Question

two years ago, marc bought a new cell phone. his purchase price was $750. the value of the phone for the next 2 years was $637.50 and $541.88, respectively. find the exponential function that represents the value of marcs phone t years after it was purchased. (1 point)

$f(t)=750cdot0.85^{t}$

$f(t)=750cdot0.85t$

$f(t)=637.50cdot0.85^{t}$

$f(t)=750cdot1.15^{t}$

Explanation:

Step1: Recall the exponential function form

The general form of an exponential function is \(f(t)=a\cdot b^{t}\), where \(a\) is the initial value. Here, the initial purchase price \(a = 750\) (when \(t = 0\), \(f(0)=750\)), so we can eliminate the option \(f(t)=637.50\cdot0.85^{t}\).

Step2: Check the non - exponential form

The function \(y = 750\cdot0.85t\) is a linear function (in the form \(y=mx\) where \(m = 750\times0.85\)), not an exponential function. So we can eliminate \(f(t)=750\cdot0.85t\).

Step3: Calculate the growth/decay factor

We know that when \(t = 1\), \(f(1)=637.50\). Substitute \(a = 750\) and \(t = 1\) into \(f(t)=a\cdot b^{t}\), we get \(637.50=750\cdot b^{1}\). Then \(b=\frac{637.50}{750}=0.85\).
If \(b = 1.15\), when \(t = 1\), \(f(1)=750\times1.15 = 862.5
eq637.50\). So we can eliminate \(f(t)=750\cdot1.15^{t}\).

Answer:

\(f(t)=750\cdot0.85^{t}\) (the first option)