Let's analyze the problem step by step.

Provided Information:

• Time (years): 0, 1, 2, 3, 4, 5
• Depreciation (dollars): 990, 500, 250, 110, 60, 25

Problem Breakdown:

1. Graph the Data: We will plot the data points (Time, Depreciation) on a coordinate plane:

   • Points: (0, 990), (1, 500), (2, 250), (3, 110), (4, 60), (5, 25)

2. Determine Behavior (Linear or Exponential): Compare the rate of change. Exponential behavior typically involves a consistent ratio between successive terms, while linear behavior involves a consistent difference.

3. Write an Equation: We will fit the data using an exponential decay function:
   $y = a \cdot b^x$
   where:

   • $a$ is the initial value at $x = 0$,
   • $b$ is the base (decay factor).

Let me perform the calculations and sketch the graph. numpy as np import matplotlib.pyplot as plt

Data points

time = np.array([0, 1, 2, 3, 4, 5]) depreciation = np.array([990, 500, 250, 110, 60, 25])

Plotting the graph

plt.figure(figsize=(8, 5)) plt.scatter(time, depreciation, color='blue', label='Data Points') plt.plot(time, depreciation, linestyle='--', color='red', label='Trend Line') plt.title("Depreciation of Computer Value Over Time") plt.xlabel("Time (Years)") plt.ylabel("Depreciation (Dollars)") plt.grid(True) plt.legend() plt.show()### Step 1: Graph of the Data The scatter plot shows the depreciation data over time. The trend line (dashed red) illustrates the decreasing value of the computer.

Step 2: Behavior Analysis

This data exhibits exponential decay rather than linear behavior because the depreciation does not decrease by a constant amount (as seen in linear behavior). Instead:

• The ratio of successive values is approximately constant. For example: $\frac{500}{990} \approx 0.505, \quad \frac{250}{500} = 0.5, \quad \frac{110}{250} \approx 0.44, \quad \frac{60}{110} \approx 0.545$

Step 3: Finding the Exponential Decay Equation

The general equation is: $y = a \cdot b^x$

• At $x = 0$, $y = a = 990$.
• To find $b$, calculate the ratio between consecutive values:
  $b \approx \frac{500}{990} \approx 0.505$

The equation becomes: $y = 990 \cdot (0.505)^x$

Would you like further details, such as step-by-step verification or a refined graph of the equation? Here are some follow-up questions to explore further:

1. Would you like me to verify the exponential equation against all data points?
2. Should I calculate the percentage error of this model?
3. Do you need a detailed explanation of why the data is not linear?
4. Would you like me to extend the graph for predictions beyond 5 years?
5. Should I create a step-by-step guide for finding the equation manually?

Tip: In exponential decay, values decrease rapidly at first and then level off, forming an asymptote.