, your kid can practice rate of change and initial value with 5-question quizzes that are checked as they go, with an explanation for every answer. When a homework problem is the trouble, Wiz gives a hint for each step instead of the answer.

Keep reading

Try Wiz with tonight’s homework

Free to start, no card needed. Each kid gets 3 homework questions, 5 practice quizzes and 1 writing review a day on the free plan.

Start free See pricing