Math: 1178f2df

ID: 1178f2df

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)

Comment: Pin down the first function, then work to solve. This is a more challenging question than it may appear to be for the following reasons:

1) There are no answer choices supplied.

2) You need to know how to model a parabola.

3) The table does not directly provide access to f(x).

4) There are a number of calculations involved, and the numbers get large.

5) There are interactions between positive and negative numbers, providing one more chance to miss something small that will make the answer wayward.

6) You have to use the calculator wisely, since, again, there is no direct input supplied.

Method 1: If y = f(x) + 4, then f(x) = y - 4. Thus, the quadratic function will have the following coordinates that we can glean from the table: (21, -8 - 4); (23, 8 - 4); (25, -8 - 4) or (21, -12); (23, 4); and (25, -12). Remember the vertex form of a parabola, which for some reason is not provided on the reference screen: y = a(x - h)^2 + k, in which (h, k) is the vertex and a shows the stretch and the direction of the basic graph y = x^2. Because parabolas have a vertical axis of symmetry, the vertex is (23, 4). To solve for a, plug in the vertex and another point on the graph of f(x).

f(x) = a(x - 23)^2 + 4

(-12) = a((21) - 23)^2 + 4

-16 = a(-2)^2

-16 = 4a

-4 = a

f(x) = -4(x - 23)^2 + 4

The y-intercept of the function is when x = 0. Substitute to solve.

f(0) = -4((0) - 23)^2 + 4

f(0) = -4(-23)^2 + 4

f(0) = -4(529) + 4

f(0) = -2116 + 4

f(0) = -2112

The correct answer is -2112.

Method 2: You can use Desmos to model the parabola in vertex form, written above, then play with the slider for a until you hit the other two points. Just make sure you are not modeling y = f(x) + 4, or everything will be shifted up 4 units. I will show the default value for a below.

Sliding a to the right makes the graph skinnier, but it never points downward, where we know we need to have y-values. When a = 0, a line emerges at y = 4. Playing around a bit once the graph begins to point downward by making a negative, you should be looking for one or both of (21, -12) and (25, -12). When a = -4, you have the correct graph, which you can ascertain by sliding the point on the parabola to 21 or 25.

From here, you can either scroll way down to see the y-intercept or, perhaps more sensibly, let the calculator work through the arithmetic when x = 0.

 As the blue box shows, the correct answer is -2112.

Method 3: One of the above methods should prove efficient for solving this question. Other methods, such as exploring the standard form of the parabola, will likely lead to wasting time and, eventually, errant guessing.