Choice D is correct. Since the value of $y$ increases by a constant factor, $4$ , for each increase of $1$ in the value of $x$ , the relationship between $x$ and $y$ is exponential. An exponential relationship between $x$ and $y$ can be represented by an equation of the form $y = a {(b)}^{x}$ , where $a$ is the value of $x$ when $y equals 0$ and $y$ increases by a factor of $b$ for each increase of $1$ in the value of $x$ . Since $y equals 200$ when $x equals 0$ , $a equals 200$ . Since $y$ increases by a factor of $4$ for each increase of $1$ in the value of $x$ , $b equals 4$ . Substituting $200$ for $a$ and $4$ for $b$ in the equation $y = a {(b)}^{x}$ yields $y equals 200 left parenthesis 4 right parenthesis Superscript x$ . Thus, the equation $y equals 200 left parenthesis 4 right parenthesis Superscript x$ represents the relationship between $x$ and $y$ .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect. This equation represents a relationship where for each increase of $1$ in the value of $x$ , the value of $y$ increases by a factor of $200$ , not $4$ , and when $x equals 0$ , $y$ is equal to $4$ , not $200$ .

Choice C is incorrect and may result from conceptual errors.