SAT Question 1f353a9e: Nonlinear Functions answer and explanation with Desmos solution

Choice D is correct. The y-intercept of a graph in the ty-plane is the point where $t=0$. For the given function $f$, the y-intercept of the graph of $y=f\left(t\right)$ in the ty-plane can be found by substituting $0$ for $t$ in the equation $y=8,000{\left(0.65\right)}^t$, which gives $y=8,000{\left(0.65\right)}^0$. This is equivalent to $y=8,000\left(1\right)$, or $y=8,000$. Therefore, the y-intercept of the graph of $y=f\left(t\right)$ is $\left(0,8,000\right)$. It's given that the function $f$ models the number of coupons a company sent to their customers at the end of each year. Therefore, $f\left(t\right)$ represents the estimated number of coupons the company sent to their customers at the end of each year. It's also given that $t$ represents the number of years since the end of $1998$. Therefore, $t=0$ represents $0$ years since the end of $1998$, or the end of $1998$. Thus, the best interpretation of the y-intercept of the graph of $y=f\left(t\right)$ is that the estimated number of coupons the company sent to their customers at the end of $1998$ was $8,000$.