Understanding Linear Functions

Linear functions have special traits that make them easy to recognize and important for learning about graphs. Let's break down what makes these functions unique.

What Does the Graph Look Like?

• The graph of a linear function is a straight line.
• This straight line shows a constant rate of change.
• The simplest way to write a linear function is with the formula y = mx + c.
  • Here, m stands for the slope, which tells us how steep the line is.
  • c is the y-intercept, which is where the line crosses the y-axis.

Understanding Slope

• The slope (m) shows how much y changes when x changes by one unit.

• If the slope is positive, the line goes up. If it’s negative, the line goes down.

• To find the slope between two points on the line, let’s say (x1, y1) and (x2, y2), you can use this simple formula:

  $m = \frac{y_2 - y_1}{x_2 - x_1}$

What is the Y-Intercept?

• The y-intercept is the value of y when x is 0.
• In the equation y = mx + c, this value is just c. It helps us start plotting the graph.

Key Features of Linear Functions

• Linear functions can be added together or multiplied by a number, and they will still be linear.
• The input values (domain) and output values (range) for linear functions can be any real number unless stated otherwise.

How to Plot a Linear Function

• To plot a linear function, first find two points:
  1. The y-intercept.
  2. Another point using the slope.
• Connect these points with a straight line that goes on forever in both directions.

These features make it easy to identify and plot linear functions. They are a basic building block of math that you will keep using in your studies!