Functions

Algebra: Functions

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

Use the same method for every math question on the SAT.  Start by reading the question carefully and identifying the bottom line; what do you need to find?  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that it matches the bottom line.

In the xy-plane, the graph of the line with equation y = a intersects the graph of the quadratic function f(x) = x² - 6x + 8 in exactly one point. What is the value of a?

Bottom Line: a = ?

Assess your Options:  You could just try plugging this into your calculator, but if you do not think carefully about what you are doing, you are likely to answer a question that was not asked.  Instead, think through every piece of information that you were given in this problem.

Attack the Problem:  What kind of graph is the function that you are given?  A parabola!  You know this because it has an x².  Picture a parabola in your mind (you know that this is a normal, upward-facing parabola because there is no negative before the x²).  Draw a u-shaped parabola on the xy-axis as part of your scratch work.

Now think about the fact that when y equals a certain number, it creates a vertical line. No matter what y equals, that vertical line will only ever intercept the graph at one point. That's not very useful! However, try flipping the given equation on its head: consider a = y. Remember that a =  is just like x =  and will create a horizontal line. Depending on what x equals, the horizontal line might cross the graph at two points, at no point at all, or at exactly one point--the vertex. You know that you must find the vertex of the parabola, so solve your function for x by setting your polynomial equal to zero and finding the roots of the equation:

x² - 6x + 8 = 0

(x – 2)(x – 4) = 0

(x – 2) = 0 and (x – 4) = 0

x = 2 and x = 4

You just found the two places where the parabola crosses the x-axis: 2 and 4.  All parabolas are symmetrical.  That means that the vertex must be halfway between these two numbers at x = 3.  You found the x value of the vertex, but you need the y value. 

Plug in 3 for the x in your original equation:

f(x) = x² - 6x + 8

f(3) = (3)² - 6(3) +8

f(3) = 9 – 18 + 8

f(3) = -1

Loop Back:  When you solve a function for the f(x), you solve for y.  In this problem, you are told that y = a.  You have solved for a, so you are ready to look down at your answer choices.

(A) -3

(B) -1

(C) 1

(D) 3

(E) 4

The correct answer is (B).

On sat.collegeboard.org, 31% of the responses were correct.

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