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

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.