1. Understanding the Slope-Intercept Form
The slope-intercept form of a line is one of the most commonly used forms because it clearly shows the slope of the line and where the line cuts the y-axis. The form of the equation is:
\( y = mx + c \)
Here, \(m\) represents the slope, and \(c\) represents the y-intercept.
1.1. Definition
The slope-intercept form is the equation of a line written as \(y = mx + c\), where the slope is \(m\) and the line intersects the y-axis at \((0, c)\).
2. Meaning of Slope and Intercept
The slope and y-intercept tell us how the line is positioned on the Cartesian plane.
2.1. Slope (m)
The slope \(m\) measures how steep the line is. It tells us the rate of change of y with respect to x.
- If \(m > 0\), the line goes upward.
- If \(m < 0\), the line goes downward.
- If \(m = 0\), the line is horizontal.
2.2. Y-Intercept (c)
The y-intercept is the point where the line crosses the y-axis. This happens when \(x = 0\). So the point is \((0, c)\).
2.2.1. Example for Intercept
If the equation of a line is \(y = 3x + 2\), then the y-intercept is 2 because the line cuts the y-axis at \((0, 2)\).
3. How to Form the Equation in Slope-Intercept Form
To write the equation of a line in slope-intercept form, you need the slope and the y-intercept. Substitute these values into the formula \(y = mx + c\).
3.1. Step-by-Step Process
- Find the slope \(m\).
- Find the y-intercept \(c\).
- Substitute the values into the form \(y = mx + c\).
4. Examples
The following examples show how to create equations in slope-intercept form.
4.1. Example 1: Given Slope and Intercept
Write the equation of a line with slope \(m = 2\) and y-intercept \(c = -3\).
\( y = 2x - 3 \)
4.2. Example 2: Negative Slope
Slope \(m = -4\) and y-intercept \(c = 5\).
\( y = -4x + 5 \)
4.3. Example 3: Zero Slope
If the slope is 0 and the y-intercept is 7:
\( y = 0x + 7 = 7 \)
The equation represents a horizontal line.
4.4. Example 4: Line Passing Through the Origin
If the line passes through the origin, the y-intercept is 0. Slope \(m = 3\):
\( y = 3x \)