BackStudy Notes: Lines and Their Equations (Section 2.3)
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Section 2.3: Lines Guided Notes
Recognizing Linear Equations
Linear equations are foundational in algebra and describe straight lines when graphed on a coordinate plane.
Definition: A linear equation in one variable includes only constants and variables that are raised to the first power.
General Form: , where a and b are constants and x is the variable.
Key Property: The graph of a linear equation in two variables () is always a straight line.
Finding Intercepts of a Graph Given an Equation
Intercepts are points where the graph crosses the axes. They are useful for graphing and understanding the behavior of lines.
y-intercept: Set in the equation and solve for .
x-intercept: Set in the equation and solve for .
Example: For :
y-intercept:
x-intercept:
Objective 1: Determining the Slope of a Line
The slope of a line measures its steepness and direction. It is a key characteristic of linear equations.
Definition: The slope m of a nonvertical line passing through two distinct points and is given by:
Interpretation: The slope tells how much changes for a unit change in .
Types of Slope:
Positive slope: Line rises from left to right.
Negative slope: Line falls from left to right.
Zero slope: Horizontal line.
Undefined slope: Vertical line (denominator is zero).
Example Calculations:
Between and :
Between and : (undefined)
Between and :
Objective 2: Graphing a Line Given a Point and Slope
Given a point and a slope, you can graph a line by plotting the point and using the slope to find additional points.
Procedure:
Plot the given point on the coordinate plane.
Use the slope to find other points. For example, if , move up 3 units and right 1 unit from the given point.
Draw a straight line through the points.
Example: Sketch the line with passing through . Find three more points by applying the slope repeatedly.
Objective 3: Finding the Equation of a Line Using the Point-Slope Form
The point-slope form is useful when you know the slope and a point on the line.
Formula:
Where is a point on the line and is the slope.
Example: Find the equation of the line with passing through :
Another Example: For and point :
Interpretation: This form is especially helpful for quickly writing the equation when a point and the slope are known.
Objective 4: Finding the Equation of a Line Using the Slope-Intercept Form
The slope-intercept form is the most common way to express the equation of a line, especially for graphing.
Formula:
Where is the slope and is the y-intercept (the value of when ).
Converting from Point-Slope to Slope-Intercept: Solve the point-slope equation for to get it into slope-intercept form.
Example: From :
Interpretation: The slope-intercept form makes it easy to identify the slope and y-intercept directly from the equation.
Summary Table: Forms of Linear Equations
Form | Equation | When to Use |
|---|---|---|
Slope-Intercept | When you know the slope and y-intercept | |
Point-Slope | When you know the slope and a point | |
Standard | General form; useful for finding intercepts |
Special Cases: Horizontal and Vertical Lines
Horizontal and vertical lines are special types of linear equations with unique properties.
Horizontal Lines:
Equation: (where is a constant)
Slope:
Graph: A straight line parallel to the x-axis
Example: The line passing through :
Vertical Lines:
Equation: (where is a constant)
Slope: Undefined
Graph: A straight line parallel to the y-axis
Example: The line passing through :
