Question

sketch a graph of the following function and verify using your graphing calculator.

$y = 6000e^{0.08x}$

choose the correct graph below.

○ a.

○ b.

○ c.

○ d.

Explanation:

Step1: Analyze the function type

The function \(y = 6000e^{0.08x}\) is an exponential growth function. The general form of an exponential function is \(y = Ae^{kx}\), where \(A>0\) and \(k>0\) for growth. Here \(A = 6000\) and \(k=0.08>0\).

Step2: Check the y - intercept

When \(x = 0\), \(y=6000e^{0}=6000\times1 = 6000\).

Step3: Analyze the behavior as \(x\to+\infty\)

As \(x\to+\infty\), since \(k = 0.08>0\), \(y = 6000e^{0.08x}\to+\infty\). Exponential growth functions increase as \(x\) increases.
Option A and C are decreasing functions (since the \(y\) - values go down as \(x\) increases, which would be the case for \(y=Ae^{-kx},k > 0\)). Option D has a \(y\) - intercept near \(0\) (when \(x = 0\), if \(y\approx0\), it does not match \(y(0)=6000\)). Option B is an increasing function (matches exponential growth) and has a non - zero \(y\) - intercept (matches \(y(0) = 6000\)).

Answer:

B.