https://file.aiquickdraw.com/mathbot/file/1737510358784blpzzkod.jpg

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.

Fitting Exponential Functions to Data. Graph the given data on a coordinate plane, determine whether the data exhibits linear or exponential behavior, and write an equation for the data.

Learn how to analyze computer value depreciation data to determine whether it exhibits linear or exponential behavior. Explore graphing, exponential decay functions, and data fitting with step-by-step guidance for Grades 9-12.

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