3.1: Slope as a Rate | Algebra 1 Honors

How it works

Slope measures change in output per unit change in input. Use the same point order in numerator and denominator. Positive slope indicates increasing output; a horizontal line has slope zero. A vertical line has undefined slope because its change in x is zero.

m=\frac{y_2-y_1}{x_2-x_1}\quad(x_2\ne x_1)

Worked examples

Example 1

Find the slope through $\displaystyle (1,2)$ and $\displaystyle (4,8)$ .

$\displaystyle m=(8-2)/(4-1)=6/3=2$ . Output increases by $\displaystyle 2$ for each added input unit.

Example 2

A tank drops from $\displaystyle 30$ liters to $\displaystyle 18$ liters in $\displaystyle 4$ minutes. Find the average rate.

The rate is $\displaystyle (18-30)/4=-3$ liters per minute. The negative sign represents decreasing volume.

Problems

Problem 3.1.1

Find the slope through $\displaystyle (-2,5)$ and $\displaystyle (2,-3)$ .

Show worked solution

$\displaystyle m=(-3-5)/(2-(-2))=-8/4=-2$ .

Problem 3.1.2

Find the slope of the line through $\displaystyle (3,1)$ and $\displaystyle (3,7)$ .

Show worked solution

The denominator is $\displaystyle 3-3=0$ , so the line is vertical and its slope is undefined.

Problem 3.1.3

Use the graphed points to find the slope and explain its sign.

A falling line passes through marked points at (−1, 4) and (2, −2). — Read the coordinates of the two marked points.

Show worked solution

The points are $\displaystyle (-1,4)$ and $\displaystyle (2,-2)$ . The slope is $\displaystyle (-2-4)/(2-(-1))=-2$ ; the line falls as $\displaystyle x$ increases.

A falling line passes through marked points at (−1, 4) and (2, −2). The slope is −2. — The slope is −2.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 3.1.1

Problem 3.1.2

Problem 3.1.3

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