Recognizing Linear Functions

a

We want to know if the function given in the following table has a constant rate of change.

We can check the rate of change algebraically by noting the change in $x$ and change in $y$ between each of the data points.

We find the rate of change between each row by dividing the change in

y

by the change in

x .

rate of change = \frac{change in y}{change in x}

Let's find the rate of change between the first two rows.

rate of change = \frac{change in y}{change in x}

change in x = 1.67

,

change in y = 20

rate of change = \frac{2 0}{1 . 6 7}

UseCalcUse a calculator

rate of change = 11.97604 \dots

RoundDecRound to

rate of change \approx 11.98

The rate of change between the first two rows is approximately

11.98 .

In the context of the exercise, this means that the runner ran

20

meters with an average speed of

11.98

m/s. We can follow the same process for the other given segment splits.

As we can see, the rate of change is not constant. The runner had a higher average speed later in the race than in the middle.

b

We have been asked to find if the function given by the following table has a constant rate of change.

Let's find the change in $x$ and change in $y$ between each of the rows.

We can find the rate of change between each set of points by using the following relationship.

rate of change = \frac{change in y}{change in x}

We will now find the rate of change for the first two points.

rate of change = \frac{change in y}{change in x}

change in x = 1.6

,

change in y = 2

rate of change = \frac{2}{1 . 6}

UseCalcUse a calculator

rate of change = 1.25

The rate of change between the first two points is

1.25 .

Let's repeat this process for the other points.

As we can see, the rate of change is constant. It tells us that the elevator's speed is $1.25$ floors/second.

Recognizing Linear Functions 1.6 - Solution