Lines: Slope and Intercept — AP® Statistics Notes

Unit 5 describes how one quantitative variable changes with another using a straight line. Before that, you need to be comfortable reading any line: what its slope says, what its intercept says, and how to find both.

The equation of a line

Statistics writes a line as

$y = a + bx,$

with the intercept first. $b$ is the slope and $a$ is the y-intercept. (Algebra classes often write $y = mx + b$ ; it is the same line with different letters.)

The line y = 12 + 3x for the cost in dollars of renting a bike for x hours, on axes from 0 to 8 hours and 0 to 40 dollars. The line starts at the intercept, 12 dollars at 0 hours. A step of 1 hour to the right (the run) goes with a rise of 3 dollars: the slope is 3 dollars per hour. — **Figure 1.** The rental cost line $y = 12 + 3x$ , with its intercept and one rise-over-run step. LevlPrep original diagram.

Slope

The slope is the change in y for each one-unit increase in x. On a graph it is rise over run: from any point on the line, move 1 unit right, and y changes by $b$ . A positive slope rises; a negative slope falls.

From two points $(x_1, y_1)$ and $(x_2, y_2)$ :

$b = \frac{y_2 - y_1}{x_2 - x_1}.$

Subtract in the same order on top and bottom. The slope has units: y units per x unit, such as dollars per hour.

Slope and intercept from two points. A line passes through $(2, 7)$ and $(6, 19)$ .

Slope: $b$ $= \frac{19 - 7}{6 - 2}$ $= \frac{12}{4}$ $= 3$ .

Intercept: write $y = a + 3x$ and use one point, $(2, 7)$ : $7 = a + 3(2) = a + 6$ , so $a = 1$ .

Equation: $y = 1 + 3x$ . Check with the other point: $1 + 3(6) = 19$ . Correct.

Intercept

The y-intercept is the value of y when $x = 0$ , where the line crosses the vertical axis. For the bike rental $y = 12 + 3x$ , it is the 12 dollar fee charged before any hours. Sometimes $x = 0$ makes no sense: a line giving adult height from shoe length has an intercept at shoe length 0, which no adult has. Then the intercept is just a number that places the line, with no meaning of its own.

Saying it in context

Interpret a line. A cooling drink’s temperature is $y = 80 - 2.5x$ degrees Fahrenheit after $x$ minutes.

Slope: "For each additional minute, the temperature of the drink decreases by 2.5 degrees."

Intercept: "At 0 minutes, when it is poured, the drink is 80 degrees." Here $x = 0$ makes sense, so the intercept has a meaning.

A value: after 6 minutes, $y$ $= 80 - 2.5(6)$ $= 80 - 15$ $= 65$ degrees.

	Slope $b$	Intercept $a$
Meaning	change in y per one-unit increase in x	y when $x = 0$
Units	y units per x unit	y units
Template	"For each additional [x unit], [y] increases (decreases) by [b] [y units]."	"When [x] is 0, [y] is [a] [y units]." (only if $x = 0$ makes sense)