Graphs, Functions, and Models

Graphing Linear Equations

Linear equations can be written in different forms, with the two most common being Slope-Intercept Form and Standard Form. Understanding how to convert between these forms and graph the equations is essential in Precalculus.

• Slope-Intercept Form: - m is the slope of the line (rise over run). - b is the y-intercept (the point where the line crosses the y-axis).

• Standard Form: - A, B, and C are integers. - This form is useful for quickly finding intercepts.

Converting Between Forms

To graph a line given in Standard Form, it is often helpful to convert it to Slope-Intercept Form:

• Start with

• Solve for :

• Now, the equation is in the form .

Finding Intercepts

• y-intercept: Set and solve for .

• x-intercept: Set and solve for .

Example 1: Converting Standard Form to Slope-Intercept Form

Given :

• Subtract from both sides:

• Divide both sides by 2:

Now, the slope is and the y-intercept is $3$.

Example 2: Graphing Using Slope-Intercept Form

Graph :

1. Plot the y-intercept on the graph.

2. From the y-intercept, use the slope : - Go down 3 units (rise = -3), right 2 units (run = 2) to plot a second point.

3. Draw a straight line through the two points.

Example 3: Finding Intercepts from Standard Form

Given :

• y-intercept: Set :

• x-intercept: Set :

• Intercepts: and

Summary Table: Forms of Linear Equations

Additional info: The handwritten calculations and graph in the image illustrate the process of converting from Standard Form to Slope-Intercept Form and plotting the line using the slope and intercepts.