Question

Damian works after school. Each
day he earns a set amount, plus
an hourly wage, as shown in the
table. Write a linear function f
that Damian can use to determine
his pay.
hours: 1, 1.5, 2, 2.5, 3
pay: 22, 28, 34, 40, 46

Answer 1

To find a linear function \( f(h) \) that represents Damian's pay based on the number of hours he works \( h \), we will first analyze the data provided:

Given the hours and corresponding pay:

\[
\begin{align*}
\text{Hours (h)} & : 1, 1.5, 2, 2.5, 3 \\
\text{Pay (f(h))} & : 22, 28, 34, 40, 46 \\
\end{align*}
\]

### Step 1: Identify the Change in Pay and Hours

We will calculate the change in pay for the change in hours to find the hourly wage.

- From 1 to 1.5 hours: \( 28 - 22 = 6 \) (Increase of 6 for 0.5 hours)
- From 1.5 to 2 hours: \( 34 - 28 = 6 \) (Increase of 6 for 0.5 hours)
- From 2 to 2.5 hours: \( 40 - 34 = 6 \) (Increase of 6 for 0.5 hours)
- From 2.5 to 3 hours: \( 46 - 40 = 6 \) (Increase of 6 for 0.5 hours)

This indicates that for every additional 0.5 hours, Damian earns an extra $6. Therefore, his hourly wage can be calculated as follows:

\[
\text{Hourly Wage} = \frac{6}{0.5} = 12
\]

### Step 2: Determine the Initial Amount (y-intercept)

To find the y-intercept (the amount Damian earns when he works 0 hours), we can use any of the data points to derive the equation. Let's calculate using the point \( (1, 22) \):

From the linear equation form:

\[
f(h) = mh + b
\]

Where:
- \( m \) is the slope (which we calculated as $12 per hour).
- \( b \) is the y-intercept (fixed amount).

Substituting \( h = 1 \) and \( f(h) = 22 \):

\[
22 = 12(1) + b
\]
\[
22 = 12 + b
\]
\[
b = 22 - 12 = 10
\]

### Step 3: Write the Function

We have determined:
- The slope \( m = 12 \)
- The y-intercept \( b = 10 \)

Thus, the linear function that represents Damian's pay based on hours worked is:

\[
f(h) = 12h + 10
\]

### Summary

The function \( f(h) = 12h + 10 \) can be used by Damian to calculate his pay based on the number of hours \( h \) he works.