Graphs of Linear Functions

What Are Linear Functions?

A linear function is a function that creates a straight line when graphed. It can be written in the form:

$$y = mx + c$$

• $m$ is the slope or gradient of the line, indicating how steep the line is.
• $c$ is the y-intercept, the point where the line crosses the y-axis.

Characteristics of Linear Functions

• Constant Rate of Change: The slope $m$ represents a constant rate of change.
• Straight Line: The graph is always a straight line.
• Intercepts: The line crosses the y-axis at $c$ and can cross the x-axis at $-\frac{c}{m}$ if $m \neq 0$.

Graphing Linear Functions

1. Identify the slope $m$ and y-intercept $c$.
2. Plot the y-intercept $(0, c)$.
3. Use the slope $m$ to find another point. From the y-intercept, move up or down and left or right according to the slope.
4. Draw the line through the points.

Examples

Example 1: Graph $y = 2x + 3$

Example 2: Graph $y = -\frac{1}{2}x + 1$