Let’s Think About It
It is Reggie’s first day on the job at The Taco Shop near his house. Reggie is looking over the menu when he notices that the restaurant now offers family or group combo orders of 12 tacos. If a customer orders 1 combo, he will get 12 tacos. If the customer orders 2 combos, he will get 24 tacos. Because his math class recently studied functions, Reggie realizes the number of tacos is a function of the number of combos ordered, as demonstrated below.
How can this data can be displayed visually with a graph?
In this concept, you will learn how to graph linear functions.
A function is a set of data that has a specific relationship. One variable in the data set is related to or depends on a different variable in the same data set. Each input matches with only one output.
This is why graphing functions is important. A graph of a function can show the relationship between thevalue and the value.
Take a look at the following table.
In this table, the lettersand are used instead of input and output. They mean the same thing, but in mathematics as you work with functions, you will use and more often.
Here,is the input value and is the output value. The value depends on the value. You can see that each value of the column matches with only one value of the column. This means that this table forms a function.
Here is another table.
In this table, thevalue of 1 is connected to two different “ values (5 and 7) at the same time. This is NOT a function. Remember, to be a function each value of the column can match with only ONE value of the column.
Let’s look at functions in a real life situation.
Felix has a job cutting grass in the summer time. He earns $10.00 per lawn that he cuts. The amount of money that Felix makes is related to the number of lawns that he cuts. Let’s look at some data about Felix and then show how this forms a function.
Felix cut the following lawns on four different days.
This data can be organized in a table with the number of lawns represented by thevalue and the amount of money earned represented by the value. The is the value that can be counted on or depended on and the value changes depending on the value.
This is represented in the following table.
You can say that the amount of money that Felix earns is a function of the number of lawns that he mows.
You can also graph functions on the coordinate grid by using the values in each column to form ordered pairs.
Notice that in an ordered pair you have anvalue and a value. Let’s write this data as ordered pairs.
Now you can graph the data.
Create a graph by plotting the values (the number of lawns mowed) on the axis and the values (the amount of money earned) on theaxis.
This graph forms a linear function. Anytime a graph forms a line like this, it is called a graph of a linear function. If the line rises from left to right, the graph of the function increases. If the line slopes downward from left to right, the graph of the linear function decreases.
On a coordinate plane, graph the linear function that is represented by the ordered pairs in the table below.
First, identify the ordered pairs by pairing thevalue with its value.
The ordered pairs shown in the table areand .
Next, plot those five points on the coordinate plane by locating the-coordinate then going up or down to the -coordinate.
Then connect them as shown below.
Notice that the graph of this function is a straight line that decreases from left to right. That is because this function is a linear function.
Answer the following questions about graphs and linear functions.
Does this graph show an increase or a decrease?
First, look at the line representing this function. The line rises as it goes from left to right. Therefore, the function increases.
The answer is the function increases.
In the graph in Example 1, is the amount of money earned represented on theaxis or the axis?
First, look at the graph. The money earned is in increments of $5. This data is on the axis.
The answer is theaxis.
According to the graph in Example 1, what is the largest number of wind chimes sold?
First, locate “wind chimes sold” on the graph. It is represented on theaxis.
Next, locate the largest -coordinate.
The answer is 5 wind chimes.
Remember Reggie and his first day at The Taco Shop?
A customer walks up to the counter and asks Reggie how many tacos come in the family combo. Reggie tells the customer that each combo contains 12 tacos. The customer tells Reggie that he needs to order enough tacos for 11 people. He tells Reggie that 5 tacos should be enough for each person, which means he needs to get at least 55 tacos. Reggie does the math in his head and figures out that 5 combo packs will be 60 tacos because the number of tacos is a function of the number of combos the man orders. Reggie walks over to the kitchen and tells the cooks to get busy.
The following chart represents this function.
How can Reggie display this function and data on a graph?
To graph this linear function, first identify the ordered pairs.
Next, plot those five points on the coordinate plane by locating the -coordinate then going up or down to the-coordinate. Then connect them as shown below.
Here is a graph representing the data from the table.
Notice that this is a linear graph showing the relationship between number of combo packs and the number of tacos.
Graph each function represented by the data in the table then determine which represent linear graphs and which ones do not.