As you prepare for the SAT, you’ll definitely want to review quadratic functions. You’ll need to know how to solve quadratic equations, as well as understand the properties of the graphs of these functions.
Here’s a challenging problem involving a quadratic function. Give it a try and then scroll down for the solution.
f(x) = a(x + b)(x – 8)
The equation above represents a quadratic function where a and b are constants. If the function is graphed in the xy-plane, its vertex is at the point (2, -54). What is the value of a + b?
(Scroll down for the solution)
In order to get this problem correct, you need to know about the symmetry properties of the graphs of quadratic functions. The graph of a quadratic function is always a parabola centered on a line called the axis of symmetry which passes through the vertex.
The x-intercepts (or “roots” or “zeroes”) of the function must be the same distance away from the axis of symmetry. In this case, the axis of symmetry is the vertical line x = 2. We know this because the vertex of the function is (2, -54).
For this function, because one of the factors is (x – 8), we know that one of the x-intercepts of the graph must be (8, 0). This point lies 6 horizontal units away from the axis of symmetry. The other x-intercept must also be 6 horizontal units away from the axis of symmetry but in the other direction, so it has to be (-4, 0). That makes the first factor (x + 4).
Here’s a graph of the function:
Now that we know the value of b, let’s see if we can figure out the value of a. We can do this using the fact that one of the points on the graph is (2, -54). Let’s plug that point into the function and solve for a.
f(x) = a(x + 4)(x – 8)
-54 = a(2 + 4)(2 – 8)
-54 = a(6)(-6)
-54 = a(-36)
-54/-36 = a
1.5 = a
So, the value of a is 1.5 and the value of b is 4. That makes a + b = 1.5 + 4 = 5.5. The correct answer is Choice B.
If you want to get some more practice working with quadratic functions and their graphs, you might want to check out our e-book, SAT Math: Focus on Quadratics & Parabolas. It’s available on Amazon for only $0.99.
If you have questions about this problem or anything else to do with the SAT, send us an email at firstname.lastname@example.org