Use the image to answer the question.

Question

A decreasing curve with an arrow on the end is plotted on a coordinate plane. The curve begins at the point left parenthesis 0 comma 700 right parenthesis.

The value of a cell phone decreases exponentially at a rate of 24% each year after its release date. If the initial value of the phone is $700, the situation can be modeled by the equation v(t)=700(0.76)t
, where v is the value of the cell phone t years since its release date. Interpret the graph of the equation and identify the true statement.

(1 point)
Responses

As t increases, v decreases quickly at first and then slowly.
As t increases, v decreases quickly at first and then slowly.

As t increases, v increases quickly at first and then slowly.
As t increases, v increases quickly at first and then slowly.

As t increases, v decreases slowly at first and then quickly.
As t increases, v decreases slowly at first and then quickly.

As t increases, v increases slowly at first and then quickly

Answer 1

The model provided for the value of a cell phone is \( v(t) = 700(0.76)^t \), which indicates that the value of the phone decreases exponentially over time at a rate of 24% per year.

To interpret the graph:

• At \( t = 0 \), the value starts at $700.
• As \( t \) increases, the exponent \( t \) causes the \( 0.76^t \) term to decrease, which means \( v(t) \) also decreases.

In an exponential decay function, the value decreases quickly at first and then slows down over time. This is characteristic of exponential functions where a percentage decrease leads to rapid initial changes that gradually taper off.

Based on this understanding, the true statement among the responses provided is:

As t increases, v decreases quickly at first and then slowly.