import numpy as np import matplotlib.pyplot as plt

Define the function f(x) = 2 * 3^x + 1

def f(x): return 2 * (3**x) + 1

Generate x values

x = np.linspace(-3, 3, 500) # Range of x values for the graph

Calculate y values

y = f(x)

Plot the function

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r'$f(x) = 2 \cdot 3^x + 1$') plt.axhline(0, color='black', linewidth=0.8, linestyle='--') plt.axvline(0, color='black', linewidth=0.8, linestyle='--') plt.title("Graph of $f(x) = 2 \cdot 3^x + 1$") plt.xlabel("x") plt.ylabel("f(x)") plt.legend() plt.grid(alpha=0.5) plt.show() <<ImageDisplayed>>successHere is the graph of the function $f(x) = 2 \cdot 3^x + 1$. Let me know if you want further details or have any questions!

Related Questions:

1. How does changing the base of the exponent (e.g., from 3 to 4) affect the graph?
2. What are the asymptotes of the function, and how do they relate to its behavior?
3. Can we find the inverse of $f(x) = 2 \cdot 3^x + 1$, and if so, how does its graph compare?
4. What happens to the graph if the multiplier (2) is increased or decreased?
5. How would adding a different constant (e.g., +5 instead of +1) affect the vertical shift?

Tip:

When analyzing exponential functions, focus on the base and the constants to understand their effect on growth rates and shifts.