numpy as np import matplotlib.pyplot as plt

Define the functions

x = np.linspace(-10, 10, 400) f_x = x h_x = 4 * x - 2

Plot the functions

plt.figure(figsize=(8, 6)) plt.plot(x, f_x, label='f(x) = x', linewidth=2) plt.plot(x, h_x, label='h(x) = 4x - 2', linewidth=2)

Add labels, legend, and grid

plt.title('Graphs of f(x) = x and h(x) = 4x - 2') plt.xlabel('x') plt.ylabel('y') plt.axhline(0, color='black',linewidth=0.5) plt.axvline(0, color='black',linewidth=0.5) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.legend()

Show the plot

plt.show()The graph displays the two functions:

• $f(x) = x$, which is a straight line passing through the origin with a slope of 1.
• $h(x) = 4x - 2$, which is a straight line with a slope of 4 and a y-intercept at -2.

Would you like to analyze these graphs further or explore specific aspects of their relationships?

Here are 5 questions to expand your understanding:

1. How do the slopes of $f(x)$ and $h(x)$ compare, and what does this tell you about their steepness?
2. At what point do $f(x)$ and $h(x)$ intersect?
3. How does the y-intercept of $h(x)$ influence its position relative to $f(x)$?
4. What is the difference in their rates of change, and how does this reflect on the graph?
5. Can you derive the equation for where $f(x)$ and $h(x)$ are equal?

Tip: When comparing functions graphically, focus on their slope and intercept to understand their behavior relative to each other.