**Key Terms**

o Algebra

o Operation

o Variable

o Function

o Graph

**Objectives**

o Learn why an understanding of algebra is important

o Begin to acquire a grasp of the jargon associated with algebra

o Prepare to study the mathematical details of algebra

*Operations*

An **operation** in algebra is nothing more than a procedure performed on one or more numbers (or variables or functions). Multiplication, for instance, is an operation that is fundamentally performed on two numbers: for instance, 4 3 is an example of a multiplication operation performed on the numbers 4 and 3.

*Variables*

One of the fundamental tools or concepts that we will regularly use in studying algebra is the **variable**, which is a symbol (typically a letter) that is used to *represent* a number. Instead of writing 2 + 9, we might write 2 + *x*, where *x* is a variable that does not have an as-yet specified value. The variable *x* could be 9, or it could be 2, or it could be any other number that you can think of. Nevertheless, the variable *x* acts in the exact same way as a number; thus, if *x* is equal to 9, then both 2 + 9 = 11 and 2 + *x* = 11. In addition, all the rules associated with arithmetic operations such as 2 + 9 also apply when variables are used. For example, 2 + *x* = *x* + 2. Variables can be used in cases where a number is too long to write out (such as a repeating decimal), or it can be used where a particular value is unknown or otherwise unspecified.

__Practice Problem:__ Use variables to write an expression for the final cost of an item given an unspecified base price and a sales tax rate of 5%.

__ Solution:__ We know that the final cost of an item is its base price plus its price multiplied by the sales tax rate. For example, if an item costs $30, then the final cost is

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We can write a general expression for this operation using a variable in place of the price, however. Let's use the variable *p* to represent the price of an item. In the example above, we used the example of $30--in this case, however, we don't know what the price is, so let's simply substitute *p* for the price.

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Note that this is a general expression: it works no matter what the price is. For example, if the price is $20.37, just substitute $20.37 for *p*. This example illustrates how we can write a general algebraic expression for a mathematical operation.

*Functions*

A relation between a given set of numbers and variables is a **function**. A function, like a variable, is often assigned a symbol: for obvious reasons, *f* is often a letter of choice for a function (but *other choices are also legitimate, depending on the details of the problem*). The function symbol is then typically followed by parentheses that contain the variables used in the expression. Following the example of the above practice problem, we can define a function *f* that converts the price *p* of an item into its final cost that includes sales tax. We would write the function as follows.

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When you choose a value for the variable, simply substitute the value for every instance of the variable in the function. Thus, if an item is priced at $60, then its final cost is *f*($60), or

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__ Practice Problem: __Write a function for the area of a rectangle with one known side of length 10.

__Solution:__ Let's start by drawing a diagram of the rectangle described in the problem.

The diagram shows the known side (of length 10) and the unknown side, whose length we can call *w*, since it is unknown. We know that the area of a rectangle is the product of the length and the width, so let's define a function *A*(*w*) that is the area of the rectangle in terms of the unknown width variable *w*. The full expression, and the solution to the problem, is then the following:

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*Graphs*

A particularly helpful tool in studying algebra is the** graph**. The term *graph* can have a range of meanings depending on the context, but in algebra, a graph is usually meant as a *coordinate graph*. A graph shows the value of a function (typically in the vertical direction) with respect to a range of values for a variable (typically in the horizontal direction).

__Practice Problem:__ Draw the graph of the function *f*(*x*) = 2*x*.

__Solution:__ This function simply involves doubling of the value of the variable. First, let's draw a set of coordinate axes, with the vertical axis corresponding to *f*(*x*) and the horizontal axis corresponding to *x*. Also, label the axes with evenly spaced tick marks (as you might see on a ruler). Finally, select some values for *x*, calculate *f*(*x*), and draw the corresponding points on the graph. If you connect these points, you will then have the graph of the function.