We are given a graph which shows the number of tokens customers received for each dollar spent.

We are asked to find the rate of change and the initial value of a function represented by this graph. Let's start with the rate of change.

Rate of Change

To find the rate of change, we will first choose two points from the graph. We can use any two points on the graph, but this solution will use $(5, 32)$ and $(6, 36) .$

Each of the chosen points represents a number of tokens received corresponding to a certain amount of dollars spent.

Point	Dollars Spent	Tokens Received
$(5, 32)$	$5$	$32$
$(6, 38)$	$6$	$38$

Now to find the rate of change, we will divide the change in the tokens received by the change in dollars spent.

Rate of change = \frac{Change in tokens received}{Change in dollars spent} ⇕ Rate of change = \frac{3 8 - 3 2}{6 - 5}

Let's simplify this expression!

Rate of change = \frac{3 8 - 3 2}{6 - 5}

SubTerms

Subtract terms

Rate of change = \frac{6}{1}

DivByOne

$\frac{a}{1} = a$

Rate of change = 6

We found that the rate of change is

6,

which means that the number of tokens received per

1

dollar is

6 .

Next, we will find the initial value.

Initial Value

The initial value is the $y -$ value when $x = 0 .$ To find it, we will need to extend the given graph.

It looks like the line crosses the

y -

axis a little above

0

but it is difficult to see its exact value. Let's think of another way to find this value. Since the rate of change is constant,

x

and

y

have a linear relationship. This means that we can represent their relationship with an equation in slope-intercept form.

y = m x + b

In this equation,

m

is the rate of change and

b

is the initial value. Earlier, we found that the rate of change is

6 .

Let's substitute

m = 6

into the equation.

y = m x + b ⇕ y = 6 x + b

Now let's find the value of

b .

We can substitute any point on the line into our partial equation and solve for

b .

Let's use

(5, 32)

again.

y = 6 x + b

SubstituteII

$x = 5$ , $y = 32$

32 = 6 (5) + b

▼

Solve for

b

Multiply

32 = 30 + b

SubEqn

$LHS - 30 = RHS - 30$

32 - 30 = b

SubTerm

Subtract term

2 = b

RearrangeEqn

Rearrange equation

b = 2

We found that the initial value

b

is equal to

2 .

We can interpret this as the first

100

customers each receive

2

additional tokens per dollar spent at the arcade.

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