Graphing Linear Functions
Grade 9 · Math · Study Guide
Graph lines using slope-intercept form, find slope from two points, and interpret graphs in real-world contexts.
A linear function creates a straight line when graphed on the coordinate plane. The most useful form for graphing is slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis). To graph, plot the y-intercept first, then use the slope to find a second point.
Slope measures the steepness and direction of a line. It is calculated as rise over run: m = (y₂ − y₁) / (x₂ − x₁). A positive slope rises from left to right; a negative slope falls. A horizontal line has slope 0; a vertical line has undefined slope.
Standard form Ax + By = C can be converted to slope-intercept form by solving for y. Two lines are parallel if they have the same slope and different y-intercepts. They are perpendicular if their slopes are negative reciprocals (m₁ × m₂ = −1).
| Slope Value | Line Direction | Example |
|---|---|---|
| Positive (m > 0) | Rises left to right | y = 2x + 1 |
| Negative (m < 0) | Falls left to right | y = −3x + 4 |
| Zero (m = 0) | Horizontal | y = 5 |
| Undefined | Vertical | x = 3 |
Key terms
- Slope-intercept form
- y = mx + b, where m is slope and b is the y-intercept
- Slope
- The rate of change of a line; rise divided by run
- Y-intercept
- The point where the line crosses the y-axis; (0, b)
- X-intercept
- The point where the line crosses the x-axis; (a, 0)
- Parallel lines
- Lines with the same slope but different y-intercepts — they never intersect
- Perpendicular lines
- Lines whose slopes are negative reciprocals; they meet at 90°
Flashcards
- Slope-intercept form
- y = mx + b; m = slope, b = y-intercept
- Slope formula
- m = (y₂ − y₁) / (x₂ − x₁)
- Y-intercept
- Where the line crosses the y-axis; the b in y = mx + b
- Positive slope
- Line rises from left to right
- Negative slope
- Line falls from left to right
- Parallel lines
- Same slope, different y-intercept — never intersect
Practice questions
What is the slope of the line y = −2x + 5?
- 5
- −2
- 2
- −5
Show answer
Answer: −2
In y = mx + b, the coefficient of x is the slope. Here m = −2.
The y-intercept of y = 3x − 4 is (0, 3).
Show answer
Answer: False
The y-intercept is (0, −4) — substituting x = 0 gives y = −4.
The slope between points (1, 2) and (3, 8) is ___.
Show answer
Answer: 3
m = (8 − 2)/(3 − 1) = 6/2 = 3.
Which equation represents a line parallel to y = 2x + 1?
- y = −2x + 3
- y = 2x − 5
- y = 1/2 x + 1
- y = −1/2 x + 4
Show answer
Answer: y = 2x − 5
Parallel lines have the same slope. Only y = 2x − 5 has slope 2.
What is the y-intercept of the line 3x + y = 9?
- (3, 0)
- (0, 9)
- (0, 3)
- (9, 0)
Show answer
Answer: (0, 9)
Set x = 0: y = 9. The y-intercept is (0, 9).
Fun facts
- Every time you look at a road grade sign (like '6% grade'), you are reading a slope — rise of 6 feet for every 100 feet traveled.
- GPS navigation systems use linear equations constantly to calculate your position between known coordinates.
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