Two variables, x and y , are related such that for each increase of 1 in

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 $y=a{\left(b\right)}^x$, where $a$ is the value of $x$ when $y=0$ and $y$ increases by a factor of $b$ for each increase of $1$ in the value of $x$. Since $y=200$ when $x=0$, $a=200$. Since $y$ increases by a factor of $4$ for each increase of $1$ in the value of $x$, $b=4$. Substituting $200$ for $a$ and $4$ for $b$ in the equation $y=a{\left(b\right)}^x$ yields $y=200{\left(4\right)}^x$. Thus, the equation $y=200{\left(4\right)}^x$ 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 $x=0$, $y$ is equal to $4$, not $200$.

Choice C is incorrect and may result from conceptual errors.