Final answer:
To determine the linear function modeling the relationship between hours worked and earnings for Esperanza, we calculate the slope from two points (6, 120) and (8, 158) and find the y-intercept. The resulting linear function is y = 19x + 6, indicating she earns $19 per hour plus a $6 fixed amount.
Step-by-step explanation:
To find the equation of the linear function that models the relationship between the number of hours Esperanza works and the amount of money she earns, we use two given points. Yesterday, Esperanza worked for 6 hours and earned $120, which gives us the point (6, 120). Today, she worked 8 hours and earned $158, corresponding to the point (8, 158). The slope (m) of the linear function can be calculated by dividing the difference in earnings by the difference in hours.
The slope m is:
m = (earnings difference) / (hours difference)
m = (158 - 120) / (8 - 6)
m = 38 / 2
m = 19
Therefore, Esperanza earns $19 per hour worked. We can now find the y-intercept (b), which represents the starting amount Esperanza earns before working any hours. To find the y-intercept, we can plug in the value of the slope and one of the points into the equation y = mx + b and solve for b.
Using the point (6, 120):
120 = 19 × 6 + b
120 = 114 + b
b = 120 - 114
b = 6
So, the y-intercept is $6. Now that we have both the slope and y-intercept, we can write the equation:
y = 19x + 6
Therefore, the linear function that models the relationship between the number of hours Esperanza works and the amount of money she earns is y = 19x + 6.