After reviewing this unit, you will be able to:
In many introductory courses you will come across characteristics or elements such as rates, outputs, income, etc., measured by numerical values. Some of these will always remain the same, and some will change. The characteristic or element that remains the same is called a constant. For example, the number of donuts in a dozen is always 12. That means the number of donuts in a dozen is a constant.
While some of these characteristics or elements remain the
same, some of these values can vary (e.g., the price of a dozen
donuts can change from $2.50 to $3.00), we call these characteristics
or elements variables. Variable is the generic term for any characteristic or
element that changes. You should be able to determine which
characteristics or elements are constants and which are variables.
Which of the following are variables and which are constants?
This is a variable. The temperature outside your home will change depending on the weather.
This is a constant. The square feet in a room 12 ft by 12 ft is always 144 square feet. It does not change.
This is a variable. The noise level changes depending on the number of people talking and yelling at any given time.
We express a relationship between two variables, which we will refer to as x and y, by stating the following: The value of the variable y depends upon the value of the variable x. We can write the relationship between variables in an equation. For instance:
is an example of a relationship between x and y variables. The equation also has an "a" and "b" in it. These are constants that help define the relationship between the two variables.
The following is an example to illustrate how these equations are constructed.
Throughout this tutorial we will use an example of a pizza shop that charges 7 dollars for a plain pizza with no toppings and 75 cents for each additional topping added. The total price of a pizza (y) depends upon the number of toppings (x) you order. So, price of a pizza is a dependent variable and number of toppings is the independent variable. In this example both the price and the number of toppings can change, therefore both are variables. The total price of the pizza also depends on the price of a plain pizza and the price per topping. In our example, the price of a plain pizza and the price per topping do not change, therefore these are constants. The relationship between the price of a pizza and the number of toppings can be expressed as an equation of the form:
If we know that x (the number of toppings) and y (the total price) represent variables, what are a and b? In our pizza example, "a" is the price of a plain pizza with no toppings and "b" is the price of each topping. They are constant. In other words, they are fixed values which specify how x relates to y.
We can set up an equation to show how the total price of pizza relates to the number of toppings ordered.
If we create a table of this particular relationship between x and y, we'll see all the combinations of x and y that fit the equation. For example, if plain pizza (a) is $7.00 and price of each topping (b) is $.75, we get:
Price of Each Topping
Number of Toppings
|$ 7.00||$ 7.00||$.75||0|
In the units that follow, we will review how this information can be displayed in the form of a graph. Before moving on, take a few moments to try the practice problem.
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