A painter charges an initial fee of $50 and $30 for each hour of labor. Write a linear function to determine the total cost. How much will it cost if the painter takes 4.5 hours to paint?

The initial fee of $50 is essentially the y intercept because this is the value when x = 0 (x is the number of labor-hours). So b = 50.

The slope is m = 30 because each increase of 1 hour leads to the cost bumping up by 30 dollars. In other words, slope = rise/run = (change in cost) / (change in hours) = 30/1

So we plug m = 30 and b = 50 into the y = mx+b formula to get y = 30x+50

Replace y with f (x) to get f (x) = 30x+50

The linear function for the cost is f (x) = 30x+50

Note: Some books may use other letters (instead of x and f (x)), but the idea is still the same

Once you know the cost function, you replace x with 4.5 to find the amount it will cost to have a painter work for 4.5 hours.

f (x) = 30x+50

f (4.5) = 30*4.5+50

f (4.5) = 135+50

f (4.5) = 185

It will cost 185 dollars to have the painter work for 4.5 hours