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Advanced Math / Nonlinear functions Difficulty: Hard

The function $f (x) = \frac{1}{9} {(x - 7)}^{2} + 3$ gives a metal ball's height above the ground $f (x)$ , in inches, $x$ seconds after it started moving on a track, where $0 \leq x \leq 10$ . Which of the following is the best interpretation of the vertex of the graph of $y = f (x)$ in the xy-plane?

The metal ball's minimum height was $3$ inches above the ground.

The metal ball's minimum height was $7$ inches above the ground.

The metal ball's height was $3$ inches above the ground when it started moving.

The metal ball's height was $7$ inches above the ground when it started moving.

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Explanation

Choice A is correct. The graph of a quadratic equation in the form $y = a {(x - h)}^{2} + k$ , where $a$ , $h$ , and $k$ are positive constants, is a parabola that opens upward with vertex $(h, k)$ . The given function $f (x) = \frac{1}{9} {(x - 7)}^{2} + 3$ is in the form $y = a {(x - h)}^{2} + k$ , where $y = f (x)$ , $a = \frac{1}{9}$ , $h equals 7$ , and $k equals 3$ . Therefore, the graph of $y = f (x)$ is a parabola that opens upward with vertex $left parenthesis 7 comma 3 right parenthesis$ . Since the parabola opens upward, the vertex is the lowest point on the graph. It follows that the y-coordinate of the vertex of the graph of $y = f (x)$ is the minimum value of $f (x)$ . Therefore, the minimum value of $f (x)$ is $3$ . It’s given that $f (x) = \frac{1}{9} {(x - 7)}^{2} + 3$ represents the metal ball’s height above the ground, in inches, $x$ seconds after it started moving on a track. Therefore, the best interpretation of the vertex of the graph of $y = f (x)$ is that the metal ball’s minimum height was $3$ inches above the ground.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.