Lesson 3: Slope

New Concept

Slope is the core concept for measuring change. It's a single number, $m = \frac{\Delta y}{\Delta x}$, that tells you how steep a line is and in what direction it's heading. We'll compute and interpret it as a rate of change.

What’s next

Now you have the basic idea. Next, you'll apply this through interactive examples, plotting data, and calculating slope on practice cards to master the concept.

Property

A rate of change is a type of ratio that measures how one variable changes with respect to another. It is calculated as:

Examples

• If a car travels 120 miles in 2 hours, its rate of change (speed) is $\frac{120 \text{ miles}}{2 \text{ hours}} = 60$ miles per hour.

• If a 300-gallon tank fills in 15 minutes, the rate of change is $\frac{300 \text{ gallons}}{15 \text{ minutes}} = 20$ gallons per minute.

Property

The Greek letter $\Delta$ ('delta') is used in mathematics to indicate change. The slope of a line is defined by the ratio

as we move from one point to another on the line. In symbols, the slope, denoted by $m$, is:

Examples

• For a line passing through points $(2, 5)$ and $(4, 9)$, the change in $y$ is $\Delta y = 9 - 5 = 4$ and the change in $x$ is $\Delta x = 4 - 2 = 2$. The slope is $m = \frac{4}{2} = 2$.

• If we move from point $A(1, 3)$ to point $B(5, 6)$, the slope is $m = \frac{\Delta y}{\Delta x} = \frac{6-3}{5-1} = \frac{3}{4}$.

Property

The slope of a line measures the rate of change of $y$ with respect to $x$. In different situations, this rate might be interpreted as a rate of growth or a speed.

Examples

• If a graph of distance (miles) vs. time (hours) has a slope of $55$, it represents an average speed of 55 miles per hour.

• If a graph of cost (dollars) vs. weight (pounds) has a slope of $3$, it means the price is 3 dollars per pound.

Property

Positive slopes correspond to lines that increase from left to right. Negative slopes correspond to lines that decrease from left to right. The larger the absolute value of the slope, the steeper the graph.

Examples

• A line with slope $m = 2$ is steeper than a line with slope $m = \frac{1}{3}$ because $|2| > |\frac{1}{3}|$.

• A line with slope $m = -3$ is steeper than a line with slope $m = -1$ because $|-3| > |-1|$. Both lines slant downwards.