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.

For more help with math, visit

Parabolas

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

Don’t let this question intimidate you just because it has a parabola.  Use the same method that you would use with any other math problem.  Read the question carefully, identify the bottom line, and choose an efficient method to solve the problem.  Then attack the problem and loop back to make sure that you solved for the bottom line. The quadratic function f is graphed in the xy-plane above. If f(x) ≤ u for all values of x, which of the following could be the coordinates of point P?

Your bottom line is which values could be the coordinates of point P, so make a note of the bottom line on your paper, and start with what you know about this point.  You are told that f(x) ≤ u for all the values of x.  That is your y value, so that is just letting you know that nothing can be higher than u, which is on point P.  If you are looking for the highest point on a downward opening parabola, what are you actually looking for?  The vertex!

Think about it this way: as the parabola extends outward from the vertex, both sides stay an equal distance from the vertex. You have just examined the information given about the y-axis, so turn your attention to the x-axis.  You are given two x values that are of equal height on your parabola, so the x value of the vertex, P, must be exactly between them.  Your highest value is 4, so you might be tempted to halve 4 and get 2.  Just be sure to remember that the first point is not at zero, but at 1.  That means that your parabola has been shifted 1 unit to the right.  To find the midpoint, use the midpoint formula, which is simply an average of the two numbers that you have.

You now have the x value of 2.5. You are not given any additional information about the limits of the y-axis, so loop back to the bottom line.  The question is not actually asking you to find both x and y coordinates.  Remember that your bottom line is what “could” be the coordinates, so this is probably enough information to find the correct answer.  Look down at your answer choices now.

(A) (2, 3.5)
(B) (2.25, 3.25)
(C) (2.5, 3)
(D) (2.75, 4)
(E) (3, 2.5)

All you know about the y value is that it must be greater than 0, so all of the y values will work, but only one of the answers has the x value of 2.5.

The correct answer is (C).

On sat.collegeboard.org, 53% of the responses were correct.

For more help with math, visit

Functions

Link of the Day

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8/16 Functions

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

Read each math question carefully so that you can identify exactly what is being asked of you.  Once you have identified the bottom line, assess your options to find an efficient way to solve the problem.  Finally, attack the problem, solve it, and loop back to make sure that your answer addresses the bottom line that you were asked to find. Which of the following could be the equation of the function graphed in the xy-plane above?

You have been given a graph, and you must find the equation that has been graphed.  You could plug all of the answer choices into your calculator, but that would take a long time and you risk making a typo.  Instead, break the graph down into its most basic components.  What shape that you have often seen does this graph most resemble?  It looks like a parabola opening upwards, so you know that f(x) = x² will be part of your equation.

Picture the f(x) = x² parabola in your mind.  It passes through the origin at (0,0).  However, the graph in this problem would extend past the point (0,0) into the negative numbers if you continued the basic curve of the parabola.  To translate the function down on the graph, you would need to subtract a number from the original function.  Now you have f(x) = x² - n, where n = any number.

There is one more step.  The basic curve of the normal parabola has been reflected across the x-axis in this problem so that all the values of the parabola are now positive.  What can you do to make sure that all of the numbers in a function are positive?  Take the absolute value of the function.  Now you have f(x) = |x² - n|.  Look down at your answer choices.

(A) y =  (-x)² + 1
(B) y = -x² + 1
(C) y = |x² + 1|
(D) y = |x² - 1|
(E) y = |(x – 1)²|

(A), (B), and (C) cannot be the answers because they all add to the equation and would result in a parabola that has been shifted above the x-axis.  (E) will not be symmetric to the y-axis, and the graph that you have remains symmetric to the y-axis; it has not been shifted to the right or the left. The (x – 1)² part of the equation in (E) shifts the entire parabola away from its original position on the y-axis.  (D) is the only answer that matches the equation you wrote for this graph.

The correct answer is (D).

On sat.collegeboard.org, 39% of the answers were correct.

For more help, visit www.myknowsys.com!