Question

sketch a graph of the following function and verify using your graphing calculator.
f(x)=9e^{-0.2x}
choose the correct graph below.
○ a.
○ b.
○ c.
○ d.

Explanation:

Step1: Analyze the function type

The function \(f(x) = 9e^{-0.2x}\) is an exponential decay function. The general form of an exponential function is \(y = Ae^{kx}\), where \(A = 9\) and \(k=- 0.2<0\).

Step2: Find the y - intercept

When \(x = 0\), \(f(0)=9e^{-0.2\times0}=9\times1 = 9\). So the y - intercept is \((0,9)\).

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

As \(x\to+\infty\), \(y = 9e^{-0.2x}=\frac{9}{e^{0.2x}}\to0\).

Step4: Analyze the behavior as \(x\to-\infty\)

As \(x\to-\infty\), \(y = 9e^{-0.2x}=9e^{|0.2x|}\to+\infty\).

Exponential decay functions have a y - intercept, and as \(x\) increases, the function value approaches \(0\). Option A is an exponential decay function with a y - intercept at \(y = 9\) (since when \(x = 0\), \(y=9\) in the graph of option A) and as \(x\) increases, \(y\) approaches \(0\). Option B is an exponential growth function (\(k>0\) in \(y = Ae^{kx}\)), option C has a y - intercept higher than \(9\) (when \(x = 0\), \(y\) in option C is \(15\)), and option D is not an exponential decay function.

Answer:

A.