The graph of the function f in the xy-plane above is a parabola. Which of the following defines f ?

Options

A. `f (x) = 4(x − 3)^2 + 1`
B. `f (x) = 4(x + 3)^2 + 1`
C. `f (x) = (x − 3)^2 + 1`
D. `f (x) = 3(x + 3)^2 + 1`

Answer

A

Explanation

The equation of a parabola in vertex form is `f(x) = a(x − h)^2 + k`, where the point (h, k) is the vertex of the parabola and a is a constant. The graph shows that the coordinates of the vertex are (3, 1), so h = 3 and k = 1. Therefore, an equation that defines f can be written as `f(x) = a(x − 3)^2 + 1`. To find a, substitute a value for x and its corresponding value for y, or f(x). For example, (4, 5) is a point on the graph of f. So a must satisfy the equation` 5 = a(4 − 3)^2 + 1`, which can be rewritten as `4 = a(1)^2`, or a = 4. An equation that defines f is therefore `f(x) = 4(x − 3)^2 + 1`.
Choice B is incorrect and may result from a sign error when writing the equation of the parabola in vertex form. Choice C is incorrect and may result from omitting the constant a from the vertex form of the equation of the parabola. Choice D is incorrect and may result from a sign error when writing the equation of the parabola in vertex form as well as by miscalculating the value of a.