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Advanced Math / Nonlinear functions Difficulty: Medium

A physics class is planning an experiment about a toy rocket. The equation $y = - 16 {(x - 5.6)}^{2} + 502$ gives the estimated height $y$ , in feet, of the toy rocket $x$ seconds after it is launched into the air. Which of the following is the best interpretation of the vertex of the graph of the equation in the xy-plane?

This toy rocket reaches an estimated maximum height of $502$ feet $16$ seconds after it is launched into the air.

This toy rocket reaches an estimated maximum height of $502$ feet $5.6$ seconds after it is launched into the air.

This toy rocket reaches an estimated maximum height of $16$ feet $502$ seconds after it is launched into the air.

This toy rocket reaches an estimated maximum height of $5.6$ feet $502$ seconds after it is launched into the air.

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Explanation

Choice B is correct. The vertex of the graph of a quadratic equation is where it reaches its minimum or maximum value. When a quadratic equation is written in the form $y = a {(x - h)}^{2} + k$ , the vertex of the parabola represented by the equation is $(x, y) = (h, k)$ . In the given equation $y = - 16 {(x - 5.6)}^{2} + 502$ , the value of $h$ is $5.6$ and the value of $k$ is $502$ . It follows that the vertex of the graph of this equation in the xy-plane is $left parenthesis x comma y right parenthesis equals left parenthesis 5.6 comma 502 right parenthesis$ . Additionally, since $a = - 16$ in the given equation, the graph of the quadratic equation opens down, and the vertex represents a maximum. It’s given that the value of $y$ represents the estimated height, in feet, of the toy rocket $x$ seconds after it is launched into the air. Therefore, this toy rocket reaches an estimated maximum height of $502$ feet $5.6$ seconds after it is launched into the air.

Choice A is incorrect. The $16$ in the equation is an indicator of how narrow the graph of the equation is rather than where it reaches its maximum.

Choice C is incorrect. The $16$ in the equation is an indicator of how narrow the graph of the equation is rather than where it reaches its maximum.

Choice D is incorrect. This is an interpretation of the vertex of the graph of the equation $y = - 16 {(x - 502)}^{2} + 5.6$ , not of the equation $y = - 16 {(x - 5.6)}^{2} + 502$ .