 3.05 Describing the slope of a line

Lesson

The slope of a line tells us how steep or slanted the line is. More specifically, it is the ratio of the change in the $y$y-coordinates (vertical change) to the change in the $x$x-coordinates (horizontal change). There are several different types of slopes that a line can have.

Positive and negative slopes

Some lines have positive slopes, like these:   If we picture the line as a hill and imagine walking on the hill from left to right we will notice that with the lines above we would be walking uphill. This is why we consider the slope of these lines to be positive.

But some lines have decreasing slopes, like these:   If we imagine walking on a hill from left to right again, for these lines we would be walking downhill.

Exploration

Use the applet below to change the slope and create lines that have positive or negative slope. Make a note of anything interesting. Try to find or imagine types of slope other than positive or negative.

Notice that changing the slope doesn't move the line up or down, it simply adjusts the steepness. The closer to $0$0 the slope is, the flatter or less steep the line is. The farther from $0$0 the slope is (in the positive or negative direction) the steeper the line is. That is to say a line with a slope of $2$2 and a line with a slope of $-2$2 are equally steep, just slanted in opposite directions and a line with a slope of $1$1 is steeper than a line with a slope of $\frac{1}{4}$14.

Worked Example

Question 1

Consider the graph shown. What type of slope does it have? Think: As we look at the graph from left to right what does the line seem to be doing?

Do: The line is going up or increasing as you read the graph from left to right. We could even imagine ourselves walking uphill on this line. Therefore, the line has a positive slope.

Slope of horizontal lines

Some lines have a slope that is neither positive nor negative.

The line shown below is horizontal. Imagine being at the beach and looking out at the horizon. The line where the ocean meets the sky is a horizontal line. The name makes a little more sense now!  Note the coordinates of points $A$A, $B$B, and $C$C on the horizontal line above. $A=\left(-4,4\right)$A=(4,4) $B=\left(2,4\right)$B=(2,4) $C=\left(4,4\right)$C=(4,4) Notice that all of the $y$y-coordinates are the same. The line has a $y$y value of $4$4 no matter what the $x$x value is.

Horizontal lines have no vertical change. The only change is from left to right so these lines look flat. The slope of a horizontal line is $0$0 because the line isn't actually sloped at all so we can say that is has no slope or $0$0 slope.

Slope of vertical lines

The line shown below is vertical. Note the coordinates of points $A$A$B$B, and $C$C on the vertical line above. $A=\left(5,-4\right)$A=(5,4) $B=\left(5,-2\right)$B=(5,2) $C=\left(5,4\right)$C=(5,4) Notice that all of the $x$x-coordinates are the same. The line has an $x$x value of $5$5 no matter what the $y$y value is.

Vertical lines are infinitely sloped. If we imagine this line as a hill there is no way we could walk up it. We would need climbing equipment so this is really more of a cliff! For this reason, the slope of a vertical line is considered undefined. We will investigate why that is more once we learn how to calculate slope mathematically.

Practice problems

Question 2

A line crosses the $x$x-axis at $-7$7, and crosses the $y$y-axis at $5$5. Is the slope of the line positive or negative?

1. Positive

A

Negative

B

Positive

A

Negative

B

Question 3

1. Which line has a slope of $0$0?

A

B

C

D

E

A

B

C

D

E
2. Which line has a slope that is undefined?

A

B

C

D

E

A

B

C

D

E

Question 4

Consider the points on the graph.

1. What is the slope of the line segment $AB$AB?

$0$0

A

$3$3

B

undefined

C

$2$2

D

$0$0

A

$3$3

B

undefined

C

$2$2

D
2. What is the slope of the line segment $BC$BC?

$5$5

A

$0$0

B

$6$6

C

undefined

D

$5$5

A

$0$0

B

$6$6

C

undefined

D

Outcomes

8.16a

Recognize and describe the graph of a linear function with a slope that is positive, negative, or zero