Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for 8th grade linear functions? Then this Grade 8 lesson on Modeling Real Situations With Linear Functions explains each idea step by step, with worked examples.
Quick answer: A linear function y = mx + b has a constant rate of change m and an initial value b. Find m from two points as change in y over change in x, then find b by substituting a point. In context, m is the change per unit and b is the starting amount.
Standards: 8.F.B.4, 8.F.B.5 · Level: Grade 8
How do you build a linear function from a description?
A linear function is y = mx + b. The rate of change m is how much y changes per 1 unit of x. The initial value b is the output when x = 0.
A hiking club fund starts at $40 and grows by $12.50 each week. The rate is 12.5 and the initial value is 40, so y = 12.5x + 40.
How do you find the function from two points, a table or a graph?
- Find the rate: m = y2 − y1x2 − x1. In a table or graph, use any two points.
- Find the initial value: substitute one point into y = mx + b and solve for b.
- Write y = mx + b and test it against a third point.
Rate: (29 − 17) ÷ (6 − 2) = 3. Initial value: 17 = 3(2) + b, so b = 11.
The function is y = 3x + 11. Check (6, 29): 3(6) + 11 = 29.
What do the rate and the initial value mean in context?
| Model | Rate of change | Initial value |
|---|---|---|
| C = 0.45m + 28 (van rental) | $0.45 per mile | $28 base fee |
| h = 120 − 8t (draining tank) | −8 gallons per minute | 120 gallons at the start |
How do you describe a graph and sketch one?
A function is increasing when the graph rises left to right, decreasing when it falls, and constant when it is level. It is linear if the graph is a straight line.
Using the y-value of the first point as the rate. The rate is a change divided by a change, never a single output.
Beep! A decreasing linear function has a negative rate. Write the minus sign: a tank draining 8 gallons per minute has rate −8.
- Rate of change m = (y₂ − y₁) ÷ (x₂ − x₁).
- Initial value b is the output when x = 0.
- Find b by substituting a known point into y = mx + b.
- Increasing means the graph rises; decreasing means it falls; constant means level.
- A linear graph is a straight line; a nonlinear graph bends.
Modeling Real Situations With Linear Functions: frequently asked questions
How do you find the initial value of a linear function?
Substitute the rate and one known point into y = mx + b and solve for b. For example, with m = 3 and the point (2, 17), 17 = 6 + b gives b = 11.
What does the rate of change tell you in a real situation?
It says how much the output changes for each 1 unit of input, such as dollars per week or gallons per minute. A negative rate means the output is decreasing.
Can a linear function be decreasing?
Yes. When the rate of change m is negative, the function falls from left to right. A draining tank with rate −5 gallons per minute is an example.
How can you tell from a table if a function is linear?
Check that equal steps in x give equal changes in y. If the changes in y are constant, the function is linear; if they grow or shrink, it is nonlinear.
What is a qualitative graph?
It shows the general shape of a relationship, such as rising, level or falling, without exact numbers. You can sketch one from a story by marking each stage.
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