To use the intercepts to graph the linear function given by the equation $2x = 4y + 4$, we need to rewrite it in a more familiar form and find the intercepts.

Rewrite the Equation

First, let's rewrite the equation in standard form $Ax + By = C$.

$2x = 4y + 4$

Subtract $4y$ from both sides:

$2x - 4y = 4$

Divide the entire equation by 2 to simplify:

$x - 2y = 2$

Finding the Intercepts

X-intercept

The x-intercept occurs when $y = 0$.

$x - 2(0) = 2 \implies x = 2$

So, the x-intercept is $(2, 0)$.

Y-intercept

The y-intercept occurs when $x = 0$.

$0 - 2y = 2 \implies -2y = 2 \implies y = -1$

So, the y-intercept is $(0, -1)$.

Graphing the Equation

1. Plot the x-intercept $(2, 0)$.
2. Plot the y-intercept $(0, -1)$.
3. Draw a straight line through these two points.

Here's the graph of the equation:

x & y & \text{Points} \\ \hline 2 & 0 & (2, 0) \\ 0 & -1 & (0, -1) \\ \end{array}$$ Now, let's plot this. ```python import matplotlib.pyplot as plt import numpy as np # Define the function to plot the line def plot_line(): x = np.linspace(-2, 4, 400) y = (1/2)*x - 1 plt.figure(figsize=(8, 6)) plt.plot(x, y, '-r', label='x - 2y = 2') plt.plot(2, 0, 'bo') # x-intercept plt.plot(0, -1, 'bo') # y-intercept plt.axhline(0, color='black',linewidth=0.5) plt.axvline(0, color='black',linewidth=0.5) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.legend(loc='best') plt.title('Graph of x - 2y = 2') plt.xlabel('x') plt.ylabel('y') plt.xlim(-2, 4) plt.ylim(-3, 3) plt.show() plot_line() ``` The graph shows the line passing through the intercepts $$(2, 0)$$ and $$(0, -1)$$. Would you like more details or have any questions? Here are 5 more questions you might want to explore: 1. How do you convert a linear equation from standard form to slope-intercept form? 2. How do you find the slope of a line given its equation? 3. What is the significance of the slope and intercepts in a linear equation? 4. How do you graph a linear equation using a table of values? 5. How do you determine the point of intersection between two linear equations? **Tip:** Always check your intercepts by substituting them back into the original equation to ensure they satisfy the equation.