Use intercepts to graph the linear function.

2 x equals 4 y plus 4

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:

\[
\begin{array}{c|c|c}
x & y & \text{Points} \\
\hline
2 & 0 & (2, 0) \\
0 & -1 & (0, -1) \\
\end{array}
\]

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.

Learn how to use intercepts to graph the linear function given by the equation 2x = 4y + 4. This problem covers graphing lines, finding intercepts, and understanding linear equations. Suitable for students in Grades 8-10.

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