Slope and Rate of Change, SAT math lesson

A road sign warning of an 8 percent grade means the road climbs 8 meters for every 100 meters you travel horizontally. That ratio of up to across is the slope of the road. On the SAT the same number shows up in two forms, as the steepness of a line on a graph and as a rate such as dollars per hour or liters per minute in a word problem.

See also Drawing Lines from Slope and Intercepts and Changing Quantities and Average Rate in Foundations for Calculus and Linear Functions and Constant Rate of Change in Precalculus.

Key terms

Rise

Definition For a move from the point $P(x_1, y_1)$ to the point $Q(x_2, y_2)$ , the rise is the vertical change

y_2 - y_1.

It is positive when $Q$ is higher than $P$ , negative when $Q$ is lower, and zero when they are at the same height.

In plain words The rise is how far you go up or down to get from the first point to the second. Going down counts as a negative rise.

Example From $P(-2, 5)$ to $Q(4, -1)$ the rise is $-1 - 5 = -6$ . You go down $6$ units.

Run

Definition For the same move from $P(x_1, y_1)$ to $Q(x_2, y_2)$ , the run is the horizontal change

x_2 - x_1.

It is positive when $Q$ is to the right of $P$ , negative when $Q$ is to the left, and zero when one point is directly above the other.

In plain words The run is how far you go left or right. Going left counts as a negative run.

Example From $P(-2, 5)$ to $Q(4, -1)$ the run is $4 - (-2) = 6$ . Subtracting a negative number adds, which is the step people most often get wrong.

Slope

Definition The slope of the line through two points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \ne x_2$ is

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}.

For a line that is not vertical this number is the same for every pair of distinct points on it. Either point may come first, as long as the same point comes first in the numerator and in the denominator. A line with positive slope rises from left to right, a line with negative slope falls from left to right, and a larger value of $\lvert m \rvert$ means a steeper line.

In plain words Slope is rise over run, the change in $y$ for each step of $1$ to the right.

Example For $P(-2, 5)$ and $Q(4, -1)$ ,

m = \frac{-1 - 5}{4 - (-2)} = \frac{-6}{6} = -1.

Starting from $Q$ instead gives the same slope,

\frac{5 - (-1)}{-2 - 4} = \frac{6}{-6} = -1.

The line falls $1$ unit for each unit to the right. Through $(1, 3)$ and $(5, 11)$ the slope is $\frac{8}{4} = 2$ , a rising line twice as steep as the first one.

Rate of change

Definition When a quantity $y$ depends linearly on a quantity $x$ , the rate of change of $y$ with respect to $x$ is

\frac{\text{change in } y}{\text{change in } x},

which is the slope of the graph of $y$ against $x$ . For a linear relationship it is the same over every interval. Its units are the units of $y$ per unit of $x$ . A positive rate means $y$ increases as $x$ increases, and a negative rate means it decreases.

In plain words The rate tells you how much the output changes for each one unit of input, and the units say exactly what that means.

Example A candle burns at a steady rate. It is $30$ centimeters tall when lit and $22$ centimeters tall $4$ hours later. Its rate of change is

\begin{gathered} \frac{22 - 30}{4 - 0} = -2 \\[0.4em] \text{centimeters per hour.} \end{gathered}

The negative sign says the height goes down, by $2$ centimeters each hour. In the same way, if $3$ kilograms of apples cost $7.50$ dollars and $8$ kilograms cost $20$ dollars at a fixed price per kilogram, the rate is $\frac{12.50}{5} = 2.50$ dollars per kilogram.

Horizontal line

Definition A horizontal line is the set of all points with the same $y$ -coordinate $k$ , and its equation is $y = k$ . Any two distinct points on it have rise $0$ and nonzero run, so its slope is $0$ . It is the graph of the constant function $f(x) = k$ .

In plain words A flat line goes neither up nor down, so its slope is zero. In a story it describes a quantity that is not changing.

Example Through $(-3, 4)$ and $(5, 4)$ the slope is

\frac{4 - 4}{5 - (-3)} = \frac{0}{8} = 0,

and the line is $y = 4$ . If the graph of a tank’s water volume against time is a horizontal segment, nothing is flowing in or out during that time.

Vertical line

Definition A vertical line is the set of all points with the same $x$ -coordinate $h$ , and its equation is $x = h$ . Any two distinct points on it have run $0$ , and division by zero is not defined, so the slope of a vertical line is undefined. A vertical line is not the graph of a function, because the input $h$ would have infinitely many outputs. A slope of zero and an undefined slope are different things.

In plain words A vertical line goes straight up, so there is no run to divide by. It has no slope at all, which is not the same as a slope of zero.

Example Through $(2, 1)$ and $(2, 6)$ the rise is $5$ and the run is $0$ . The quotient $\frac{5}{0}$ is undefined, so the line $x = 2$ has undefined slope.

Check yourself

What is the slope of the line through $(-1, 7)$ and $(3, -5)$ ? (A) $-3$ (B) $-\frac{1}{3}$ (C) $\frac{1}{3}$ (D) $3$

Show the answer

The correct choice is (A). The slope is

\frac{-5 - 7}{3 - (-1)} = \frac{-12}{4} = -3.

Choice (D) comes from subtracting in opposite orders on the top and bottom, which gives $\frac{-12}{-4}$ . Choice (B) puts the run over the rise, and (C) does both.

A water tank is drained at a constant rate. It holds $340$ liters $2$ days after draining starts and $215$ liters $7$ days after draining starts. By how many liters does the amount of water decrease each day?

Show the answer

The answer is $25$ . The rate of change is

\begin{gathered} \frac{215 - 340}{7 - 2} = \frac{-125}{5} = -25 \\[0.4em] \text{liters per day.} \end{gathered}

The negative sign means a decrease, so the amount goes down by $25$ liters each day, and you enter the positive number $25$ .

Line $\ell$ passes through the points $(4, -2)$ and $(4, 9)$ . Which statement is true? (A) The slope of $\ell$ is $0$ . (B) The slope of $\ell$ is $11$ . (C) The slope of $\ell$ is undefined. (D) An equation of $\ell$ is $y = 4$ .

Show the answer

The correct choice is (C). Both points have $x$ -coordinate $4$ , so the run is $0$ and the rise is $11$ . Division by $0$ is undefined. Choice (A) mixes up vertical and horizontal lines, (B) uses the rise without dividing by the run, and (D) names the horizontal line through height $4$ . An equation of $\ell$ is $x = 4$ .

Slope and Rate of Change

Theory, part 21

Practice, part 22

Key terms

Rise

Run

Slope

Rate of change

Horizontal line

Vertical line

Check yourself

How this lesson connects

Builds on

Used later in