To find the equation of the linear function that models the relationship between the number of hours worked by Esperanza and the amount of money she earned, we can use the information given about her earnings.
Let's define:
- \( x \) as the number of hours worked
- \( y \) as the amount of money earned
From the information provided, we have two points:
- For 6 hours worked: \( (6, 120) \)
- For 8 hours worked: \( (8, 156) \)
Step 1: Find the slope (m).
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( (6, 120) \) and \( (8, 156) \):
\[ m = \frac{156 - 120}{8 - 6} = \frac{36}{2} = 18 \]
Step 2: Use the slope and one of the points to find the equation.
Now we have the slope \( m = 18 \). We can use the point-slope form of the equation \( y - y_1 = m(x - x_1) \). Let's use the point \( (6, 120) \):
\[ y - 120 = 18(x - 6) \]
Step 3: Simplify the equation.
Distributing the slope on the right side:
\[ y - 120 = 18x - 108 \]
Now, adding 120 to both sides:
\[ y = 18x + 12 \]
Step 4: Write the final equation.
Thus, the equation of the linear function in \( y = mx + b \) form is:
\[ y = 18x + 12 \]
This means that Esperanza earns $18 for every hour worked and has a base earning of $12.