SAT Math: A Challenging Functions Problem

As my students and I have prepared for the new SAT, one of the things we’ve noticed is that the more challenging Math problems aren’t just computational or problems where you can throw something into your calculator to get the answer. They are more theoretical — you have to really know the math in order to come up with the correct answer.

Let’s consider a challenging problem involving functions.

A function is defined by the equation below.

f(x) = a(x – 2)(x – 10)

If the minimum value of the function is -8, what is the value of a?

A)  -2
B)  1/4
C)  1/2
D)  2

Scroll down for the solution
.
.
.
.
The first thing to recognize is that if you were to multiple out the right side of the function equation, you would get something that has an x² as its highest power. The function is quadratic and its graph would be a parabola.

You can also see that f(x) is equal to zero when x = 2 and when x = 10. That means that the graph will cross the x-axis at (2, 0) and (10, 0). Because of the symmetry properties of parabolas, the axis of symmetry must be exactly halfway between these two points. If you average the x-coordinates, you’ll see that the axis of symmetry for this graph is x = 6.

Additionally, the minimum value of the function must be the vertex. Because the vertex is on the axis of symmetry, its coordinates must be (6, -8).

Functions 5-4

You can now plug those coordinates into the function equation to determine the value of a.

f(x) = a(x – 2)(x – 10)
-8 = a(6 – 2)(6 – 10)
-8 = a(4)(-4)
-8 = -16a
-8/-16 = -16a/-16
1/2 = a

The correct answer is Choice C.

If you have questions about this problem or anything else to do with the SAT, send us an email at info@cardinalec.com

Leave a Reply

Your email address will not be published. Required fields are marked *