91Ó°ÊÓ

To understand how to find the slope of a linear function, picture a straight line on a graph. The slope measures how steep the line is. You can think of it as the 'rise over run,' which tells you how much the line goes up (or down) for a given distance across.
In mathematical terms, the slope, denoted as \( m \), is calculated using two points on the line: \( (x_1, y_1) \) and \( (x_2, y_2) \). The formula is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• \(y_2 - y_1\): The difference in the y-coordinates (vertical change).
• \(x_2 - x_1\): The difference in the x-coordinates (horizontal change).

Continuing with our example points \((4, -5)\) and \((6, -10)\), we substitute these values into the formula:
\( m = \frac{-10 - (-5)}{6 - 4} \), which simplifies to \( m = \frac{-5}{2} = -2.5 \).
Now, you know that the slope of our line is \( -2.5 \). This negative value means the line is sloping downward as it moves from left to right.

The slope-intercept form of a linear equation is very useful. It’s written as \( y = mx + b \). Here’s what each symbol means:

• \( y\): The y-coordinate of a point on the line.
• \( x\): The x-coordinate of a point on the line.
• \( m\): The slope of the line.
• \( b\): The y-intercept of the line.

This form is great because it directly shows the slope and the y-intercept.
Using our calculated slope \(-2.5\) and inserting it into the slope-intercept formula, we have:
\( y = -2.5x + b \).
Next, we need to find \( b \), the y-intercept, to complete the equation.

The y-intercept \( b \) is where the line crosses the y-axis of the graph. When a line crosses the y-axis, the x-coordinate is 0. To find \( b \), you can use one of the points on the line and the slope, substituting them into the slope-intercept form.

In our example, we use the point \( (4, -5) \) and the slope \(-2.5\). So the equation is: \( -5 = -2.5(4) + b \).
This simplifies to:
\( -5 = -10 + b \)
Adding 10 to both sides, we find:
\( b = 5 \)
Now, we have both \( m \) and \( b \).
Substituting these back into the slope-intercept form gives us the final equation of our line:
\( y = -2.5x + 5 \).
This is the linear function that includes the points \( (4, -5) \) and \( (6,-10) \). Now you can easily see both the slope and the y-intercept in the equation!