Understanding Slope

Show your work for each problem.

1.
What does slope measure?
1. a)The length of a line
2. b)The steepness of a line
3. c)Where the line crosses the y-axis
4. d)The position of a line
Answer: The steepness of a line
Slope measures how steep a line is - how much it rises or falls as you move from left to right.
2.
In the slope formula $m = \frac{\text{rise}}{\text{run}}$ , what goes in the numerator?
1. a)The vertical change (rise)
2. b)The x-coordinate
3. c)The y-intercept
4. d)The horizontal change (run)
Answer: The vertical change (rise)
The rise (vertical change, or change in y) goes in the numerator (on top). The formula is $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$ .
3.
A line rises 6 units and runs 2 units. What is its slope?
Answer: 3
Slope $= \frac{\text{rise}}{\text{run}} = \frac{6}{2} = 3$ . The line goes up 3 units for every 1 unit to the right.
4.
Find the slope of the line through $(1, 2)$ and $(3, 8)$ .
Answer: 3
$m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$
5.
Find the slope of the line through $(2, 5)$ and $(6, 13)$ .
Answer: 2
1. What is $y_2 - y_1$ ? 8
2. What is $x_2 - x_1$ ? 4
3. What is the slope (rise/run)? 2
6.
Find the slope of the line through $(3, 10)$ and $(7, 2)$ .
Answer: -2
$m = \frac{2 - 10}{7 - 3} = \frac{-8}{4} = -2$ . The negative slope means the line goes down from left to right.
7.
A horizontal line has a slope of:
1. a)1
2. b)0
3. c)-1
4. d)Undefined
Answer: 0
A horizontal line has rise = 0. Slope = $\frac{0}{\text{run}} = 0$ . No matter how far you go horizontally, you don't go up or down.
8.
The slope of a vertical line is:
1. a)Undefined
2. b)Infinity
3. c)0
4. d)1
Answer: Undefined
A vertical line has run = 0. Slope = $\frac{\text{rise}}{0}$ , which is undefined because you cannot divide by zero.
9.
Find the slope of the line through $(-2, 3)$ and $(4, -9)$ .
Answer: -2
1. What is $y_2 - y_1$ ? -12
2. What is $x_2 - x_1$ ? 6
3. What is the slope? -2
10.
If a line goes downward from left to right, its slope is:
1. a)Positive
2. b)Zero
3. c)Negative
4. d)Undefined
Answer: Negative
When a line goes down from left to right, the change in y is negative while the change in x is positive. Negative divided by positive equals negative.
11.
A road rises 150 meters over a horizontal distance of 2500 meters. What is the grade as a percentage?
Answer: 6
1. What is the rise? 150
2. What is the run? 2500
3. Calculate rise/run as a decimal: 0.06
4. Convert to percentage (multiply by 100): 6
12.
Find the slope of the line through $(-3, -5)$ and $(2, 10)$ .
Answer: 3
$m = \frac{10 - (-5)}{2 - (-3)} = \frac{10 + 5}{2 + 3} = \frac{15}{5} = 3$
13.
A taxi ride costs 5 dollars for 0 km (the starting fee) and 25 dollars for 10 km. What is the rate in dollars per km (the slope)?
Answer: 2
1. What is the change in cost (rise)? 20
2. What is the change in distance (run)? 10
3. What is the rate per km (slope)? 2
14.
What is the slope of the line through $(4, 7)$ and $(4, 15)$ ?
1. a)-8
2. b)0
3. c)Undefined
4. d)8
Answer: Undefined
The rise is $15 - 7 = 8$ and the run is $4 - 4 = 0$ . Since we cannot divide by zero, the slope is undefined: this is a vertical line. (0 is the run divided by the rise, upside down.)
15.
A water tank is being filled. At 2 minutes, it has 15 liters. At 8 minutes, it has 45 liters. What is the filling rate (slope) in liters per minute?
Answer: 5
1. What is the change in volume (rise)? 30
2. What is the change in time (run)? 6
3. What is the filling rate (slope)? 5
16.
Which slope represents the steepest line?
1. a) $m = \frac{1}{2}$
2. b) $m = -3$
3. c) $m = 0$
4. d) $m = 5$
Answer: $m = 5$
$m = 5$ is the steepest because $|5| = 5$ is the largest absolute value. $m = -3$ falls, but with $|-3| = 3$ it is less steep. A slope of 5 means the line rises 5 units for every 1 unit to the right.