Slope and Linear Functions
Grade 8 · Math · Study Guide
Understand slope as a rate of change and connect it to linear equations and graphs.
Slope measures how steep a line is — it is the rate of change, telling you how much y changes for every 1 unit increase in x. The slope formula is m = (y2 - y1) / (x2 - x1), also described as rise over run. Positive slope goes up left to right; negative slope goes down; zero slope is horizontal; undefined slope is vertical.
Slope-intercept form y = mx + b is the most useful format. Here m is slope and b is the y-intercept (where the line crosses the y-axis). To graph from this form: plot the y-intercept first, then use rise/run to find a second point, then draw the line.
Linear functions have a constant rate of change — that is what makes them linear. You can check if a relationship is linear by verifying the rate of change between any two points is always the same. Real-world examples include distance-time graphs and cost-per-item pricing.
| Slope Type | Value of m | Description | Example equation |
|---|---|---|---|
| Positive | m > 0 | Line rises left to right | y = 2x + 1 |
| Negative | m < 0 | Line falls left to right | y = -3x + 4 |
| Zero | m = 0 | Horizontal line | y = 5 |
| Undefined | No value | Vertical line | x = 2 |
Key terms
- Slope (m)
- Rate of change of a line; m = (y2-y1)/(x2-x1) = rise/run.
- Y-intercept (b)
- Where a line crosses the y-axis; y-value when x = 0.
- Slope-intercept form
- y = mx + b; m is slope and b is y-intercept.
- Rise over run
- Slope as vertical change divided by horizontal change.
- Rate of change
- How much one quantity changes per unit of another; constant for linear functions.
- X-intercept
- Where a line crosses the x-axis; value of x when y = 0.
Flashcards
- Slope formula
- m = (y2 - y1) / (x2 - x1) = rise / run
- Slope-intercept form
- y = mx + b; m = slope, b = y-intercept.
- Positive slope
- Line rises from left to right; m > 0.
- Negative slope
- Line falls from left to right; m < 0.
- Zero slope
- Horizontal line; y never changes; m = 0.
- Undefined slope
- Vertical line; x never changes; slope is undefined.
Practice questions
What is the slope through (1, 3) and (4, 9)?
- 1
- 2
- 3
- 6
Show answer
Answer: 2
m = (9-3)/(4-1) = 6/3 = 2.
In y = -4x + 7, the slope is 7.
Show answer
Answer: False
In y = mx + b, m is slope. Here m = -4 and b = 7. The slope is -4.
In y = mx + b, the variable b represents the ___.
Show answer
Answer: y-intercept
In y = mx + b, b is the y-intercept — where the line crosses the y-axis.
Which equation has slope 3 and y-intercept -2?
- y = -2x + 3
- y = 3x - 2
- y = 2x + 3
- y = 3x + 2
Show answer
Answer: y = 3x - 2
m = 3 and b = -2 gives y = 3x - 2.
A horizontal line has slope:
- 1
- -1
- 0
- undefined
Show answer
Answer: 0
Horizontal line has no rise for any run: slope = 0/run = 0.
Fun facts
- Road grades are slopes as percentages! A 6% grade means the road rises 6 feet for every 100 feet of horizontal distance — a slope of 0.06.
- The steepest street in the world is in Dunedin, New Zealand with a 35% grade — a slope of 0.35!
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