91Ó°ÊÓ

The slope-intercept form is one of the most popular formats for expressing the equation of a straight line. Its general structure is \[ y = mx + b \] where:

• \( m \) represents the slope of the line.
  
• \( b \) represents the y-intercept of the line, which is the point where the line crosses the y-axis.
  

This form makes it easy to identify both the slope and y-intercept just by looking at the equation.
It's particularly useful when you want to graph a line quickly. In the context of our exercise, we transitioned from the point-slope to slope-intercept form resulting in:\[ y = \frac{1}{4}x - 1 \] which reveals that the slope is \( \frac{1}{4} \) and the y-intercept is \( -1 \).
This means the line crosses the y-axis at (0, -1).
Understanding how to use the slope-intercept form helps in quickly visualizing the basic framework of the line in a graph.

The point-slope form of a linear equation is particularly handy when you know a point on the line and the slope. The general formula is:\[ y - y_1 = m(x - x_1) \] where:

• \( m \) is the slope.
  
• \( (x_1, y_1) \) are the coordinates of a known point on the line.
  

This form is exceptionally useful for quickly writing the equation of a line. In our problem, the point-slope form is applied with the slope \( \frac{1}{4} \) and the point \( (1, -\frac{3}{4}) \), leading to:\[ y + \frac{3}{4} = \frac{1}{4}(x - 1) \] From this, the equation was manipulated into slope-intercept form through algebraic steps. Understanding the point-slope form helps in converting real-world scenarios into understandable linear models.

Graphing linear functions involves plotting a straight line on a coordinate plane. Begin with identifying the slope and y-intercept from the line's equation. For our equation, \[ y = \frac{1}{4}x - 1 \]

• The slope \( \frac{1}{4} \) tells us that for every 4 units the line travels horizontally, it moves up 1 unit vertically.
  
• The y-intercept \( -1 \) indicates the point where the line crosses the y-axis.
  

To graph this, you:

• Start at the y-intercept (0, -1).
  
• Use the slope to find the next point. From (0, -1), move 1 unit up and 4 units to the right to reach (4, 0).
  
• Draw a line through those points extending it in both directions.
  

Graphing not only helps in visualizing the relationship between variables but also aids in understanding the concept of slope and intercept.