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.