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The slope-intercept form of a linear equation is a way of writing the equation so that you can easily identify the slope and the y-intercept. It is written as: \[ y = mx + b \] Where:

• \( y \) is the dependent variable (usually on the vertical axis).
• \( x \) is the independent variable (usually on the horizontal axis).
• \( m \) is the slope of the line.
• \( b \) is the y-intercept, where the line crosses the y-axis.

In the given equation \( y = 4x - 4 \), the slope-intercept form tells us that \( m = 4 \) and \( b = -4 \). This makes it straightforward to graph because you know exactly where the line starts on the y-axis and how it rises or falls.

The slope of a line indicates its steepness and direction. It is represented by \( m \) in the equation \( y = mx + b \). The slope is the ratio of the change in the y-values to the change in the x-values between any two points on the line. Mathematically, it is: \[ m = \frac{ \text{rise} }{ \text{run} } \] Where:

• \( \text{rise} \) is the change in y-values.
• \( \text{run} \) is the change in x-values.

In the equation \( y = 4x - 4 \), the slope \( m \) is 4. This means for every 1 unit you move to the right along the x-axis, the y-value increases by 4. The slope helps us determine another point on the graph simply by moving horizontally and vertically from the y-intercept.

The y-intercept is where the line crosses the y-axis. It is represented by \( b \) in the equation \( y = mx + b \). To find the y-intercept, you set \( x = 0 \) because the y-axis corresponds to \( x = 0 \). In the equation \( y = 4x - 4 \), the y-intercept \( b \) is -4. This means the line crosses the y-axis at the point \( (0, -4) \). Plotting this point is the first step in graphing the equation because it gives us a starting point for where the line touches the y-axis.

Plotting points on a graph is essential to visualizing the equation you are working with. Here’s how to plot points for the equation \( y = 4x - 4 \): 1. **Start with the y-intercept**: First, plot the point \( (0, -4) \) on the y-axis. This is where the line crosses the y-axis.2. **Use the slope**: From the y-intercept, use the slope to find another point. Since the slope \( m = 4 \), move 1 unit right (positive direction on x-axis) and 4 units up (positive direction on y-axis) to reach the point \( (1, 0) \). 3. **Draw the line**: Once you have at least two points, draw a straight line through them extending in both directions. This line represents the equation \( y = 4x - 4 \). By plotting these key points and extending the line, you'll have a clear graph of the linear equation.