Piecewise linear functions

Question

Derek earns $25 per hour (or part of an hour) for a regular 40-hour work week. Sometimes, he works over 40 hours a week and earns “double time” for the extra hours worked. (That means that his hourly rate for the additional hours over 40 hours per week is double his normal hourly rate.) Per work regulations, Derek can work at most 50 hours a week. 

A) If Derek works 46 hours in one week, how much would he earn that week? (Make sure you include units)

B) Create a piecewise linear function to represent the amount Derek will earn in one week if he works xxhours. (Make sure to define your variables and use proper notation)

C) Graph this function in an appropriate window. (Make sure to label your axes, scale and any important points.) 

D) Is your graph continuous?

Answer 1

A) Derek works 40 hours at $25 per hour and 6 additional hours at $50 per hour. His earnings are:

Earnings = 40·$25 + 6·$50 = ______?

B) If Derek works x hours, the first 40 are paid at $25 per hour and the amount over 40 hours, x - 40, is paid at $50 per hours:

$25x if 0 < x ≤ 40 hours

Earnings =

$25·40 + $50·(x-40) if x > 40 hours

You can simplify $25·40 + $50·(x-40) to $50x - 1000.

C) I'll let you graph it. For values of x between 0 and 40, graph 25x. For values of x greater than 40, graph 50x - 1000. The two are continuous if there is no break or gap in the graph where they join at x = 40. Alternatively, if 25x = 50x-1000 for x = 40, they are continuous.