import numpy as np import matplotlib.pyplot as plt

Define the functions

x = np.linspace(-2, 3, 500) # A range of x values y1 = np.exp(x) # e^x y2 = x**2 - 2 # x^2 - 2

Plot the functions

plt.figure(figsize=(8, 6)) plt.plot(x, y1, label=r"$e^x$") plt.plot(x, y2, label=r"$x^2 - 2$") plt.axhline(0, color='black', linewidth=0.8, linestyle='--') # x-axis plt.axvline(0, color='black', linewidth=0.8, linestyle='--') # y-axis

Add labels, title, legend

plt.title("Graphs of $e^x$ and $x^2 - 2$", fontsize=14) plt.xlabel("$x$", fontsize=12) plt.ylabel("$y$", fontsize=12) plt.legend(fontsize=12) plt.grid(alpha=0.5) plt.show() <<ImageDisplayed>>successHere are the graphs of $e^x$ and $x^2 - 2$. The two functions are plotted over the range $[-2, 3]$.

Let me know if you need any further analysis or questions about the behavior of these functions!

Here are 5 questions for further exploration:

1. Where do these two functions intersect?
2. What is the slope of $e^x$ at $x = 1$?
3. Find the x-intercepts of $x^2 - 2$.
4. What happens to $e^x$ and $x^2 - 2$ as $x \to \infty$?
5. How does the concavity of $e^x$ compare to $x^2 - 2$?

Tip: For better insights into these graphs, you can analyze their derivatives or solve the equations for intersections!