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In linear algebra, the slope-intercept form of a linear function is a way to represent a line with the equation: \( f(x) = mx + b \). This format is particularly useful for graphing, as it provides immediate information about the line's slope and its y-intercept.
Here’s what the equation means:

• \( m \) represents the slope. The slope indicates how steep the line is and can also describe the rate of change. It shows how much \( y \) changes for a unit change in \( x \).
• \( b \) represents the y-intercept. This is the point where the line crosses the y-axis. In other words, it is the value of \( f(x) \) when \( x = 0 \).

Understanding the slope-intercept form makes it easy to quickly draw linear graphs and understand how changes in parameters affect the line.

Finding the slope of a line helps describe how two variables are related. It's the "rise over run" or the amount that \( y \) changes as \( x \) changes by one unit. The slope is calculated using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]You take two points on the line, denoted as \((x_1, y_1)\) and \((x_2, y_2)\), and use their coordinates to find \( m \).
In our exercise, points \((-2, 7)\) and \((4, -2)\) are used:
Apply the formula to get:

• \( m = \frac{-2 - 7}{4 - (-2)} = \frac{-9}{6} \)

This simplifies to \( m = -\frac{3}{2} \), representing a downward sloping line, meaning for each 2 units increased in \( x \), the value of \( y \) decreases by 3 units.

The y-intercept is an essential component of the linear equation. It is where the line crosses the y-axis.
Finding the y-intercept comes after determining the slope, following this process:
1. Use the slope \( m \) that you calculated.
2. Substitute \( m \) and one point's coordinates into the linear equation \( f(x) = mx + b \).
For example, using point \((-2, 7)\):

• \( 7 = -\frac{3}{2}(-2) + b \)
• Simplify to find \( b \): \( 7 = 3 + b \)
• \( b = 4 \)

Thus, \( b = 4 \) is the y-intercept, signifying that the line crosses the y-axis at \( (0, 4) \).
Knowing the y-intercept helps you plot one point immediately on a graph.

The point-slope calculation is a helpful technique when tasked with finding the equation of a line while given just two points.
This method involves starting with the point-slope form: \[y - y_1 = m(x - x_1)\]Here, \( m \) is the slope and \((x_1, y_1)\) is a point on the line.
Using this form provides an efficient way to derive the line's equation.
For instance, given a slope \( m = -\frac{3}{2} \) and a point \((-2, 7)\), substitute these into the point-slope formula:

• \( y - 7 = -\frac{3}{2}(x + 2) \)

Then, rearrange it to slope-intercept form \( f(x) = mx + b \) to get: \[f(x) = -\frac{3}{2}x + 4\]Point-slope calculation is versatile and reliable in constructing the equation of the line from minimal information, especially handy in many real-world math problems.