Recognizing Linear Functions

a

Let's start by drawing the graph and mark the $x$ -coordinates of the points $A$ and $B .$

We can see that there are

4

units between the

x

-coordinates making

Δ x = 4 .

We can also calculate the change in

x

using the relationship

Δ x = x_{2} - x_{1} .

Δ x = x_{2} - x_{1}

x_{2} = 7

,

x_{1} = 3

Δ x = 7 - 3

SubTermSubtract term

Δ x = 4

In the same way we determine the change in the

y

-direction,

Δ y,

by finding the vertical distance between the points

A

and

B .

We can see that the vertical distance between the points is

2

giving us

Δ y = 2 .

We can also insert the

y

-values in the formula

Δ y = y_{2} - y_{1} .

Δ y = y_{2} - y_{1}

y_{2} = 6

,

y_{1} = 4

Δ y = 6 - 4

SubTermSubtract term

Δ y = 2

The answer is that the change in the

x

-direction,

Δ x,

is

4

and the change in the

y

-direction,

Δ y,

is

2

.

b

We can find the line's rate of change using the following equation.

rate of change = \frac{Δ y}{Δ x}

Let's calculate the rate of change. In Part A we found that

Δ x = 4

and

Δ y = 2 .

m = \frac{Δ y}{Δ x}

Δ x = 4

,

Δ y = 2

m = \frac{2}{4}

\frac{a}{b} = \frac{a / 2}{b / 2}

m = \frac{1}{2}

WriteDecWrite as a decimal

m = 0.5

The line's rate of change is

0.5 .

Recognizing Linear Functions 1.4 - Solution