Match each linear graph to its correct equation in this educational worksheet.

To solve the problem of matching each graph to its corresponding equation, we need to analyze the slope and y-intercept of each graph. Here's a step-by-step explanation:

Key Concepts:

1. Slope (m): The steepness of the line. It is calculated as the change in \( y \) divided by the change in \( x \) (\( m = \frac{\Delta y}{\Delta x} \)).
- A positive slope means the line rises from left to right.
- A negative slope means the line falls from left to right.
2. y-intercept (b): The point where the line crosses the y-axis (\( y \)-value when \( x = 0 \)).

Given Equations:

- \( y = 2x - 4 \)
- \( y = 3x - 4 \)
- \( y = \frac{1}{2}x - 1 \)
- \( y = -2 - \frac{1}{4}x \)
- \( y = x \)
- \( y = 1 - \frac{1}{2}x \)
- \( y = 4x - 2 \)

Analysis of Each Graph:

#### Graph A:
- Slope: The line rises steeply. By counting the grid, the slope appears to be 2 (for every 1 unit increase in \( x \), \( y \) increases by 2).
- y-intercept: The line crosses the y-axis at \( y = -4 \).
- Match: \( y = 2x - 4 \).

#### Graph B:
- Slope: The line rises very steeply. By counting the grid, the slope appears to be 4 (for every 1 unit increase in \( x \), \( y \) increases by 4).
- y-intercept: The line crosses the y-axis at \( y = -2 \).
- Match: \( y = 4x - 2 \).

#### Graph C:
- Slope: The line rises gently. By counting the grid, the slope appears to be \( \frac{1}{2} \) (for every 2 units increase in \( x \), \( y \) increases by 1).
- y-intercept: The line crosses the y-axis at \( y = -1 \).
- Match: \( y = \frac{1}{2}x - 1 \).

#### Graph D:
- Slope: The line rises at a moderate rate. By counting the grid, the slope appears to be 1 (for every 1 unit increase in \( x \), \( y \) increases by 1).
- y-intercept: The line crosses the y-axis at \( y = 0 \).
- Match: \( y = x \).

#### Graph E:
- Slope: The line rises very steeply. By counting the grid, the slope appears to be 3 (for every 1 unit increase in \( x \), \( y \) increases by 3).
- y-intercept: The line crosses the y-axis at \( y = -4 \).
- Match: \( y = 3x - 4 \).

#### Graph F:
- Slope: The line falls gently. By counting the grid, the slope appears to be \( -\frac{1}{2} \) (for every 2 units increase in \( x \), \( y \) decreases by 1).
- y-intercept: The line crosses the y-axis at \( y = 1 \).
- Match: \( y = 1 - \frac{1}{2}x \).

#### Graph G:
- Slope: The line rises moderately. By counting the grid, the slope appears to be \( \frac{1}{4} \) (for every 4 units increase in \( x \), \( y \) increases by 1).
- y-intercept: The line crosses the y-axis at \( y = -2 \).
- Match: \( y = -2 + \frac{1}{4}x \) (Note: This is equivalent to \( y = \frac{1}{4}x - 2 \), but the given equation is \( y = -2 - \frac{1}{4}x \), which is incorrect. However, based on the options, this is the closest match).

#### Graph H:
- Slope: The line falls moderately. By counting the grid, the slope appears to be \( -\frac{1}{4} \) (for every 4 units increase in \( x \), \( y \) decreases by 1).
- y-intercept: The line crosses the y-axis at \( y = -2 \).
- Match: \( y = -2 - \frac{1}{4}x \).

Final Matches:

- A: \( y = 2x - 4 \)
- B: \( y = 4x - 2 \)
- C: \( y = \frac{1}{2}x - 1 \)
- D: \( y = x \)
- E: \( y = 3x - 4 \)
- F: \( y = 1 - \frac{1}{2}x \)
- G: \( y = -2 + \frac{1}{4}x \) (closest match, though not exact)
- H: \( y = -2 - \frac{1}{4}x \)

Boxed Answer:

\[
\boxed{
\begin{array}{ll}
A & y = 2x - 4 \\
B & y = 4x - 2 \\
C & y = \frac{1}{2}x - 1 \\
D & y = x \\
E & y = 3x - 4 \\
F & y = 1 - \frac{1}{2}x \\
G & y = -2 + \frac{1}{4}x \\
H & y = -2 - \frac{1}{4}x \\
\end{array}
}
\]