Finding Slope from Two Points

Find the slope of the line passing through $(2, 3)$ and $(5, 9)$.

Identify the coordinates: $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (5, 9)$ = Points labeled

Find the rise (change in y): $y_2 - y_1 = 9 - 3 = 6$ = Rise = 6

Find the run (change in x): $x_2 - x_1 = 5 - 2 = 3$ = Run = 3

Calculate slope: $m = \frac{\text{rise}}{\text{run}} = \frac{6}{3} = 2$ = $m = 2$

Answer: The slope is $m = 2$. This means the line rises 2 units for every 1 unit it moves right.

Negative Slope

Find the slope of the line through $(1, 8)$ and $(4, 2)$.

Identify the coordinates: $(x_1, y_1) = (1, 8)$ and $(x_2, y_2) = (4, 2)$ = Points labeled

Find the rise: $y_2 - y_1 = 2 - 8 = -6$ = Rise = $-6$

Find the run: $x_2 - x_1 = 4 - 1 = 3$ = Run = 3

Calculate slope: $m = \frac{-6}{3} = -2$ = $m = -2$

Answer: The slope is $m = -2$. The negative sign tells us the line goes downward from left to right.

Slope from a Graph

A line passes through $(0, 1)$ and $(3, 7)$. What is its slope?

Count the rise: From $y = 1$ to $y = 7$: rise = $7 - 1 = 6$ = Rise = 6

Count the run: From $x = 0$ to $x = 3$: run = $3 - 0 = 3$ = Run = 3

Calculate slope: $m = \frac{6}{3} = 2$ = $m = 2$

Answer: The slope is $m = 2$. On the graph, you'd go up 2 squares and right 1 square to trace the line.

Zero and Undefined Slope

Find the slope of lines through: (a) $(2, 5)$ and $(7, 5)$, (b) $(3, 1)$ and $(3, 6)$.

For (a): Find rise and run: Rise = $5 - 5 = 0$, Run = $7 - 2 = 5$ = Rise = 0, Run = 5

Calculate slope for (a): $m = \frac{0}{5} = 0$ = Horizontal line: $m = 0$

For (b): Find rise and run: Rise = $6 - 1 = 5$, Run = $3 - 3 = 0$ = Rise = 5, Run = 0

Calculate slope for (b): $m = \frac{5}{0}$ = undefined (division by zero!) = Vertical line: undefined

Answer: (a) $m = 0$ (horizontal line), (b) Slope is undefined (vertical line). Remember: horizontal = zero slope, vertical = no slope.

Mistake: Mixing up rise and run in the formula

Why: Students sometimes calculate $\frac{\text{run}}{\text{rise}}$ instead of $\frac{\text{rise}}{\text{run}}$.

Correct: Remember: slope = $\frac{\text{rise}}{\text{run}}$ = $\frac{y_2 - y_1}{x_2 - x_1}$. The y-values go on top!

Mistake: Subtracting coordinates in different orders

Why: Using $(y_2 - y_1)$ but $(x_1 - x_2)$ gives the wrong sign.

Correct: Always subtract in the same order: if you do $y_2 - y_1$, you must also do $x_2 - x_1$.

Mistake: Saying vertical lines have slope = 0

Why: Confusing "no slope" with "zero slope."

Correct: Horizontal lines have slope = 0. Vertical lines have undefined slope (cannot divide by zero).

Mistake: Forgetting the negative sign

Why: Not noticing when the line goes downward.

Correct: If the line falls from left to right, the slope is negative. Always check your sign!

Road Grades and Highways

Road signs show grade as a percentage, which is slope expressed as rise per 100 units of run.

A 7% grade means the road rises 7 meters for every 100 meters of horizontal distance: slope = $\frac{7}{100} = 0.07$

Phone Data Usage

Your phone plan charges a constant rate per gigabyte. The slope of your bill graph is the cost per GB.

If your bill goes from 20 euros to 35 euros when you use 3 extra GB, the slope is $\frac{35-20}{3} = 5$ euros per GB.

Filling a Pool

When filling a pool at a constant rate, the slope of the water level graph tells you liters per minute.

If the water level rises from 50 cm to 90 cm in 20 minutes, the slope is $\frac{40}{20} = 2$ cm per minute.

Q: Does it matter which point I call $(x_1, y_1)$?

A: No! You can pick either point as point 1. Just be consistent: subtract in the same order for both x and y.

Q: What if I get a fraction for slope?

A: That's perfectly fine! A slope of $\frac{2}{3}$ means "rise 2, run 3." You can leave it as a fraction or convert to a decimal (about 0.67).

Q: Is a steeper line always better?

A: It depends on context! A steeper hill is harder to climb. A steeper savings rate means you save faster. Interpret slope based on what the graph represents.