8.F.4: Model a Linear Relationship
Also written as 8.F.B.4 and CCSS.MATH.CONTENT.8.F.B.4 · Cluster: Use functions to model relationships between quantities.
The Official Wording
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Common Core State Standards, Grade 8, Functions · Read it at the source · Checked October 14, 2026
What 8.F.4 Means in Plain English
Eighth graders write the rule behind a real situation. A plumber charges $95 for 1 hour and $155 for 3 hours, so each hour costs (155 − 95) ÷ 2 = $30 and the call-out fee is $65: C = 30h + 65.
They read the rate and starting value off tables and graphs too, and say what each means: $30 for every extra hour, $65 before any work starts. High school builds functions this way for any pattern (HSF-BF.1) and linear ones from two points (HSF-LE.2).
8.F.4 I Can Statements
- I can find the rate of change from two points or a table.
- I can find the starting value, even when it is not given.
- I can explain what the slope and starting value mean in the story.
8.F.4 Worked Example
Two readings from a draining pool and nothing else, turned into a function with every number explained.
A pool drains at a steady rate. After 2 hours it holds 7,600 gallons, and after 5 hours it holds 5,800 gallons. Write a function for the water W left after t hours, and say what its numbers mean.
- Write the two facts as (x, y) pairs, time first.(2, 7600) and (5, 5800)
- Rate of change: the change in water over the change in time. It comes out negative because the water goes down.(5800 − 7600) ÷ (5 − 2) = −1800 ÷ 3 = −600 gallons per hour
- Initial value: the pool at t = 0, not at t = 2. Walk back two hours, adding 600 gallons for each one.7600 + 2 × 600 = 8800 gallons at t = 0
- Put the rate and the starting value into y = mx + b.W = −600t + 8800
- Interpret both numbers in the story and on the graph: where the line meets the vertical axis, and how steeply it falls.8800 = gallons at the start, −600 = gallons lost each hour
Check it: Substitute t = 5: −600 × 5 + 8800 = −3000 + 8800 = 5800, the second reading. Then t = 2 gives −1200 + 8800 = 7600, the first.
Common Mistakes with 8.F.4
Using a reading as the starting value
W = −600t + 7600, because 7,600 was the first number given
Try this: Ask when the first reading was taken. At t = 2 it is not the start, so walk back to t = 0 first.
Losing the negative sign
W = 600t + 8800 for a pool that is draining
Try this: Say the direction out loud before computing: water going down means a negative rate and a line falling from left to right.
Dividing the wrong way up
(5 − 2) ÷ (5800 − 7600), a rate of −1/600
Try this: Write the units of the rate before dividing. Gallons per hour means the change in gallons goes on top and hours go underneath.
Where 8.F.4 Fits
Practice 8.F.4
Quick Check
- A gym charges $40 to join plus $25 a month. Write the total cost C after m months, and say what the 25 means.
- A line passes through (0, 3) and (4, 11). What is its function?
- A table shows that 3 tickets cost $41 and 7 tickets cost $77. What is the price per ticket, and what is the fixed fee?