Describe the rate of change of a line

Increasing and Decreasing

We have already been working with the idea of the slope of a line.  

We have seen that some lines have increasing slopes.

And some have decreasing slopes. (we sometimes call this a negative rise)

This applet will show you positive and negative slopes.

Slope

The slope of a line is a measure of how steep it is.  In mathematics we call this the SLOPE.

A slope is a single value that describes:

• if a line is increasing (has positive slope)
• if a line is decreasing (has negative slope)
• how far up or down the line moves (change in the $y$y value) with every unit step to the right (for every 1 unit increase in $x$x)

Take a look at this line for example. I've highlighted the horizontal and vertical steps.  

We call the horizontal measurement the RUN and the vertical measurement the RISE.  

Here, for every 1 step across (run of 1), the line goes up 2 (rise of 2).  This line has a slope of 2.

Sometimes it is difficult to measure how far the line goes up or down (how much the $y$y value changes) in 1 horizontal unit. In this case we calculate the slope by using a formula. 

$\text{Slope }=\frac{\text{rise }}{\text{run }}$Slope =rise run ​

Where you take any two points whose coordinates are known or can be easily found, and look for the rise and run between them.

Finding the slope from a graph

If you have a graph, like this one, you can find the rise and run by drawing a right triangle created by any two points on the line. The line itself becomes the hypotenuse. 

This line has a slope of $\frac{\text{rise }}{\text{run }}=\frac{4}{3}$rise run ​=43​

In this case, the slope is positive because, over the $3$3 unit increase in the $x$x values, the $y$y value has increased.

This applet allows you to see the rise and run between two points on a line of your choosing.

Finding the slope from a pair of coordinates

If you have a pair of coordinates, such as $\left(3,6\right)$(3,6) and $\left(7,-2\right)$(7,−2) we can find the slope of the line between these points also using the rule.

$\text{Slope }=\frac{\text{rise }}{\text{run }}$Slope =rise run ​

Firstly, it is good mathematical practice to draw a quick sketch of the points.  This helps us quickly identify what the line looks like (we can see that it will be decreasing in nature, so it will have negative slope).  

The rise is the difference in the $y$y values of the points. In this case the difference between $-2$−2 and $6$6 is $-8$−8 

($-2-6=-8$−2−6=−8). 

The run is the difference in the $x$x values of the points. In this case the difference between $7$7 and $3$3 is $4$4 

($7-3=4$7−3=4)

Notice that we subtracted the $x$x values and the $y$y values in the same order.

$\text{Slope }$Slope	$=$=	$\frac{\text{rise }}{\text{run }}$rise run ​
	$=$=	$\frac{-8}{4}$−84​
	$=$=	$-2$−2

 

Slope of horizontal and vertical lines

Horizontal lines have NO rise value.  The $rise=0$rise=0.  

So the slope of a horizontal line is $\text{Slope }=\frac{\text{rise }}{\text{run }}$Slope =rise run ​ =$\frac{0}{run}$0run​ = $0$0  

 It doesn't matter what the run is, the slope will be 0.

Vertical lines have NO run value.  The $run=0$run=0.  

So the slope of a vertical line is $\text{Slope }=\frac{\text{rise }}{\text{run }}$Slope =rise run ​ =$\frac{\text{rise }}{0}$rise 0​

It doesn't matter what the rise is, any division by 0 results in the value being undefined. 

 

Let's have a look at some worked examples.

 

Question 1

Question 2

Question 3