Slope: Definition and Calculation
Define slope as the ratio of vertical change to horizontal change (rise/run) and calculate it from two points using the slope formula. Classify lines as having positive, negative, zero, or undefined slope and connect slope to rate of change.
A lesson of Algebra 1, in Linear Functions, about 30 minutes.
Slope: What It Is and How to Find It
By the end of this lesson, you will be able to work out the slope between any two points, read it off a graph, say what it means in a real situation, and explain why a vertical line has no slope at all.
Slope Is a Rate, Not a Steepness
The last unit said a linear function has a constant rate of change. Slope is that rate, given a number.
$$m = \frac{\text{change in } y}{\text{change in } x} = \frac{\text{rise}}{\text{run}}$$
Between any two points on a line:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Read the formula as a sentence rather than a pattern: how much the output changed, divided by how much the input changed. A slope of $3$ means "every step right, three steps up". A slope of $-\tfrac12$ means "every two steps right, one step down".
Why it is a rate and not just steepness. On a graph of distance against time, the slope is speed, in miles per hour. On cost against quantity it is price per unit. The units come from the axes, and reading a slope back with its units is where most of the marks in an applied question are.
The Four Cases
| Slope | The line | Reading it |
|---|---|---|
| Positive | Climbs left to right | Output grows as input grows |
| Negative | Falls left to right | Output shrinks as input grows |
| Zero | Horizontal | Output never changes |
| Undefined | Vertical | Input never changes |
The last two are the ones that get swapped, and the formula settles it.
Horizontal: the $y$ values are equal, so the top of the fraction is zero. $\frac{0}{\text{something}} = 0$. Slope zero.
Vertical: the $x$ values are equal, so the bottom is zero. $\frac{\text{something}}{0}$ is not a number. Slope undefined, which is not the same as zero and not the same as "no slope written down".
Example 1: Between Two Points
Here is the whole method on one pair, worked all the way through. Find the slope between $(2, 3)$ and $(6, 11)$.
- Work out the change in y, top minus bottom 8
- Work out the change in x, in the SAME order 4
- Divide to get the slope 2
Worked Example 1: Your Turn
Find the slope between $(-1, 7)$ and $(3, -1)$.
Plot It: See the Rise and the Run
Plot the two points from Example 1 and look at the triangle between them.
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