numpy as np import matplotlib.pyplot as plt

Define the functions

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

def g(x): return 2*x + 3

Create a range of x values

x_values = np.linspace(-10, 10, 400)

Calculate the corresponding y values for both functions

y_f = f(x_values) y_g = g(x_values)

Find the intersection by solving 3x + 2 = 2x + 3

3x - 2x = 3 - 2 -> x = 1

x_intersection = 1 y_intersection = f(x_intersection)

Plot the graphs

plt.figure(figsize=(8, 6)) plt.plot(x_values, y_f, label=r'$f(x) = 3x + 2$', color='blue') plt.plot(x_values, y_g, label=r'$g(x) = 2x + 3$', color='green')

Highlight the intersection point

plt.scatter(x_intersection, y_intersection, color='red', zorder=5) plt.text(x_intersection, y_intersection + 1, f'({x_intersection}, {y_intersection})', color='red')

Labels and title

plt.axhline(0, color='black',linewidth=1) plt.axvline(0, color='black',linewidth=1) plt.title('Graphs of $f(x) = 3x + 2$ and $g(x) = 2x + 3$') plt.xlabel('x') plt.ylabel('y') plt.legend()

Show plot

plt.grid(True) plt.show()Here is the graph of the functions $f(x) = 3x + 2$ and $g(x) = 2x + 3$ on the same Cartesian plane. The two functions intersect at the point $(1, 5)$, which is highlighted in red.

Would you like further details about the intersection or any other part of the graph?

Here are some related questions:

1. How do you find the point of intersection of two linear functions?
2. What is the slope of each line in the graph?
3. How would the graph change if the coefficient of $x$ in $f(x)$ was increased?
4. What would happen if $f(x)$ and $g(x)$ were non-linear functions instead of linear ones?
5. How do you graphically solve a system of linear equations?

Tip: To find the point of intersection algebraically, set the two functions equal to each other and solve for $x$.