Ontario 09 Applied (MFM1P)
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Interpret rate of change
Lesson

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 slope, steepness or slope of a line. Further, we look at the equations in slope-intercept form (that is, $y=mx+b$y=mx+b, where $m$m is the slope), 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 increasing or decreasing.

The lines below have increasing slopes. Notice how as the values on the $x$x axis increase, the values on the $y$y axis also increase.

These next graphs have decreasing rates of change. Unlike graphs with a positive slope, 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 slope.

The rate of change 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$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 slope by using a formula. 

$\text{Slope }=\frac{\text{rise }}{\text{run }}$Slope =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 slope of the line between these points also using the rule.

$m=\frac{y_2-y_1}{x_2-x_1}$m=y2y1x2x1

Let's substitute our coordinates into this formula:

$m$m $=$= $\frac{y_2-y_1}{x_2-x_1}$y2y1x2x1
  $=$= $\frac{6-\left(-2\right)}{3-7}$6(2)37
  $=$= $\frac{6+2}{3-7}$6+237
  $=$= $\frac{8}{-4}$84
  $=$= $-2$2

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

 

Slope 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 slope 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

 

Remember

Description of rate of change:     $\text{Rate of change }=\frac{\text{rise }}{\text{run }}$Rate of change =rise run

Slope of Vertical Line = undefined

Slope of Horizontal Line = $0$0

 

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

Examples

Question 1

What kind of slope does the following line have?

  1. Positive

    A

    Negative

    B

    Undefined

    C

    Zero

    D

    Positive

    A

    Negative

    B

    Undefined

    C

    Zero

    D

Question 2

After Mae starts running, her heartbeat increases at a constant rate.

  1. Complete the table.

    Number of minutes passed ($x$x) $0$0 $2$2 $4$4 $6$6 $8$8 $10$10 $12$12
    Heart rate ($y$y) $49$49 $55$55 $61$61 $67$67 $73$73 $79$79 $\editable{}$
  2. What is the unit change in $y$y for the above table?

  3. Form an equation that describes the relationship between the number of minutes passed ($x$x) and Mae’s heartbeat ($y$y).

  4. In the equation, $y=3x+49$y=3x+49, what does $3$3 represent?

    The change in one minute of Mae’s heartbeat.

    A

    The total time Mae has run.

    B

    The total distance Mae has run.

    C

    The change in one minute of Mae’s heartbeat.

    A

    The total time Mae has run.

    B

    The total distance Mae has run.

    C

Question 3

Gas costs a certain amount per litre. The table shows the cost of various amounts of gas.

Number of litres ($x$x) $0$0 $10$10 $20$20 $30$30 $40$40
Cost of gas ($y$y) $0$0 $16.40$16.40 $32.80$32.80 $49.20$49.20 $65.60$65.60
  1. Write an equation linking the number of litres of gas pumped ($x$x) and the cost of the gas ($y$y).

  2. How much does gas cost per litre?

  3. How much would $47$47 litres of gas cost at this unit price?

  4. In the equation, $y=1.64x$y=1.64x, what does $1.64$1.64 represent?

    The unit rate of cost of gas per litre.

    A

    The number of litres of gas pumped.

    B

    The total cost of gas pumped.

    C

    The unit rate of cost of gas per litre.

    A

    The number of litres of gas pumped.

    B

    The total cost of gas pumped.

    C

 

 

 

 

 

Outcomes

9P.LR3.01

Determine, through investigation, that the rate of change of a linear relation can be found by choosing any two points on the line that represents the relation, finding the vertical change between the points (i.e., the rise) and the horizontal change between the points (i.e., the run), and writing the ratio

9P.LR3.02

Determine, through investigation, connections among the representations of a constant rate of change of a linear relation

9P.LR3.04

Express a linear relation as an equation in two variables, using the rate of change and the initial value

9P.LR3.05

Describe the meaning of the rate of change and the initial value for a linear relation arising from a realistic situation and describe a situation that could be modelled by a given linear equation

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