To graph the equation \( y = e^{-2x} \), follow these steps:

### 1. **Understand the equation**:
   - The equation is in the form \( y = ae^{bx} \), where \( a = 1 \) and \( b = -2 \).
   - This is an **exponential decay** function because the base of the exponent is \( e \), which is approximately 2.718, and the exponent is negative (\( -2x \)).

### 2. **Identify key characteristics**:
   - **Asymptote**: The graph will approach but never cross the **x-axis** (y = 0). This means there is a **horizontal asymptote** at \( y = 0 \).
   - **Intercept**: When \( x = 0 \), \( y = e^0 = 1 \). So the curve will pass through the point (0, 1).
   - **Decay**: Since the exponent is negative, the function decreases as \( x \) increases.
   
### 3. **Plotting the graph**:
   - **Step 1**: Start by plotting the **y-intercept** at \( (0, 1) \).
   - **Step 2**: Choose several values for \( x \) (positive and negative), and calculate the corresponding \( y \)-values.
     - For \( x = -1 \): \( y = e^{-2(-1)} = e^2 \approx 7.389 \)
     - For \( x = 1 \): \( y = e^{-2(1)} = e^{-2} \approx 0.135 \)
     - For \( x = 2 \): \( y = e^{-2(2)} = e^{-4} \approx 0.018 \)
   - **Step 3**: Plot these points: \( (-1, 7.389) \), \( (0, 1) \), \( (1, 0.135) \), \( (2, 0.018) \), etc.
   - **Step 4**: Draw a smooth curve through these points, making sure it approaches the x-axis as \( x \to \infty \) (going to the right) and increases rapidly as \( x \to -\infty \) (going to the left).

### 4. **Behavior**:
   - As \( x \to \infty \), the value of \( y \) gets closer and closer to 0, but never quite reaches it.
   - As \( x \to -\infty \), the value of \( y \) grows exponentially large.

### 5. **Final Graph**: 
   - The curve starts high (on the left), passes through \( (0,1) \), and gradually decreases toward \( y = 0 \) as \( x \) increases. The graph will be **smooth and continuous**, and it will never cross the x-axis.

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Would you like any further details on how to graph exponential functions? Or have any questions about specific steps?

### Related Questions:
1. What are the differences between exponential growth and decay functions?
2. How does changing the value of \( b \) in \( y = e^{bx} \) affect the graph?
3. How can you solve equations involving exponential functions like \( e^{-2x} = 5 \)?
4. What happens if the base of the exponential changes (e.g., \( y = 2^{-2x} \))?
5. Can you graph the inverse of \( y = e^{-2x} \)?

### Tip:
When graphing exponential functions, always remember to look for the horizontal asymptote (in this case, \( y = 0 \)) and identify the intercept for clarity.

Learn how to graph the exponential decay function y = e^-2x with step-by-step instructions. This guide covers exponential functions, identifying asymptotes, and plotting key points. Perfect for high school students in Grades 9-12.

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