3.5Rate of Change & Slope
Slope is the language of change. Whether you're reading a graph, scanning a table, or working from two points, rise over run tells you exactly how fast — and in what direction — a quantity is changing.
Why This Matters
Slope measures how fast something changes — speed, steepness, growth rate. Engineers use it to design roads and ramps, economists use it to analyze trends, and physicists use it to calculate velocity. It's one of the most applied concepts in all of math.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does slope describe the rate of change of a linear relationship, and how can you find it from a graph, a table, or two points?
Lesson Overview
The rate of change describes how much one quantity changes relative to another. In a linear relationship, the rate of change is constant — it stays the same between any two points. This constant rate of change is called the slope. Slope measures the steepness and direction of a line and is calculated as rise over run: the vertical change (rise) divided by the horizontal change (run) between any two points on the line. You can find slope three ways: (1) from a graph by counting rise and run; (2) from a table by dividing Δy by Δx; and (3) from two points using the slope formula m = (y₂ − y₁) / (x₂ − x₁).
The Slope Formula
m = Δy / Δx
= rise / run
m = (y₂−y₁) / (x₂−x₁)
two-point formula
Slope types:
↗ Positive: rises left→right
↘ Negative: falls left→right
→ Zero: horizontal line
↕ Undefined: vertical line
Four Types of Slope
Positive
rises left → right
Negative
falls left → right
Zero
horizontal line
Undefined
vertical line
Slope Triangles — Rise over Run
Points (1,2) and (5,10)
m = (10 − 2) / (5 − 1) = 8 / 4 = 2
Rise = 8 (up), Run = 4 (right). Positive slope — line rises.
Points (2,6) and (6,2)
m = (2 − 6) / (6 − 2) = −4 / 4 = −1
Rise = −4 (down), Run = 4 (right). Negative slope — line falls.
Points (3,4) and (3,9)
m = (9 − 4) / (3 − 3) = 5 / 0 = undefined
Run = 0 → division by zero. Vertical line x = 3. Not a function.
Points (1,5) and (6,5)
m = (5 − 5) / (6 − 1) = 0 / 5 = 0
Rise = 0 → horizontal line y = 5. Zero slope — no vertical change.
Finding Slope from a Table
Strategy: Δy ÷ Δx between consecutive rows
| x | y | Δy | Δx | Δy/Δx |
|---|---|---|---|---|
| 0 | 3 | — | — | — |
| 2 | 7 | +4 | +2 | 2 |
| 4 | 11 | +4 | +2 | 2 |
| 6 | 15 | +4 | +2 | 2 |
Δy is constant (+4) and Δx is constant (+2) across all rows.
m = Δy / Δx = 4 / 2 = 2
The slope is 2 — for every 2 units right, y increases by 4 (or equivalently, for every 1 unit right, y increases by 2).
⚠ If Δy/Δx is not constant, the relationship is not linear.
Rate of Change in Real-World Contexts
Distance vs. Time
Slope = 50 mph
For every 1 hour, distance increases by 50 miles.
m = 150/3 = 50
Cost vs. Quantity
Slope = $4 per item
Each additional item costs $4 more. $5 base fee.
m = (25−5)/(5−0) = 4
Earnings vs. Hours
Slope = $15/hr
Earnings increase by $15 for every hour worked.
m = 90/6 = 15
Comparing Rates of Change
Three lines — different rates
m = 3: steepest — y increases 3 per unit of x
m = 1.5: moderate — y increases 1.5 per unit of x
m = 0.5: gentlest — y increases 0.5 per unit of x
Larger |m| → steeper line → faster rate of change.
Math Tips
- Slope is a ratio, not a point: m = 2 means "2 units up per 1 unit right" — it describes direction and steepness, not a location on the plane.
- Consistent subtraction order: Always subtract in the same order: (y₂ − y₁) and (x₂ − x₁). Mixing orders gives the wrong sign.
- Negative slope trick: Write m = −3 as −3/1. Count 3 down, 1 right — or 3 up, 1 left. Both give the same line.
- Table shortcut: If x increases by 1 each row, the constant change in y is the slope directly.
- Graph shortcut: Pick two points where the line crosses grid intersections (lattice points) to avoid estimation errors.
- Units matter: In real-world problems, always attach units to slope: miles/hour, $/item, meters/second.
- Steeper = larger |m|: Compare absolute values. |−5| > |2|, so a slope of −5 is steeper than a slope of 2.
Worked Examples
Positive slope: rise=6, run=3, m=2
Negative slope: rise=−4, run=4, m=−1
Find the slope of the line through (1, 2) and (5, 10).
Label the points: (x₁, y₁) = (1, 2) and (x₂, y₂) = (5, 10).
Apply the slope formula: m = (y₂ − y₁) / (x₂ − x₁)
m = (10 − 2) / (5 − 1)
m = 8 / 4
m = 2
Check: from (1,2) go up 8, right 4 → (5, 10) ✓. Positive slope — line rises.
Example 1 — m = 2
Find the slope of the line through (2, 6) and (6, 2).
Label: (x₁, y₁) = (2, 6) and (x₂, y₂) = (6, 2).
m = (y₂ − y₁) / (x₂ − x₁) = (2 − 6) / (6 − 2)
m = −4 / 4
m = −1
Check: from (2,6) go down 4, right 4 → (6, 2) ✓. Negative slope — line falls.
Example 2 — m = −1
Find the slope of the line through (3, 4) and (3, 9).
Label: (x₁, y₁) = (3, 4) and (x₂, y₂) = (3, 9).
m = (9 − 4) / (3 − 3) = 5 / 0
Division by zero is undefined.
Both points have x = 3, so this is a vertical line: x = 3.
Vertical lines have undefined slope and are NOT functions.
Find the slope of the line through (1, 5) and (6, 5).
Label: (x₁, y₁) = (1, 5) and (x₂, y₂) = (6, 5).
m = (5 − 5) / (6 − 1) = 0 / 5 = 0
Both points have y = 5, so this is a horizontal line: y = 5.
Horizontal lines have slope = 0.
Find the slope from the table: x = 0, 2, 4, 6 and y = 1, 5, 9, 13.
Find Δy and Δx between consecutive rows.
Row 1→2: Δy = 5−1 = 4, Δx = 2−0 = 2 → ratio = 4/2 = 2
Row 2→3: Δy = 9−5 = 4, Δx = 4−2 = 2 → ratio = 4/2 = 2
Row 3→4: Δy = 13−9 = 4, Δx = 6−4 = 2 → ratio = 4/2 = 2
The ratio is constant → linear relationship.
Slope m = 2.
Find the slope from the graph of a line passing through (0, 4) and (3, −2).
Identify two clear points on the graph: (0, 4) and (3, −2).
Rise = y₂ − y₁ = −2 − 4 = −6 (moving down 6 units)
Run = x₂ − x₁ = 3 − 0 = 3 (moving right 3 units)
m = rise / run = −6 / 3 = −2
Negative slope confirms the line falls from left to right.
Example 6 — m = −2
Common Mistakes
Subtracting the y-values and x-values in different orders — e.g., (y₂ − y₁)/(x₁ − x₂).
Always subtract in the same order: slope = (y₂ − y₁)/(x₂ − x₁). Pick one point as (x₁, y₁) and stay consistent.
Confusing a zero slope (horizontal line) with an undefined slope (vertical line).
Horizontal lines have slope 0 (no rise). Vertical lines have undefined slope (no run — division by zero).
Forgetting that slope represents a rate of change with units — e.g., reading slope as just a number without context.
In context, slope = (change in output)/(change in input). Always attach units: dollars per hour, miles per gallon, etc.
Assuming a steeper-looking line always has a larger slope — without checking the scale of the axes.
Slope depends on the actual rise and run values, not just visual appearance. Check the axis scale before comparing.
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: Rate of Change and Slope
Watch the guided practice walkthrough for rate of change and slope, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Find the slope through (0, 3) and (4, 11).
Hint: m = (11 − 3) / (4 − 0) = 8 / 4 = 2.
Find the slope through (−2, 5) and (4, −1).
Hint: m = (−1 − 5) / (4 − (−2)) = −6 / 6 = −1.
Find the slope from the table: x = 0, 1, 2, 3; y = 4, 7, 10, 13.
Hint: Δy = 3 each step, Δx = 1 each step. m = 3/1 = 3.
A line passes through (5, 2) and (5, −3). What is the slope?
Hint: Run = 5 − 5 = 0. Division by zero → undefined slope. Vertical line x = 5.
A line passes through (−3, 7) and (4, 7). What is the slope?
Hint: Rise = 7 − 7 = 0. m = 0 / 7 = 0. Horizontal line y = 7.
Constant Δy=3 → slope = 3
Positive slope: m = 2
Key Vocabulary
Rate of Change
How much one quantity changes relative to another. In a linear relationship, the rate of change equals the slope.
Example: A car going 60 mph has a rate of change of 60 miles per hour.
Slope (m)
The constant rate of change of a line: m = rise / run = (y₂ − y₁) / (x₂ − x₁).
Example: Through (1,2) and (5,10): m = (10−2)/(5−1) = 2.
Rise
The vertical change between two points on a line (Δy = y₂ − y₁). Positive = up; negative = down.
Example: From (1,2) to (5,10): rise = 10 − 2 = 8.
Run
The horizontal change between two points on a line (Δx = x₂ − x₁). Always measured left to right.
Example: From (1,2) to (5,10): run = 5 − 1 = 4.
Zero Slope
A horizontal line. Rise = 0, so m = 0/run = 0.
Example: Through (−3,5) and (4,5): m = 0.
Undefined Slope
A vertical line. Run = 0, so m = rise/0 — division by zero is undefined.
Example: Through (3,1) and (3,7): m = undefined.
Interactive Practice — 5 Questions
What is the slope of the line through (1, 3) and (5, 11)?
A line passes through (4, 7) and (4, −2). What is the slope?
A table shows: x = 0, 2, 4, 6 and y = 3, 7, 11, 15. What is the slope?
Which slope is steepest?
A car travels 120 miles in 2 hours. What is the rate of change?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Find the slope through (0, 0) and (3, 9).
Find the slope through (2, 10) and (6, 2).
Find the slope from the table: x = 0, 1, 2, 3; y = 2, 5, 8, 11.
Find the slope through (4, 6) and (4, −2). Classify the slope.
A car travels 180 miles in 3 hours. What is the rate of change? What does it represent?
Common Mistakes
Mixing subtraction order: writing (y₁ − y₂)/(x₂ − x₁) instead of (y₂ − y₁)/(x₂ − x₁).
Always subtract in the same order in both numerator and denominator. Swapping one but not the other flips the sign.
Confusing rise and run: using horizontal change as rise and vertical change as run.
Rise is vertical (Δy); run is horizontal (Δx). Rise over run, not run over rise.
Saying a horizontal line has undefined slope and a vertical line has zero slope.
Horizontal line → slope = 0 (rise is zero). Vertical line → slope is undefined (run is zero, division by zero).
Treating slope as a point: thinking m = 3 means 'the point (3, 0)'.
Slope m = 3 means 'go up 3 for every 1 right.' It describes direction and steepness, not a location.
Math Tips
Slope formula: m = (y₂ − y₁) / (x₂ − x₁). Subtract y-values first (rise), then x-values (run).
From a table: if x increases by equal steps, slope = Δy ÷ Δx between any two consecutive rows.
Steepness: larger |m| means steeper line. m = −5 is steeper than m = 2 because |−5| > |2|.
Real-world slope always has units: miles per hour, dollars per item, meters per second.
Check: if Δy/Δx is not constant across all rows of a table, the relationship is not linear.