Interpret Rate of Change

We have already learnt that a rate is a ratio between two measurements with different units.

When we graph these rates, the rate of change can be understood as the gradient, steepness or slope of a line. Further, we look at the equations in gradient-intercept form (that is, $y=mx+b$y=mx+b, where $m$m is the gradient), the larger the absolute value of $m$m, the steeper the slope of the line.

For example, a line with a slope of of $4$4 is steeper than a line with a slope of $\frac{2}{3}$23​. Similarly, a line with a slope of $-2$−2 is steeper than a line with a slope of $1$1, even though one is positive and one is negative.

 

Increasing or Decreasing?

The rate of change in a graph can be positive or negative.

The lines below have positive rates of change. Notice how as the values on the $x$x-axis increase, the values on the $y$y-axis also increase.

These next graphs have negative rates of change. Unlike graphs with a positive gradient, as the values on the $x$x-axis increase, the values on the $y$y-axis decrease.

 

Rate of Change

The rate of change of a line is a measure of how steep it is.  In mathematics we also call this the gradient.

The rate of change is a single value that describes:

• if a line is increasing (has positive gradient)
• if a line is decreasing (has negative gradient)
• 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$1 step across (run of $1$1), the line goes up $2$2 (rise of $2$2).  This line has a rate of change of $2$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 gradient by using a formula. 

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

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

Finding the rate of change 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 gradient of the line between these points also using the rule.

$m=\frac{y_2-y_1}{x_2-x_1}$m=y2​−y1​x2​−x1​​

Let's substitute our coordinates into this formula:

$m$m	$=$=	$\frac{y_2-y_1}{x_2-x_1}$y2​−y1​x2​−x1​​
	$=$=	$\frac{6-\left(-2\right)}{3-7}$6−(−2)3−7​
	$=$=	$\frac{6+2}{3-7}$6+23−7​
	$=$=	$\frac{8}{-4}$8−4​
	$=$=	$-2$−2

So the rate of change between these coordinates is $-2$−2.

 

Gradient of horizontal and vertical lines

Horizontal lines

horizontal lines have a rate of change of 0

Why?

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

$\text{Rate of change }$Rate of change	$=$=	$\frac{\text{Rise }}{\text{Run }}$Rise Run ​
	$=$=	$\frac{0}{\text{Run }}$0Run ​
	$=$=	$0$0

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

 

Vertical Lines

The rate of change of vertical lines is undefined

Why?

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

$\text{Rate of Change }$Rate of Change	$=$=	$\frac{\text{Rise }}{\text{Run }}$Rise Run ​
	$=$=	$\frac{\text{Rise }}{0}$Rise 0​

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

 

 

Rate of change can also be seen in everyday situations. Let's work through some worked examples.

Examples

Question 1

Question 2

Question 3