Draw a graph of f(x)=-2x+8/2x+4  by first placing the horizontal and vertical asymptotes, then plotting an additional point on the graph.

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.

Draw a graph of f(x)=-2x+8/2x+4 by first placing the horizontal and vertical asymptotes, then plotting an additional point on the graph.

Learn how to graph the rational function f(x) = (-2x + 8) / (2x + 4) by identifying its horizontal and vertical asymptotes and plotting key points. Includes a step-by-step solution with visual examples. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation