Recognizing Linear Functions

We want to determine if the rate of change in cost is constant with respect to the number of pencils bought. To do so, we will calculate the rate of change between each consecutive pair of points. We will use the rate of change formula to find the rate of change

r

between two points.

r = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

The first two points are

(1, 0.25)

and

(4, 1) .

Let's substitute them in the above formula.

r = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePointsSubstitute

(1, 0.25)

&

(4, 1)

r = \frac{1 - 0 . 2 5}{4 - 1}

SubTermsSubtract terms

r = \frac{0 . 7 5}{3}

UseCalcUse a calculator

r = 0.25

The rate of change between the first two points is

0.25 .

We will use the same method to find the rate of change

r

between

(4, 1)

and

(7, 1.75),

and between

(7, 1.75)

and

(12, 3) .

Pair of Points	$\frac{y _{2} - y _{1}}{x _{2} - x _{1}}$	$r = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
$(1, 0.25)$ $&$ $(4, 1)$	$\frac{1 - 0 . 2 5}{4 - 1}$	$0.25$
$(4, 1)$ $&$ $(7, 1.75)$	$\frac{1 . 7 5 - 1}{7 - 4}$	$0.25$
$(7, 1.75)$ $&$ $(12, 3)$	$\frac{3 - 1 . 7 5}{1 2 - 7}$	$0.25$

As shown, all the rates are the same. This means that the rate of change is constant.

Recognizing Linear Functions 1.8 - Solution