numpy as np import matplotlib.pyplot as plt

Define the function

def f(x): return (x * (x - 1)) / (x + 2)

Generate x values

x = np.linspace(-10, 10, 1000)

Avoid division by zero

x = x[x != -2]

Generate y values

y = f(x)

Plot the graph

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$\frac{x(x-1)}{x+2}$", color="blue") plt.axhline(0, color='black', linewidth=0.8, linestyle='--') plt.axvline(0, color='black', linewidth=0.8, linestyle='--')

Highlight the asymptote at x = -2

plt.axvline(-2, color='red', linestyle='--', label="Asymptote $x = -2$")

Add labels and legend

plt.title("Graph of $\frac{x(x-1)}{x+2}$") plt.xlabel("$x$") plt.ylabel("$y$") plt.legend() plt.grid(alpha=0.5) plt.ylim(-10, 10) # Adjust y-axis to avoid extreme values near the asymptote

plt.show()Here is the graph of the function $f(x) = \frac{x(x-1)}{x+2}$:

Key features:

1. Vertical asymptote: At $x = -2$, the function is undefined.
2. X-intercepts: $x = 0$ and $x = 1$, where the numerator equals zero.
3. Behavior as $x \to \pm\infty$: The function approaches $y = x - 1$ as an oblique asymptote.
4. Critical points: The graph changes direction near these points, influenced by the derivative of the function.

Would you like a deeper analysis, such as finding the critical points or explaining the oblique asymptote?

Related Questions:

1. What are the steps to find the vertical and horizontal asymptotes of a function?
2. How do you calculate the derivative of $f(x) = \frac{x(x-1)}{x+2}$?
3. What does the behavior of $f(x)$ near its asymptotes tell us about the function?
4. How can we determine concavity and points of inflection for this function?
5. How does the numerator or denominator being zero affect the graph?

Tip:

When sketching graphs of rational functions, always identify asymptotes and intercepts first—they provide a clear structure for your graph.