Linear functions and rate of change
How does a line's change per step appear in its equation, table, and graph?
Adjust slope and intercept while the graph, coordinates, and value table update together; see what each parameter controls in y = mx + b.
Learning goals
- Interpret slope as the change in y when x increases by 1, using the m = 0 constant case as a comparison.
- Identify the y-intercept (0, b) and calculate coordinates from y = mx + b.
- Connect an equation, a value table, and a straight-line graph.
Before you begin
- Read and plot ordered pairs on Cartesian axes.
- Substitute positive and negative numbers and calculate products and sums.
Linear function experiment
- 1Start at intercept
- 2Positive slope
- 3Negative slope
- 4Parallel lines
| x | y |
|---|---|
| -2 | 2.0 |
| -1 | 2.0 |
| 0 | 2.0 |
| 1 | 2.0 |
| 2 | 2.0 |
Observe, then explain
Work through four steps. Each button moves the interactive model to the matching observation state.
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Begin with m = 0: every input gives the same output b, making a horizontal line through (0, b). This constant case isolates the role of slope; the Chinese middle-school definition of a first-degree function requires m ≠ 0.
Your task: Change only b to move the horizontal line, then make m non-zero to compare it with a linear function.
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m = 2 means y rises by 2 whenever x rises by 1. Adjacent table outputs have the same difference and the line rises left to right.
Your task: Set m = 2 and b = 1; verify (0,1), (1,3), and (2,5).
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When m is negative, y decreases as x increases. The sign sets direction while |m| gives the size of each step.
Your task: Set m = −1.5 and compare y at x = 0, 1, and 2.
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Keeping m fixed while changing b adds the same amount to every y-value, producing parallel lines with the same steepness.
Your task: Keep m = 1, vary b, and name what stays the same and what changes.
Build an explanation
Slope is a ratio
Between two points, m = Δy/Δx. A linear function has constant slope, so equal x-steps produce equal y-changes.
b locates the start
At x = 0, mx = 0, so y = b. The coordinate (0,b) is where the line crosses the y-axis.
Three representations, one relationship
The equation is a rule, the table samples inputs and outputs, and the graph shows all real coordinates satisfying the rule. A finite table is not the whole line, but every row should lie on it.
Check your understanding
Printable worksheet
Complete the worksheet first. Answers and explanations are on a separate page.
Mathematics
Linear functions and rate of change
How does a line's change per step appear in its equation, table, and graph?
- In y = 3x − 2, what happens to y when x increases by 1?
A. It increases by 3 B. It decreases by 2 C. It increases by 1
- Where does y = −2x + 4 cross the y-axis?
A. (0, 4) B. (4, 0) C. (0, −2)
- For y = 1.5x + 1, what is y when x = 2?
A. 2.5 B. 4 C. 5
- Which different line is parallel to y = 2x + 1?
A. y = 2x − 3 B. y = −2x + 1 C. y = 3x + 1
Answer page
Linear functions and rate of change
- It increases by 3
The slope m = 3 is Δy for Δx = 1. The intercept −2 sets the line's vertical position.
- (0, 4)
On the y-axis x = 0; substituting gives y = 4, so the intercept is (0,4).
- 4
Substitute x = 2: y = 1.5 × 2 + 1 = 4.
- y = 2x − 3
Distinct parallel lines have equal slopes and different intercepts; both equations have slope 2.