# Functions

## 10/18 Functions

Read the following SAT test question and select the correct answer.

For every SAT math problem, read the problem carefully so that you know exactly what information you are given.  Then identify the bottom line, the information that you must find.  Assess your options for solving the problem, and choose the most efficient method to get to the answer.  Attack the problem to find the answer, and loop back to your bottom line to make sure that your answer matches what you were supposed to find.

A manager estimates that if the company charges p dollars for their new product, where 0 ≤ p ≤ 200, then the revenue from the product will be r(p) = 2,000p – 10p² dollars each week. According to this model, for which of the following values of p would the company’s weekly revenue for the product be the greatest?

Bottom Line:  Which of the following values of p will result in the greatest revenue?

Assess your options:  You could work backwards by plugging in all of the answer choices to r(p) = 2,000p – 10p², but that will take time.  Instead, use what you know about functions to determine the answer.

Attack the problem:  You know what the graph of x² looks like: a parabola that makes a “u.”  What happens to that graph when it is -x²?  That “u” turns upside-down and the parabola looks like a hill.  That is what you have for your function r(p) = 2,000p – 10p².  Now simplify your function by pulling out the numbers and variables that your two terms have in common so that r(p) = 10p(200 – p).  If you set each part of this equation equal to zero, you will find where the parabola crosses the x-axis.  If 10p = 0 and 200 – p = 0, then p = 0 and 200.  The parabola crosses the x-axis at 0 and 200.  That makes sense because you were told in the problem that 0 ≤ p ≤ 200.  Think about the characteristics of parabolas once more.  All parabolas are symmetrical.  Where will your greatest value for the revenue be?  It will be at the top of that “hill” exactly between 0 and 200.  What is the midpoint between 0 and 200?  100.

Loop back: Your bottom line was the value of p that would have the greatest revenue.  Although your function used r(p) rather than f(x),  that p value had to be on the x-axis.  You solved for the bottom line, so you are ready to look down at the answer choices.

(A)  10
(B)  20
(C)  50
(D)  100
(E)  200

The correct answer is (D).

On sat.collegeboard.org, 37% of the responses were correct.

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