4. Linear Equations, Proportional Relationships & Lines

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.

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*.

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.

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.

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.

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.

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?

Positive

ANegative

BPositive

ANegative

B

Answer the following.

Which line has a slope of $0$0?

Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ELoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...EWhich line has a slope that is undefined?

Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ELoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...E

Consider the points on the graph.

Loading Graph...

What is the slope of the line segment $AB$

`A``B`?$0$0

A$3$3

Bundefined

C$2$2

D$0$0

A$3$3

Bundefined

C$2$2

DWhat is the slope of the line segment $BC$

`B``C`?$5$5

A$0$0

B$6$6

Cundefined

D$5$5

A$0$0

B$6$6

Cundefined

D