Digital PSAT Math Practice Question 279: Answer and Explanation

Question: 279

The vertex form of parabola f is given by the function f (x) = x² + 3. The function g(x) is given by the equation g(x) = -f (x) + 2. If the vertex of f (x) is point (a, b), what is the vertex of g(x) in terms of a and b?

• A. (a, b + 2)
• B. (-a, -b + 4)
• C. (2a, b - 2)
• D. (a, b - 4)

Correct Answer: D

Explanation:

(D) Note that in general, the vertex form of a parabola is given by (x - h)² + k, where (h, k) is the vertex of the parabola. You are told that the vertex form of parabola f is f(x) = x² + 3, which can be rewritten as f (x) = (x - 0)² + 3. Thus, the vertex of f is (0, 3) = (a, b). Now, you want to know how the vertex of g(x) compares to (a, b). Note that g(x) = - f(x) + 2 = - (x² + 3) + 2 = - x² - 3 + 2 = x² - 1. Thus, the vertex of g is (0, -1) = (a, -1). So, you just need to determine how b compares to -1. Since b = 3 and -1 = 3 - 4 = b - 4, it follows that the vertex of g is (a, b - 4) when written in terms of a and b, which matches choice (D).

Alternatively, you can think about what is happening in terms of transformations. Suppose (a, b) is the vertex of f. Then -f (x) reflects f across the x-axis, so the vertex (a, b) becomes (a, -b) when reflected. Then g(x) = -f (x) + 2 shifts -f (x) up vertically by 2, so the vertex of g(x) is (a, -b + 2). Note that this is not an answer choice. However, the vertex of f is (0, 3) since f shifts the parabola x² up 3 units. Thus, b = 3. So the vertex of g is (a, -b + 2) = (a, -3 + 2) = (a, -1). Plugging 3 in for b in the answer choices, you see that choice (D) is the only choice that results in vertex (a, -1).