Math Problem Statement

Use a graphing utility to construct a table of values for the function f(x) = 4e^(-0.6x) and sketch the graph of the function.

Solution

The given function is f(x)=4e−0.6xf(x) = 4e^{-0.6x}. We are asked to calculate values of f(x)f(x) for specific values of xx, and then identify the correct graph from the options provided.

Step 1: Compute the values of f(x)f(x) at the specified points:

  • For x=−2x = -2: f(−2)=4e−0.6(−2)=4e1.2≈4×3.3201≈13.280f(-2) = 4e^{-0.6(-2)} = 4e^{1.2} \approx 4 \times 3.3201 \approx 13.280

  • For x=−1x = -1: f(−1)=4e−0.6(−1)=4e0.6≈4×1.8221≈7.288f(-1) = 4e^{-0.6(-1)} = 4e^{0.6} \approx 4 \times 1.8221 \approx 7.288

  • For x=0x = 0: f(0)=4e−0.6(0)=4e0=4f(0) = 4e^{-0.6(0)} = 4e^0 = 4

  • For x=1x = 1: f(1)=4e−0.6(1)=4e−0.6≈4×0.5488≈2.195f(1) = 4e^{-0.6(1)} = 4e^{-0.6} \approx 4 \times 0.5488 \approx 2.195

  • For x=2x = 2: f(2)=4e−0.6(2)=4e−1.2≈4×0.3012≈1.205f(2) = 4e^{-0.6(2)} = 4e^{-1.2} \approx 4 \times 0.3012 \approx 1.205

Thus, the table of values is:

x & f(x) \\ \hline -2 & 13.280 \\ -1 & 7.288 \\ 0 & 4 \\ 1 & 2.195 \\ 2 & 1.205 \\ \end{array}$$ ### Step 2: Choose the correct graph. The function $$f(x) = 4e^{-0.6x}$$ is an exponential decay function, starting from a high value and decreasing as $$x$$ increases. Based on the characteristics: - The function is always positive (since the exponential function never reaches zero or becomes negative). - As $$x \to \infty$$, $$f(x)$$ approaches zero. - As $$x \to -\infty$$, $$f(x)$$ grows rapidly towards infinity. Looking at the options, the correct graph is the one in the bottom right, showing exponential decay as $$x$$ increases and rapid growth as $$x$$ becomes negative. Would you like to see more details or have any questions? Here are five related questions: 1. How do we interpret the rate of decay from the exponent $$-0.6x$$? 2. What happens to the graph if we increase the decay constant (e.g., make it $$-0.8x$$)? 3. How would the graph change if the coefficient $$4$$ were larger or smaller? 4. How do exponential functions differ from polynomial or linear functions in behavior? 5. What real-life phenomena can be modeled using exponential decay functions? **Tip:** Exponential functions, like $$e^{-kx}$$, decay faster as the constant $$k$$ increases.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Decay Functions
Graphing

Formulas

f(x) = a * e^(kx)
Exponential decay: f(x) = 4e^(-0.6x)

Theorems

Properties of exponential functions

Suitable Grade Level

Grades 10-12

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