Intro
Slope of a Line
Definition: The slope, denoted as m, measures the steepness and direction of a line. It shows how much the y-value (vertical change) changes for each unit increase in the x-value (horizontal change).
Formula: For any two points on a line, ($x_1, y_1$) and ($x_2, y_2$), the slope is calculated as:
$m = \dfrac{y_2 − y_1}{x_2 − x_1}$
Interpretation of Slope Values:
- Positive Slope ($m$ > 0): The line rises from left to right.
- Negative Slope ($m$ < 0): The line falls from left to right.
- Zero Slope ($m$ = 0): Horizontal line, with no vertical change as $x$ changes.
- Undefined Slope: Vertical line, where division by zero occurs in the formula.
Example: To find the slope of a line passing through (3, 4) and (7, 10), calculate:
$m = \dfrac{(10 − 4)}{(7 − 3)} = 1.5$
This slope of 1.5 indicates a rise by 1.5 units for each unit increase in $x$.
Distance Formula
Purpose: The distance formula calculates the straight-line distance between two points on a coordinate plane.
Formula: For two points ($x_1, y_1$) and ($x_2, y_2$), the distance $d$ is given by:
$d= \sqrt{(x_2 − x_1)^2+(y_2 − y_1)^2}$
Example: To find the distance between (1, 2) and (5, 6), substitute into the formula:
$d= \sqrt{(5 − 1)^2+(6 − 2)^2} $ $ = 4 \sqrt{2}$
Equation of a Line
Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
Deriving the Equation: Use the slope and one known point to substitute into the form $y = mx + b$ and solve for $b$.
Example: If slope = 2, point (1, 3), then y = 2x + 1.
Other Forms:
- Point-Slope Form: $y − y1 = m(x − x1)$
- Standard Form: $Ax + By = C$, with integers for $A$, $B$, and $C$.
Perpendicular and Parallel Lines
Parallel Lines: Same slope, different intercepts.
Perpendicular Lines: Product of slopes is −1; if one line has slope $m$, a perpendicular line has slope $\frac{−1}{m}$.
Example: Line with slope $\frac{2}{3}$ is perpendicular to line with slope $−\frac{3}{2}$.
Key Tip: Make sure you can calculate slope and distance. Practice using the formulas to reinforce learning.