Slope and Rate of Change

What Is Slope? 📈

Slope measures the steepness and direction of a line. It answers a simple question: as you move to the right, how much does the line go up or down?

Think of slope as:

• How much a line rises or falls
• How fast something is changing
• The tilt of the line

We usually use the letter $m$ to stand for slope.

A line with a big slope is steep, like a ski jump. A line with a small slope is gentle, like a wheelchair ramp. A line that is perfectly flat has a slope of zero — it doesn't go up or down at all.

Rise Over Run 🧗

To measure slope we compare two kinds of movement between two points on the line:

• Rise = the vertical change (how far up or down) — this is the change in $y$.
• Run = the horizontal change (how far left or right) — this is the change in $x$.

$m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}$

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope formula is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

You may also see it written with the Greek letter $\Delta$ (delta), which means "change in":

$m = \frac{\Delta y}{\Delta x}$

Important: Up and right are positive; down and left are negative. Always subtract the coordinates in the same order on the top and bottom.

A First Example 🔍

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

Step 1 — Label the points. $(x_1, y_1) = (1, 2) \qquad (x_2, y_2) = (4, 8)$

Step 2 — Apply the formula. $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$

Answer: $m = 2$.

This means the line rises $2$ units for every $1$ unit you move to the right. The rise ($6$) and the run ($3$) both describe the same tilt — and the slope $2$ is the simplest way to state it.

Quick Concept Check ✅

Let's make sure the big idea is clear before we practice.

Working Through the Formula ✍️

Let's carefully work two more examples, including one with negative numbers.

Example A — Find the slope through $(-2, 5)$ and $(3, -5)$.

Step 1 — Label the points. $(x_1, y_1) = (-2, 5) \qquad (x_2, y_2) = (3, -5)$

Step 2 — Substitute into the formula. $m = \frac{-5 - 5}{3 - (-2)} = \frac{-10}{3 + 2} = \frac{-10}{5} = -2$

Answer: $m = -2$. The line falls $2$ units for every $1$ unit to the right.

Example B — Find the slope through $(0, 4)$ and $(5, 4)$.

$m = \frac{4 - 4}{5 - 0} = \frac{0}{5} = 0$

Answer: $m = 0$. Both points have the same $y$-value, so there is no rise — this is a horizontal line.

⚠️ Watch Your Subtraction

The most common slope mistake is mixing up the order of subtraction. Follow these tips:

• Be consistent. Whatever point you use first on top, use first on the bottom too.
• Subtracting a negative flips it to addition: $3 - (-2) = 3 + 2 = 5$.
• It does not matter which point you call first — as long as you stay consistent, you get the same slope.

Both ways give $m = -2$ — same answer! The danger is mixing the orders, like putting $y_2 - y_1$ on top but $x_1 - x_2$ on the bottom. Pick one order and stick with it.

Your Turn: Find Each Slope ✍️

Use $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Type each answer as a single number, like 3 or -2.

1. Slope through $(1, 1)$ and $(4, 10)$
2. Slope through $(2, 7)$ and $(5, 1)$
3. Slope through $(-3, 0)$ and $(1, 8)$

The Four Types of Slope 🛝

Every line falls into one of four categories based on its slope:

Why is a vertical line undefined? Both points have the same $x$-value, so the run is $0$. The slope becomes $\frac{\text{rise}}{0}$, and dividing by zero is undefined in math.

A quick way to remember the last two: a Horizontal line is "0" (flat ground), while a Vertical line is "undefined" (you can't walk straight up a wall!).

Multiple Choice 🎯

Identify the type and value of slope.

Match the Line to Its Slope 🔽

For each description, choose the slope type that fits.

Slope as a Rate of Change 🚗

In the real world, slope is often called a rate of change — it tells you how fast one quantity changes compared to another. The units make the meaning clear.

Speed (a rate of change of distance): A car travels $150$ miles in $3$ hours at a steady speed. Treat time as $x$ and distance as $y$: $m = \frac{150 \text{ miles}}{3 \text{ hours}} = 50 \text{ miles per hour}$

The slope $50$ is exactly the car's speed!

Cost (a rate of change of money): A taxi charges based on miles driven. If a $4$-mile ride costs $10 and an $8$-mile ride costs $18: $m = \frac{18 - 10}{8 - 4} = \frac{8}{4} = 2 \text{ dollars per mile}$

Reading the units: "miles per hour," "dollars per mile," "degrees per hour" — the word "per" is a big hint that you are looking at a slope.

Word Problem Practice ✍️

Find each rate of change (slope). Type each answer as a single number.

1. A plant grows from $6$ cm to $18$ cm over $4$ weeks. What is its growth rate in cm per week? ($\frac{18 - 6}{4}$)
2. A pool drains from $80$ gallons to $20$ gallons in $5$ minutes. What is the rate of change in gallons per minute (use a negative for draining)? ($\frac{20 - 80}{5}$)
3. A runner covers $12$ km in $2$ hours at a steady pace. What is the speed in km per hour? ($\frac{12}{2}$)

Application Check 🎯

Putting It All Together 🧠

You now know how to find slope and how to interpret it. Here is the full summary:

Remember these key ideas:

• Subtract the $y$'s on top and the $x$'s on the bottom — same order for both.
• Horizontal $= 0$, vertical $=$ undefined.
• A slope is just a rate of change dressed up with units.

Ready for the final challenge? It mixes everything together!

🏆 Mixed Challenge

Use everything you've learned about slope.