Functions and Graphs

Linear Functions and Slope

Linear functions are fundamental in algebra, representing relationships with constant rates of change. The slope of a line quantifies this rate, and several forms of linear equations allow for flexible modeling and graphing.

Definition of Slope

The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on the line:

• Formula:

• Rise: (vertical change)

• Run: (horizontal change)

• Condition:

Example: Calculating Slope

Find the slope of the line passing through the points (4, -2) and (-1, 5):

• The negative slope indicates the line decreases as x increases.

Point-Slope Form of a Line

The point-slope form is useful for writing the equation of a line when a point and the slope are known:

• Formula:

• Where is a point on the line and is the slope.

Example: Writing Point-Slope Form

Write the equation for a line with slope 6 passing through (2, -5):

• Solve for y:

Slope-Intercept Form of a Line

The slope-intercept form is widely used for graphing and modeling:

• Formula:

• is the slope, is the y-intercept (where the line crosses the y-axis).

Graphing y = mx + b

To graph a linear function:

1. Plot the y-intercept .

2. Use the slope (rise/run) to find a second point.

3. Draw a line through both points, extending in both directions.

Horizontal and Vertical Lines

• Horizontal line: (slope = 0)

• Vertical line: (slope is undefined)

General Form of a Line

Any line can be written in the general form:

• A, B, and C are real numbers; A and B are not both zero.

Finding Slope and Intercepts from General Form

To find the slope and y-intercept:

• Rewrite as

• Slope:

• Y-intercept:

Graphing Using Intercepts

To graph :

1. Find x-intercept: Set , solve for .

2. Find y-intercept: Set , solve for .

3. Plot both points and draw the line.

Modeling Data with Linear Functions

Linear functions can model real-world data, such as temperature versus carbon dioxide concentration. The slope represents the rate of change, and the y-intercept gives the starting value.

• Given two data points, use the slope formula to find .

• Write the equation in point-slope or slope-intercept form.

More on Slope

Parallel and Perpendicular Lines

Understanding relationships between lines is essential for geometry and algebra.

Parallel Lines

• Nonvertical parallel lines have equal slopes.

• Vertical lines (undefined slope) are parallel to each other.

Perpendicular Lines

• Nonvertical perpendicular lines: Product of slopes is .

• Horizontal lines (slope 0) are perpendicular to vertical lines (undefined slope).

Slope as Rate of Change

The slope of a linear function represents the rate of change of the dependent variable with respect to the independent variable. This concept is widely used in modeling and interpreting data.

• For example, if the slope is 0.59, then the dependent variable increases by 0.59 units for each unit increase in the independent variable.

Average Rate of Change of a Function

The average rate of change of a function between two points and is:

• This is the slope of the secant line connecting the points and .

Example: Average Rate of Change

For from to :

• Average rate of change: