Learning Objectives
Find and interpret slope from two points.
Write a line equation and determine its intercepts.
Use slope to compare lines and the distance formula to measure a segment.
Prerequisites: ordered pairs, linear equations, and square roots. A coordinate graph links algebra to position: the first coordinate measures horizontal location, and the second measures vertical location.
1. Slope Is a Ratio of Changes
For points (x₁, y₁) and (x₂, y₂) with x₂ ≠ x₁, slope is m = (y₂ − y₁)/(x₂ − x₁). Keep the subtraction order consistent in numerator and denominator. Reversing both orders gives the same ratio; reversing only one changes its sign.
Worked Example: Slope and an Equation
Find the line through (2, 5) and (6, 13). The slope is (13 − 5)/(6 − 2) = 8/4 = 2.
Use point-slope form with the first point: y − 5 = 2(x − 2). Expanding gives y = 2x + 1.
Check the second point: 2(6) + 1 = 13. The y-intercept is (0, 1), and the output rises by 2 for each increase of 1 in x.
2. Intercepts and Special Lines
To find a y-intercept, set x = 0. To find an x-intercept, set y = 0. For y = 2x + 1, the x-intercept satisfies 0 = 2x + 1, so it is (−1/2, 0). An intercept is a point, not just a slope or a coefficient.
Line | Slope | Meaning |
|---|---|---|
y = k | 0 | Horizontal line |
x = k | Undefined | Vertical line |
y = mx + b | m | Nonvertical line with y-intercept b |
A vertical line is not the graph of a function y = f(x), because one input x corresponds to multiple outputs. A horizontal line is a function: each input has the same single output.
3. Compare Directions
Distinct nonvertical parallel lines have equal slopes. Nonvertical perpendicular lines with nonzero slopes have slopes whose product is −1, so each is the negative reciprocal of the other. The separate special case is a horizontal line perpendicular to a vertical line.
Worked Example: A Perpendicular Line
Find a line through (3, 1) perpendicular to y = (1/2)x + 4. Its slope must be −2, since (1/2)(−2) = −1.
Point-slope form gives y − 1 = −2(x − 3), so y = −2x + 7.
Check the point: −2(3) + 7 = 1. Equal slopes would have produced a parallel direction instead.
4. Distance and Midpoint
The distance between two points is √[(x₂ − x₁)² + (y₂ − y₁)²]. It comes from the Pythagorean theorem applied to the horizontal and vertical changes. Their midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2), found by averaging each coordinate separately.
Example: A Segment
For endpoints (−1, 2) and (5, 10), the changes are 6 horizontally and 8 vertically. The distance is √(36 + 64) = 10.
The midpoint is ((−1 + 5)/2, (2 + 10)/2) = (2, 6). Its displacement from each endpoint has length 5.
Common Mistakes
Do not confuse zero slope with undefined slope.
A negative reciprocal changes both the sign and the numerator-denominator order.
Subtract coordinates for distance, but average coordinates for midpoint.
5. Practice
1. Find the slope through (−2, 7) and (4, −5).
2. Find both intercepts of 3x + 2y = 12.
3. Write the line through (1, −3) parallel to y = 4x + 2.
4. Find the distance and midpoint between (0, 0) and (8, 6).
Worked Solutions
1. m = (−5 − 7)/(4 − (−2)) = −12/6 = −2.
2. Set y = 0 to obtain x = 4, and set x = 0 to obtain y = 6. The intercepts are (4, 0) and (0, 6).
3. The slope is 4. Hence y + 3 = 4(x − 1), or y = 4x − 7.
4. The distance is √(64 + 36) = 10. The midpoint is (4, 3).
6. Summary
- Slope compares vertical change with horizontal change.
- Use a known point and slope to write an equation, then verify it.
- Treat vertical and horizontal lines explicitly when comparing directions.
- Distance uses coordinate differences; midpoint uses coordinate averages.