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

Answer 1

To interpret the graph of the equation \( v(t) = 700(0.76)^t \) where \( v \) is the value of the cell phone and \( t \) is the number of years since its release date, we need to analyze the behavior of the function over time.

The function is an example of exponential decay because the base of the exponent \( 0.76 \) is less than 1. This means that as \( t \) increases (i.e., as time passes), the value \( v(t) \) will decrease.

1. Behavior of Exponential Decay: In the early years, the value of the phone will be significantly reduced because a larger percentage of the original value is being lost. Over time, since the phone’s value is already low, the absolute dollar amount that is lost each year becomes smaller, leading to a slower rate of decrease.

2. Summary of Behavior: Therefore, as time \( t \) increases, the function \( v(t) \) decreases quickly at first (when the phone is still relatively new and has a higher value) and then decreases slowly over time as the value approaching zero.

Given these points, the correct interpretation of the graph based on the provided options is:

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