import numpy as np import matplotlib.pyplot as plt

Define the function

def f(x): return (-2 * x + 8) / (2 * x + 4)

Define x values avoiding the vertical asymptote x = -2

x_values = np.linspace(-10, 10, 1000) x_values = x_values[x_values != -2]

Compute y values

y_values = f(x_values)

Plot the function

plt.figure(figsize=(8, 6)) plt.plot(x_values, y_values, label="f(x) = (-2x + 8) / (2x + 4)", color="blue")

Plot vertical asymptote (x = -2)

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

Plot horizontal asymptote (y = -1)

plt.axhline(y=-1, color='green', linestyle='--', label='Horizontal Asymptote y = -1')

Add an additional point

additional_point_x = 0 additional_point_y = f(additional_point_x) plt.scatter(additional_point_x, additional_point_y, color='purple', label=f'Point (0, {additional_point_y})')

Configure plot

plt.title("Graph of f(x) = (-2x + 8) / (2x + 4)") plt.xlabel("x") plt.ylabel("f(x)") plt.legend() plt.grid(True) plt.ylim(-10, 10) # Limit y-axis for better visualization

Show plot

plt.show() <<ImageDisplayed>>successHere is the graph of $f(x) = \frac{-2x + 8}{2x + 4}$, which includes:

1. Horizontal asymptote at $y = -1$.
2. Vertical asymptote at $x = -2$.
3. An additional point plotted at $(0, f(0)) = (0, 2)$.

Let me know if you would like further details or have any questions about this graph.

Related Questions:

1. How are horizontal and vertical asymptotes determined for a rational function?
2. What happens to $f(x)$ as $x$ approaches the vertical asymptote?
3. How does adding or subtracting constants in $f(x)$ affect the graph's asymptotes?
4. What is the significance of the additional point in understanding the graph?
5. How can we find intercepts for this function algebraically?

Tip:

To identify horizontal asymptotes, compare the degrees of the numerator and denominator in the rational function.