Slope-Intercept Form and Graphing Linear Equations

Graphs, Functions, and Models

Slope-Intercept Form of a Line

The slope-intercept form of a linear equation is a fundamental concept in precalculus, used to describe straight lines on the Cartesian plane. This form allows for easy identification of the slope and y-intercept, and facilitates graphing.

• Definition: The slope-intercept form of a line is given by the equation: $y = mx + b$ where m is the slope and b is the y-intercept.

• Slope (m): The rate of change of the line, representing how much y increases or decreases as x increases by 1.

• Y-intercept (b): The value of y where the line crosses the y-axis (when x = 0).

Example: Writing the Equation of a Line

• Given: Slope $m = 4$, passes through point $(1, 2)$.

• Step 1: Substitute the point and slope into the slope-intercept form to solve for $b$:

$y = mx + b$ $2 = 4(1) + b$ $2 = 4 + b$ $b = 2 - 4 = -2$

• Step 2: Write the equation:

$y = 4x - 2$

• Step 3: Graph the equation by plotting the y-intercept $(0, -2)$ and using the slope to find another point (rise 4, run 1).

Practice Problem

• Instruction: Write the point-slope form of the equation of a line with a slope of 4 that passes through $(1, 2)$. Then graph the equation.

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Substitute: $y - 2 = 4(x - 1)$

• Convert to Slope-Intercept Form:

$y - 2 = 4(x - 1)$ $y - 2 = 4x - 4$ $y = 4x - 2$

• Graph: Plot the y-intercept at $(0, -2)$ and use the slope to plot additional points.

Summary Table: Forms of Linear Equations

Form	Equation	When to Use
Slope-Intercept	$y = mx + b$	When you know the slope and y-intercept
Point-Slope	$y - y_1 = m(x - x_1)$	When you know the slope and a point

Additional info: The worksheet also includes graphing practice, reinforcing the connection between algebraic and graphical representations of lines.