BackGraphs of Linear Equations and Slope Concepts
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Graphs, Functions, and Models
Graphing Linear Equations
Linear equations are fundamental in precalculus and are represented graphically as straight lines. The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
Slope (m): The slope of a line measures its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between two points on the line.
Y-intercept (b): The y-intercept is the point where the line crosses the y-axis (when x = 0).
Equation of a Line:
If the slope m > 0, the line rises from left to right.
If the slope m < 0, the line falls from left to right.
If the slope m = 0, the line is horizontal.
If the equation is in the form x = a, the line is vertical and the slope is undefined.
Examples
Example 1: For the equation , the slope is 2 and the y-intercept is 1. The line rises steeply from left to right.
Example 2: For the equation , the slope is -1 and the y-intercept is 3. The line falls from left to right.
Interpreting Graphs
To graph a line, plot the y-intercept, then use the slope to find a second point.
Draw a straight line through the two points.
Summary Table: Slope and Line Direction
Slope (m) | Line Direction |
|---|---|
m > 0 | Rises left to right |
m < 0 | Falls left to right |
m = 0 | Horizontal |
Undefined | Vertical |
Additional info: The notes visually demonstrate how to plot lines using slope and y-intercept, and how to interpret the direction of the line based on the sign of the slope.
