To find the equation of the linear function that models the relationship between the number of hours Esperanza works (let's denote this as \(x\)) and the amount of money she earns (let's denote this as \(y\)), we first need to determine the slope (\(m\)) and the y-intercept (\(b\)) of the line.
From the information provided:
- When \(x = 6\) hours, \(y = 120\) dollars.
- When \(x = 8\) hours, \(y = 156\) dollars.
Step 1: Calculate the slope (\(m\)).
The formula for slope (\(m\)) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_1, y_1) = (6, 120) \) and \( (x_2, y_2) = (8, 156) \).
Now plug in the values: \[ m = \frac{156 - 120}{8 - 6} = \frac{36}{2} = 18 \]
Step 2: Use the slope and one point to find the y-intercept (\(b\)).
We can use the point \( (6, 120) \) and the slope \( m = 18 \) in the equation \( y = mx + b \).
Substituting \(x = 6\) and \(y = 120\): \[ 120 = 18(6) + b \]
Calculating \(18(6)\): \[ 120 = 108 + b \]
Now, solve for \(b\): \[ b = 120 - 108 = 12 \]
Step 3: Write the equation of the linear function.
Now that we have both \(m\) and \(b\), we can write the equation: \[ y = 18x + 12 \]
So, the equation of the linear function that models the relationship between the number of hours Esperanza works and the amount of money she earns is: \[ \boxed{y = 18x + 12} \]