To graph the function $y = e^{2x} - 3e^x + 2$, follow these steps:

Step 1: Understand the behavior of the function.

The function consists of exponential terms. Let's break it down:

• $e^{2x}$: This grows rapidly as $x$ increases, since the exponent is $2x$.
• $-3e^x$: This term decays less rapidly than $e^{2x}$, as the exponent is linear in $x$.
• $+2$: This is a constant shift upward by 2 units.

Step 2: Find the intercepts.

• Y-intercept (when $x = 0$): $y(0) = e^{2(0)} - 3e^{0} + 2 = e^0 - 3e^0 + 2 = 1 - 3 + 2 = 0$ So, the y-intercept is at $(0, 0)$.

• X-intercepts (when $y = 0$): To find the x-intercepts, solve: $e^{2x} - 3e^x + 2 = 0$ Let $u = e^x$, so $e^{2x} = u^2$. The equation becomes: $u^2 - 3u + 2 = 0$ Factor the quadratic: $(u - 1)(u - 2) = 0$ So, $u = 1$ or $u = 2$. Therefore:

  • If $u = 1$, $e^x = 1$, so $x = 0$.
  • If $u = 2$, $e^x = 2$, so $x = \ln(2) \approx 0.693$.

Thus, the x-intercepts are at $x = 0$ and $x \approx 0.693$.

Step 3: Find the asymptotic behavior.

• As $x \to \infty$, $e^{2x}$ grows much faster than $e^x$ or the constant 2. Therefore, the function will grow without bound as $x \to \infty$.
• As $x \to -\infty$, both $e^{2x}$ and $e^x$ approach 0, so the function approaches 2 (the constant term).

Thus, the horizontal asymptote is $y = 2$.

Step 4: Plot key points and asymptotes.

• Plot the intercepts $(0, 0)$ and $(\ln(2), 0)$.
• Plot some additional points for negative and positive values of $x$ (e.g., for $x = -1, 1, 2$).
• Draw the horizontal asymptote at $y = 2$.

Step 5: Sketch the graph.

• The graph should start near the horizontal asymptote of $y = 2$ as $x \to -\infty$.
• It passes through the intercepts, with the function rising steeply as $x$ increases, due to the $e^{2x}$ term dominating for large positive $x$.

Summary:

• Intercepts: $(0, 0)$ and $(\ln(2), 0)$.
• Horizontal asymptote: $y = 2$.
• End behavior: As $x \to -\infty$, $y \to 2$, and as $x \to \infty$, $y \to \infty$.

Would you like to explore more about this function, or have any further questions?

Here are some related questions:

1. How do exponential functions behave for different values of the base (e.g., $a^x$ vs $e^x$)?
2. What are the differences in the graphs of $y = e^x$ and $y = e^{2x}$?
3. How can you use derivatives to analyze the shape of the graph?
4. What are the applications of the function $y = e^{2x} - 3e^x + 2$?
5. How does the horizontal asymptote relate to the function's behavior at large values of $x$?

Tip: To graph any exponential function accurately, always consider its asymptotes and behavior at extreme values of $x$.