1. Understanding the Point-Slope Form
The point-slope form is useful when we know the slope of a line and one point through which the line passes. Instead of finding the intercept first, we use the given point directly to form the equation.
This method is convenient in many problems, especially when only one point and the slope are given.
1.1. Definition
The point-slope form of the equation of a line passing through the point \((x_1, y_1)\) with slope \(m\) is:
\( y - y_1 = m(x - x_1) \)
2. Meaning of the Formula
The point-slope formula simply tells us how much the value of y changes when x changes, starting from a known point on the line.
2.1. Why It Works
We know the slope represents the ratio of vertical change to horizontal change. So from a known point, if x changes by a certain amount, y must change in proportion to the slope.
2.2. Geometric Interpretation
The formula describes a line that passes through \((x_1, y_1)\) and rises or falls based on the slope \(m\).
3. How to Use the Point-Slope Form
To form the equation of a line using point-slope form, we simply substitute the slope \(m\) and the coordinates of a known point \((x_1, y_1)\) into the formula.
3.1. Steps
- Identify a point on the line (\(x_1, y_1\)).
- Find or use the given slope \(m\).
- Substitute into the formula:
\( y - y_1 = m(x - x_1) \)
- Simplify further if needed.
4. Examples
The following examples show how to apply the point-slope formula in different situations.
4.1. Example 1: Basic Example
Find the equation of the line passing through \((2, 3)\) with slope \(m = 4\).
\( y - 3 = 4(x - 2) \)
4.2. Example 2: Negative Slope
Point: \((5, -1)\), slope \(m = -2\).
\( y + 1 = -2(x - 5) \)
4.3. Example 3: Converting to Slope-Intercept Form
Using the equation from Example 1:
\( y - 3 = 4(x - 2) \)
Expand:
\( y - 3 = 4x - 8 \)
So:
\( y = 4x - 5 \)
4.4. Example 4: Vertical and Horizontal Considerations
Point-slope form works for all non-vertical lines. For vertical lines, slope is undefined, so this form cannot be used.