Graph Slope-Intercept Form of a Linear Function - Algebra-2

1. Fundamental Concepts

Definition: Graphing the slope-intercept form of a linear function () involves plotting the line on a coordinate plane using its slope (m) and y-intercept (b).
Core Elements: The process relies on two key components from the equation:
- The y-intercept : a specific point where the line crosses the y-axis.
- The slope : the ratio of vertical change (rise) to horizontal change (run) between two points on the line, which determines the line’s steepness and direction.

Graphical Interpretation of Slope and Intercept:
- The y-intercept is the starting point for graphing, as it provides a known point on the line.
- The slope m dictates how to move from the y-intercept to find a second point:
  - Positive slope (): Move up (positive rise) and right (positive run).
  - Negative slope (): Move down (negative rise) and right (positive run), or up (positive rise) and left (negative run).
  - Zero slope (): The line is horizontal, parallel to the x-axis, passing through .
Uniqueness: A linear function in slope-intercept form corresponds to exactly one straight line, so two points are sufficient to graph it.

Simple: Graph the line .
- Step 1: Identify the y-intercept and plot it.
- Step 2: Interpret the slope (rise = 2, run = 1). From , move up 2 units and right 1 unit to reach , then plot this point.
- Step 3: Draw a straight line through and .
Medium: Graph the line .
- Step 1: Identify the y-intercept and plot it.
- Step 2: Interpret the slope (rise = -1, run = 2). From , move down 1 unit and right 2 units to reach , then plot this point.
- Step 3: Draw a straight line through and .
Hard: Graph the line (first rewrite in slope-intercept form).
- Step 1: Rewrite the equation: Divide by 3 to get .
- Step 2: Identify the y-intercept and plot it.
- Step 3: Interpret the slope . From , move up 2 units and right 1 unit to reach ; plot this point.
- Step 4: To confirm, find a third point: from , move up 2 units and right 1 unit to , then plot it.
- Step 5: Draw a straight line through , , and .

Step 1: Ensure the equation is in slope-intercept form: If given a linear equation in another form (e.g., standard form ), rewrite it as by isolating y (solve for y using inverse operations).
Step 2: Identify and plot the y-intercept: Locate on the coordinate plane (where and ) and mark this point.
Step 3: Use the slope to find a second point:
- Express the slope as a fraction (e.g., becomes , remains ).
- From the y-intercept, move vertically by the "rise" (up if positive, down if negative) and horizontally by the "run" (always right for positive run) to find a second point; mark this point.
Step 4: Draw the line: Connect the two plotted points with a straight line, extending it infinitely in both directions.
Step 5 (Optional verification): Find a third point using the slope from the second point to ensure it lies on the line, confirming accuracy.