Slope and Rate of Change

The slope of a line describes how steep the line is — it measures how much

y

changes when

x

increases by 1. In other words, slope represents the rate of change between two quantities.

\text{Slope } (m) = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}

If you imagine moving along a line: - The numerator

(y_2 - y_1)

represents the “rise” (vertical change). - The denominator

(x_2 - x_1)

represents the “run” (horizontal change).

Slope as Rate: On the SAT, slope often represents a real-world rate such as:

\text{miles per hour}, \quad \text{dollars per item}, \quad \text{points per game}, \quad \text{etc.}

Types of Slope

\begin{aligned} m > 0 &\Rightarrow \text{line rises left to right (positive slope)} \\ m < 0 &\Rightarrow \text{line falls left to right (negative slope)} \\ m = 0 &\Rightarrow \text{horizontal line} \\ \text{Undefined slope} &\Rightarrow \text{vertical line} \end{aligned}

Slope Relationships

Parallel lines have the same slope.
Perpendicular lines have slopes that are negative reciprocals:
$m_1 \cdot m_2 = -1$

Core Skills

Find slope from two points, a graph, or an equation.
Interpret slope as a rate of change in context.
Recognize slope patterns for parallel and perpendicular lines.
Identify slope units in real-world models.

Example 1: Finding Slope from Two Points

Find the slope of the line passing through the points

(2, 5)

and

(6, 13)

. Step 1: Label the points.

(x_1, y_1) = (2, 5)

(x_2, y_2) = (6, 13)

Step 2: Use the slope formula.

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2

Final Answer:

\boxed{m = 2}

Interpretation: For every 1 unit increase in

x

y

increases by 2 units.

Example 2: Slope as a Rate of Change

A car travels 120 miles in 3 hours. What is its average rate of change in miles per hour? Step 1: Identify the change in distance and change in time.

\text{Change in distance} = 120 \text{ miles}, \quad \text{Change in time} = 3 \text{ hours}

Step 2: Compute the rate.

m = \frac{\text{change in distance}}{\text{change in time}} = \frac{120}{3} = 40

Final Answer:

\boxed{40 \text{ miles per hour}}

Interpretation: The slope of the line on a distance–time graph would be 40, showing a steady rate of travel.

Key Takeaways

Slope measures the rate of change between two variables.
Use $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ to find slope from points.
Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes.
In word problems, slope often represents a real-world rate such as “cost per item” or “distance per time.”