Functions and Graphs

Linear Functions and Slope

This section introduces the concept of linear functions, focusing on the slope of a line, various forms of linear equations, and methods for graphing lines. Understanding these concepts is fundamental for analyzing and modeling relationships in algebra.

• Objective 1: Calculate a line’s slope.

• Objective 2: Write the point-slope form of the equation of a line.

• Objective 3: Write and graph the slope-intercept form of the equation of a line.

• Objective 4: Graph horizontal or vertical lines.

• Objective 5: Recognize and use the general form of a line’s equation.

• Objective 6: Use intercepts to graph the general form of a line’s equation.

• Objective 7: Model data with linear functions and make predictions.

Definition of Slope

The slope of a line measures its steepness and direction. It is defined for a line passing through two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) as:

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

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

Point-Slope Form of the Equation of a Line

The point-slope form is useful for writing the equation of a line when you know its slope and a point it passes through:

Example: Write an equation in point-slope form for the line with slope 6 that passes through the point (2, –5). Then solve for y.

Point-slope form: Solved for y:

Slope-Intercept Form of the Equation of a Line

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

• m: Slope of the line

• b: y-intercept (where the line crosses the y-axis)

Graphing y = mx + b Using the Slope and y-Intercept

To graph a line given in slope-intercept form:

1. Plot the y-intercept (0, b) on the y-axis.

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

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

Example: Graph the linear function .

• Step 1: Plot (0, 1).

• Step 2: From (0, 1), move up 3 units and right 5 units to plot (5, 4).

• Step 3: Draw the line through these points.

Equation of a Horizontal Line

A horizontal line has the equation:

• The slope is zero.

• All points have the same y-coordinate.

Example: Graph in the coordinate system.

Equation of a Vertical Line

A vertical line has the equation:

• The slope is undefined.

• All points have the same x-coordinate.

General Form of the Equation of a Line

The general form of a line is:

• A, B, and C are real numbers.

• A and B are not both zero.

Finding the Slope and the y-Intercept from General Form

To find the slope and y-intercept from :

• Solve for y to get slope-intercept form.

• Slope:

• y-intercept:

Example: For , slope is , y-intercept is 2.

Using Intercepts to Graph Ax + By + C = 0

To graph a line in general form:

1. Find the x-intercept: Let y = 0, solve for x.

2. Find the y-intercept: Let x = 0, solve for y.

3. Plot both intercepts and draw the line through them.

Example: Graph using intercepts.

• Step 1: x-intercept: y = 0, → (6, 0)

• Step 2: y-intercept: x = 0, → → (0, -2)

• Step 3: Draw the line through (6, 0) and (0, -2).

Modeling Data with Linear Functions

Linear functions can be used to model real-world data and make predictions. Given two data points, you can find the equation of the line that best fits the data.

Example: Use the data points (317, 57.04) and (354, 57.64) to obtain a linear function that models average global temperature, f(x), for an atmospheric carbon dioxide concentration of x parts per million.

Step 1: Find the slope:

Step 2: Use point-slope form and solve for y:

Step 3: Find b using one of the points:

Final function: (rounded to one decimal place)

Additional info: The included image is the cover of the Blitzer College Algebra textbook, which is directly relevant as it visually identifies the source and context of these study notes.