These 23 problems practice Common Core standards 8.F.B.4 and 8.F.B.5. Find the rate of change and initial value, write linear functions from tables, graphs, points and stories, and describe or sketch graphs.
The last six are word problems that ask for a model, an interpretation and a prediction. Work from easiest to hardest.
Problem 1: Rate from a table
The table shows a linear function. Find the rate of change.
| \(x\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|
| \(y\) | \(4\) | \(7\) | \(10\) | \(13\) |
Problem 2: Read the equation
The equation \(y=5x+8\) is linear. Identify (a) the rate of change and (b) the initial value.
Problem 3: Slope between two points
Find the rate of change of the line through \((1,3)\) and \((4,12)\).
Problem 4: A negative rate
Find the rate of change of the line through \((2,9)\) and \((6,1)\). Is the function increasing or decreasing?
Problem 5: Linear or nonlinear?
Decide whether each table shows a linear function. Give a reason.
Table A:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| \(y\) | \(2\) | \(4\) | \(8\) | \(16\) |
Table B:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| \(y\) | \(3\) | \(7\) | \(11\) | \(15\) |
Problem 6: Increasing or decreasing
For each graph, state whether the function is increasing or decreasing, and whether it is linear or nonlinear.
Problem 7: Equation from a table
The table shows a linear function. Write an equation in the form \(y=mx+b\).
Problem 8: Equation from a graph
Write the equation of the line shown in the form \(y=mx+b\).
Problem 9: Two points to equation
Write the equation of the line through \((1,5)\) and \((3,11)\).
Problem 10: A fractional rate
Write the equation of the line through \((3,4)\) and \((9,8)\).
Problem 11: A point and a rate
A line has rate of change \(-2\) and passes through \((4,10)\). Write its equation.
Problem 12: The burning candle
A candle is 12 inches tall when lit and burns down 1.5 inches each hour. Write a function for the height \(h\) (in inches) after \(t\) hours, then find the height after 4 hours.
Problem 13: Read a story graph
The graph shows the depth of water in a bathtub. Describe what happens in each part and find the rate of change of the depth during the first 4 minutes and during the last 2 minutes of drainage.
Problem 14: Match the graph to the story
Mei walks away from home at a steady pace, stops to rest, and then walks straight home. She walks home faster than she walked away. Which graph shows her trip? Explain.
Problem 15: Sketch a trip
Priya walks 0.5 mile from home to the library in 10 minutes at a steady pace, stays 20 minutes, and walks home in 10 minutes. Sketch a graph of her distance from home against time.
Problem 16: Sketch a cooling curve
A bowl of soup cools quickly at first and then more and more slowly until it reaches room temperature. Sketch a graph of temperature against time and tell whether the function is linear.
Problem 17: Taxi fare
A taxi charges a flat fee of $3.50 plus $2 per mile. (a) Write a function for the cost \(C\) of a ride of \(m\) miles. (b) Identify the rate and initial value. (c) Find the cost of a 9-mile ride.
Problem 18: Ethan's savings
Ethan saves the same amount every week. After 3 weeks he has $120, and after 6 weeks he has $165. (a) Write a function for his savings \(y\) after \(x\) weeks. (b) What do the rate and initial value mean? (c) After how many weeks will he have $300?
Problem 19: A draining tank
A tank drains at a constant rate. After 3 hours it holds 240 gallons, and after 7 hours it holds 160 gallons. (a) Write a function for the gallons \(y\) after \(x\) hours. (b) How much water was in the tank at the start? (c) When will the tank be empty?
Problem 20: Kayak rental graph
The graph shows the cost of renting a kayak. (a) Find the rate of change and the initial value. (b) Write the function. (c) Find the cost of renting for 7 hours.
Problem 21: Filling a pool
The table shows the gallons \(g\) in a small pool after \(t\) minutes of filling.
| \(t\) | \(2\) | \(4\) | \(6\) | \(8\) |
|---|---|---|---|---|
| \(g\) | \(50\) | \(80\) | \(110\) | \(140\) |
(a) Show that the function is linear. (b) Write the function. (c) Predict the gallons after 25 minutes.
Problem 22: Two dog walkers
Sofia earns \(S=9h+12\) dollars for \(h\) hours of dog walking. Noah’s earnings \(N\) are shown below.
| \(h\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|
| \(N\) | \(\$17\) | \(\$24\) | \(\$31\) |
Write a function for Noah, then find who earns more for 10 hours and by how much.
Problem 23: Area or perimeter?
A square has side length \(s\) inches.
| \(s\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| Perimeter \(P\) | \(4\) | \(8\) | \(12\) | \(16\) |
| Area \(A\) | \(1\) | \(4\) | \(9\) | \(16\) |
Which relationship is linear? Write the linear function and name its rate of change and initial value. Explain why the other is nonlinear.
See every chapter on the Grade 8 math page.