Thank you, Mike, for your ever-awesome explanations! Here’s a question from April 2017 SAT, Section 4.22:

The graphs in the xy-plane of the following quadratic equations each have x-intercepts of -2 and 4. The graph of which equation has its vertex farthest from the x-axis?

A) y = -7 (x + 2)(x – 4)

B) y = 1/10 (x + 2)(x – 4)

C) y = -1/2 (x + 2)(x – 4)

D) y = 5 (x + 2)(x – 4)

It’s useful to note that all these equations are the same after the leading coefficient. So, for example, we could just say and then the choices would just be:

A)

B)

C)

D)

The reason I point that out is that the effect of multiplying a function by a constant, in general, is that the larger the *absolute value* of the constant is, the more you’ll be stretching that function away from the *x*-axis. If you know that little fact, then you can immediately pick the answer choice with the coefficient with the largest absolute value, which is choice A.

If you *don’t* know that little fact, but you have a graphing calculator, you’re still in good shape. Just graph each one (on the same screen, if possible) to see which one has its vertex farthest from the *x*-axis.

## Comments (1)

Thank you, Mike!

Follow-up question:

Could we also solve by putting the equations in Vertex Form y = a (x-h)^2 + k, where (h,k) is the vertex and then we get a [ (x-1)^2 – 9]

The vertex furthest from the x-axis is the one with the largest absolute value k in (h,k), which will be the choice with the largest a-value, since k will equal (-9). Answer choice A, with an a-value of 7, when multipled by (-9), gives the largest absolute value number for k in the vertext point (h,k).