Graphing Linear Equations: Slope-Intercept and Standard Form

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: $y = mx + b$ - 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: $Ax + By = C$ - 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 $Ax + By = C$

• Solve for $y$:

$By = -Ax + C$

$y = -\frac{A}{B}x + \frac{C}{B}$

• Now, the equation is in the form $y = mx + b$.

Finding Intercepts

• y-intercept: Set $x = 0$ and solve for $y$.

• x-intercept: Set $y = 0$ and solve for $x$.

Example 1: Converting Standard Form to Slope-Intercept Form

Given $3x + 2y = 6$:

• Subtract $3x$ from both sides: $2y = -3x + 6$

• Divide both sides by 2: $y = -\frac{3}{2}x + 3$

Now, the slope is $-\frac{3}{2}$ and the y-intercept is $3$.

Example 2: Graphing Using Slope-Intercept Form

Graph $y = -\frac{3}{2}x + 3$:

1. Plot the y-intercept $(0, 3)$ on the graph.

2. From the y-intercept, use the slope $-\frac{3}{2}$: - 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 $3x + 2y = 6$:

• y-intercept: Set $x = 0$:

$3(0) + 2y = 6 \implies 2y = 6 \implies y = 3$

• x-intercept: Set $y = 0$:

$3x + 2(0) = 6 \implies 3x = 6 \implies x = 2$

• Intercepts: $(2, 0)$ and $(0, 3)$

Summary Table: Forms of Linear Equations

Form	Equation	Key Features
Slope-Intercept	$y = mx + b$	Slope ($m$), y-intercept ($b$)
Standard	$Ax + By = C$	Intercepts, easy to rearrange

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.