Closed Captioned Video: Algebra Applications: Linear Functions, Segment 2: Cycling

Linear Expressions, Equations, and Functions

Linear Expressions

Linear expressions include a variable whose exponent is 1. Here are some examples of linear expressions:

Be sure you know how to translate a verbal expression into a linear expression like the ones shown above. To see examples of how to do this, click on this link to see a slide show.

Expressions aren’t equations but they are an important component of linear equations. Also, you can add and subtract linear expressions and still have a linear expression. Multiplying or dividing linear expressions will result in a non-linear expression. Here are some examples:

Linear Equations

Linear equations include a linear expression equal to a number or another linear expression. A linear equation can have one or more variables, but all terms must be linear.

Here are examples of linear equations with one variable.

If you need more practice in solving one-variable equations, click on the following links:

• Solving One-Step Addition Equations
• Solving One-Step Subtraction Equations
• Solving One-Step Multiplication Equations
• Solving One-Step Division Equations
• Solving Two-Step Multiplication and Addition Equations
• Solving Two-Step Multiplication and Subtraction Equations
• Solving Two-Step Division and Addition Equations
• Solving Two-Step Division and Subtraction Equations

Another skill related to equation solving is the ability to rewrite an equation in an equivalent form. See the examples below.

Make sure you are comfortable with the properties of equality and the structure of an equation. For a quick review, click on this link.

Linear Functions

Before studying what a linear function is, make sure you are comfortable with the following concepts, which we will also review:

• What a function is
• Independent variable
• Dependent variable
• Domain
• Range
• Different representations of functions

Brief Review of Functions 

What Is a Function? A function is a one-to-one mapping of input values (the independent variable) to output values (the dependent variable). Click on this link to see a quick tutorial on what a function is. This slide show goes over the following key points:

• For every input value (x), there is a unique output value, f(x).
• Functions can be represented as equations, tables, and graphs.
• A function machine is a useful visual representation of the input/output nature of functions.

Dependent/Independent Variables. When one variable depends on another, then it is the dependent variable. For example, the faster your speed, the farther you travel. Suppose that speed is represented by the variable s and the distance traveled is represented by the variable d. 

Here’s how to describe the relationship between s and d:

The faster the speed, the more distance traveled.

Distance is dependent on speed.

Distance is a function of speed.

d = f(s)

When studying functions, make sure you are comfortable telling the difference between the independent variable and dependent variable. Get comfortable using function notation. To learn more about function notation, click on this link.

Domain and Range. A function shows the relationship between two variables, the independent variable and the dependent variable. The domain is the allowed values for the independent variable. The range is the allowed values for the dependent variable. The domain and range influence what the graph of the function looks like.

For a detailed review of what domain and range are, click on this link to learn more. You’ll see definitions of the terms domain and range, as well as examples of how to find the domain and range for given functions.

Multiple Representations of Functions. We mentioned previously that functions can be represented in different ways. In fact, any function can be represented by an equation, usually f(x) equal to some expression; a table; or a graph. For a detailed review of multiple representations of functions, click on this link, to see a slide show that includes examples of these multiple representations.

Slope-Intercept Form

The most important form of a linear function is the slope-intercept form. 

Given the slope, m, and the y-intercept, b, for a linear function, you can easily construct the equation and graph of the linear function. To see examples of graphing linear functions in slope-intercept form, click on this link. This slide show also includes a video tutorial.

Point-Slope Form

Another method of finding the slope-intercept form involves using the point-slope form. In this case you are given the slope, m, of the line and one set of coordinates, (x, y), on the line. This is what the point-slope form looks like. 

To see examples of deriving the linear function using the  in point-slope form, click on this link. This slide show tutorial walks you through the process and provides several worked-out examples.

A number of SAT questions will test your understanding of linear functions in slope-intercept and point-slope form.

Slope-Based Questions

First, let’s review the basics of slope. This is the slope formula:

Given two coordinates, the slope of the line connecting the two points is found using the slope formula.

The key to understanding this is the slope-intercept form for parallel and perpendicular lines. Review these definitions:

Basically, lines that are parallel have the same slope. Lines that are perpendicular have slopes that are negative reciprocals. Look at the following examples.

To see examples of finding the equations of parallel and perpendicular lines, click on the following links:

• Parallel Lines
• Perpendicular Lines

The relationship between slope and grade in cycling is explored. Go on a tour of Italy through the mountains of Tuscany and apply students' understanding of slope.

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