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Advanced Math / Nonlinear functions Difficulty: Hard

$f left parenthesis t right parenthesis equals 8,000 left parenthesis 0.65 right parenthesis Superscript t$

The given function $f$ models the number of coupons a company sent to their customers at the end of each year, where $t$ represents the number of years since the end of $1998$ , and $0 \leq t \leq 5$ . If $y = f (t)$ is graphed in the ty-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?

The minimum estimated number of coupons the company sent to their customers during the $5$ years was $1,428$ .

The minimum estimated number of coupons the company sent to their customers during the $5$ years was $8,000$ .

The estimated number of coupons the company sent to their customers at the end of $1998$ was $1,428$ .

The estimated number of coupons the company sent to their customers at the end of $1998$ was $8,000$ .

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Explanation

Choice D is correct. The y-intercept of a graph in the ty-plane is the point where $t equals 0$ . For the given function $f$ , the y-intercept of the graph of $y = f (t)$ in the ty-plane can be found by substituting $0$ for $t$ in the equation $y equals 8,000 left parenthesis 0 period 65 right parenthesis Superscript t$ , which gives $y equals 8,000 left parenthesis 0 period 65 right parenthesis Superscript 0$ . This is equivalent to $y equals 8,000 left parenthesis 1 right parenthesis$ , or $y equals 8,000$ . Therefore, the y-intercept of the graph of $y = f (t)$ is $left parenthesis 0 comma 8,000 right parenthesis$ . 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, $f (t)$ 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, $t equals 0$ 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 $y = f (t)$ is that the estimated number of coupons the company sent to their customers at the end of $1998$ was $8,000$ .

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

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

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