91Ó°ÊÓ

In the world of linear functions, the slope-intercept form is a handy way to express a line. It's written as \( y = mx + b \). Here, \( m \) represents the slope, which tells us how steep the line is, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
To find the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In our exercise, the points are \( \left(\frac{2}{3}, -\frac{15}{2}\right) \) and \( (-4, -11) \). By plugging these into the formula, we found the slope to be \( \frac{3}{4} \).
The y-intercept \( b \) can be determined once we know \( m \) and at least one point on the line. Using the slope \( \frac{3}{4} \) and the point \( \left(\frac{2}{3}, -\frac{15}{2}\right) \), the calculation shows \( b = -8 \).
So, the line's equation is \( y = \frac{3}{4}x - 8 \). This equation is useful for determining any point on this line.

Function values are essentially the outputs of a function, which are also known as the y-values in linear equations like \( y = mx + b \). They show what value is produced by running an input through the function. In this exercise, two function values were given: \( f\left(\frac{2}{3}\right)=-\frac{15}{2} \) and \( f(-4)=-11 \).
These indicate that when \( \frac{2}{3} \) is input into the function, the corresponding output is \(-\frac{15}{2} \) and similarly, \(-4\) gives \(-11\).
This relationship highlights the linearity of the function; every change in \( x \) leads to a consistent and proportional change in \( y \). Once we establish a function's formula, we can easily find these values by plugging \( x \) into \( y = \frac{3}{4}x - 8 \) and solving.

Graphing linear equations involves plotting points and drawing a straight line that passes through them. This gives a visual representation of a linear function like \( y = \frac{3}{4}x - 8 \).
To graph this:

• Start by finding the y-intercept, \( -8 \), and place a point on the y-axis at that value.
• Using the slope \( \frac{3}{4} \), go up 3 units and to the right 4 units from that point. This gives us another point on the line.
• Draw a straight line through these points to establish the graph of the equation.

This method clearly shows the relationship between the variables and the linearity ensures that all the points lie perfectly on the straight line. Graphing is excellent for understanding the function's behavior and immediate visual analysis is possible once the graph is complete.