← Back to the Linear Functions worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: 8.F.B.4, 8.F.B.5 · Level: Grade 8
Problem 1 Read the equation
(a) Compare with mx + b: rate 6, initial value 15. (b) Rewrite as c = −8t + 120: rate −8, initial value 120.
(a) 6 and 15 (b) −8 and 120
Problem 2 Table to function
The outputs rise by 6 for every step of 1, so the rate is 6. At x = 0 the output is 20.
Rate 6; initial value 20; y = 6x + 20
Problem 3 Two points
Rate = (21 − 9) ÷ (5 − 2) = 4. Substitute (2, 9): 9 = 4(2) + b, so b = 1. Check (5, 21): 4(5) + 1 = 21.
y = 4x + 1
Problem 4 Gym fees
The rate is $14 per month and the initial value is $20. So y = 14x + 20. For x = 6: 14(6) + 20 = 104.
y = 14x + 20; $104
Problem 5 Equation from a graph
The line crosses the y-axis at (0, 7), so b = 7. The rise from (0, 7) to (3, 13) is 6 over a run of 3, so m = 2.
y = 2x + 7
Problem 6 Describe the function
(a) The rate −3 is negative, so it is decreasing and linear. (b) Values 0, 2, 8, 18 grow by increasing amounts: increasing and nonlinear. (c) The output never changes: constant, which is linear.
(a) decreasing, linear (b) increasing, nonlinear (c) constant, linear
Problem 7 Negative coordinates
Rate = (−7 − 11) ÷ (4 − (−2)) = −18 ÷ 6 = −3. Substitute (−2, 11): 11 = −3(−2) + b = 6 + b, so b = 5. Check (4, −7): −12 + 5 = −7.
y = −3x + 5
Problem 8 Table with gaps
From x = 1 to 3 the output rises 10 over a run of 2, so the rate is 5. Then 9 = 5(1) + b gives b = 4. Check: 5(7) + 4 = 39.
y = 5x + 4
Problem 9 Van rental meaning
The 0.45 is the rate: each mile costs $0.45. The 28 is the initial value: a $28 base fee even at 0 miles. For 120 miles: 0.45(120) + 28 = 54 + 28 = 82.
$0.45 per mile; $28 base fee; $82
Problem 10 Draining tank
Rate = (35 − 60) ÷ (5 − 0) = −5 gallons per minute. Initial value 60. So y = −5x + 60. The tank is empty when −5x + 60 = 0, so x = 12.
Rate −5; initial 60; y = −5x + 60; empty after 12 minutes
Problem 11 Drone climb
Rate = (54 − 30) ÷ (5 − 2) = 8 feet per second. Then 30 = 8(2) + b gives b = 14. So y = 8x + 14, and at 10 seconds: 8(10) + 14 = 94.
y = 8x + 14; 94 feet
Problem 12 Cyclist’s trip
A: distance increases steadily. B: the graph is level, so the cyclist is resting. C: distance increases again, and the segment is steeper than A, so the cyclist rides faster.
A increasing, B constant, C increasing faster
Problem 13 Sketch a story
The height starts high and falls at a steady rate, so the graph is a straight line sloping down from the vertical axis to the horizontal axis.
Decreasing and linear
Problem 14 Missing values
From 7 = 4(1) + b, b = 3, so y = 4x + 3. At x = 6: 24 + 3 = 27. For y = 51: 4x + 3 = 51, so 4x = 48 and x = 12.
y = 27; x = 12
Problem 15 Ride-share fare
Rate = (19 − 11.5) ÷ (6 − 3) = 7.5 ÷ 3 = 2.50. Then 11.5 = 2.5(3) + b gives b = 4. So y = 2.5x + 4, and 14 miles cost 2.5(14) + 4 = 39.
$2.50 per mile; $4.00 starting fee; $39.00
Problem 16 Phone battery
Rate = (58 − 82) ÷ (5 − 2) = −8 percent per hour. Then 82 = −8(2) + b gives b = 98. So y = −8x + 98. After 10 hours: −80 + 98 = 18. At 0%: x = 98 ÷ 8 = 12.25.
y = −8x + 98; 18%; after 12.25 hours
Problem 17 Is a line a good model?
The changes in y are 3, 6 and 12. They are not constant, so a linear function does not fit. Each output doubles, so the function is increasing and nonlinear.
Not linear; increasing and nonlinear (outputs double)
Problem 18 A ball in the air
The height increases quickly at first, then more slowly, reaches the highest point, and then decreases. The graph bends, so the function is nonlinear. At the highest point the ball stops rising and starts to fall.
Increasing then decreasing; nonlinear
Problem 19 Savings race
Noah’s rate is (100 − 20) ÷ 4 = 20 and his initial value is 20, so y = 20x + 20. Set 15x + 60 = 20x + 20. Then 40 = 5x and x = 8. Savings: 15(8) + 60 = 180.
Week 8; $180 each
Problem 20 Candle burn time
Rate = (13.5 − 18) ÷ 3 = −1.5 inches per hour, so h = −1.5t + 18. The height is 0 when t = 18 ÷ 1.5 = 12 hours. At 15 hours the formula gives 18 − 22.5 = −4.5, a negative height, which is impossible.
h = −1.5t + 18; 12 hours; the model only applies for 0 ≤ t ≤ 12