Have you ever been to an OmniTheater? Take a look at this dilemma.
Tasha and Uniqua think that their seventh grade class should go to the Omni Theater to see a film on the rainforest. By calling the Science Museum where the theater is located, they discover the following information.
The cost of a ticket is $5.00, but there is a $2.00 service fee per ticket.
“That is a lot of money,” Tasha says.
“Well, it depends on how many students actually go,” Uniqua says.
“Let’s figure it out. There are 22 students in our class and we could have anywhere from 22 to 18 students go based on the number of students absent. Now we need to do the math,” Tasha says taking out a piece of paper.
You will need to use a function to solve this dilemma. To figure out the range of costs with this trip, you will need to write an equation and create a table to show how the cost changes based on the number of students who attend the trip. By the end of this Concept, you will know how to go about solving this problem.
Guidance
Do you remember how to identify a function and a function rule?
A function is a relation in which each member of the domain is paired with exactly one member of the range.
In other words, a number in the domain cannot have two values for the range. When we look at the values in the domain and the range, we can figure out if the relation is a function or not.
Functions can be represented with values in a table. A function table is an input/output table where the input is the domain and the output is the range.
There are function rules that go with function tables. Do you know what a function rule is?
A function rule can be written in words or in the form of an equation. The function rule tells you what operation or operations to perform with the input to get the output.
You can also start with equations or rules and then see how these equations can help us to figure out ordered pairs.
Let’s start by thinking about the following equation.
\begin{align*}3+2=5\end{align*}
This is a true equation. You probably remember equations like this one from back in your elementary school days. However, we can look at this equation in a new way. Here we have the statement that three plus two is equal to five. Well, there are other values that could also be added together to equal 5. We could add positive and negative numbers to equal five. Therefore, there are many possible values that could be added together to equal five. Let’s change this equation to one where that is clear.
\begin{align*}x+y=5\end{align*}
Now we have used the values \begin{align*}x\end{align*} and \begin{align*}y\end{align*} to show that we have two different values that can be added together to equal \begin{align*}y\end{align*}.
Think about ordered pairs. An ordered pair has an \begin{align*}x\end{align*} and a \begin{align*}y\end{align*} value. If we were to find values that would make this a true statement, then we could also say that we had ordered pairs that would make this a true statement.
One answer for this equation is the ordered pair (2, 3) where the \begin{align*}x\end{align*} value is 2 and the \begin{align*}y\end{align*} value is 3. The sum is equal to five.
Let’s look at another situation.
Find three solutions to the equation \begin{align*}2x+y=12\end{align*} and write them in ordered pairs.
\begin{align*}2 \cdot 2+8=12\end{align*} so the ordered pair is (2, 8).
\begin{align*}2 \cdot 3+6=12\end{align*} so the ordered pair is (3, 6).
\begin{align*}2 \cdot -5+22=12\end{align*} so the ordered pair is (-5, 22).
When an equation is written in a form where \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are added together to equal a third value, we call that standard form. We can say that standard form is \begin{align*}Ax+By=C\end{align*}.
Write the definition for standard form and its equation in your notebook.
You just learned how to identify an equation in standard form. We can also write equations in function form. Function form is when the \begin{align*}y\end{align*} value is equal to the rest of the equation.
\begin{align*}y=2x+1\end{align*}
This is an equation in function form. We can see that the \begin{align*}y\end{align*} value is a function of \begin{align*}2x\end{align*} plus one. This means that the value of \begin{align*}y\end{align*} will change based on what the \begin{align*}x\end{align*} value is. We can also use \begin{align*}f(x)\end{align*} to show that the \begin{align*}y\end{align*} value is a function of the rest of the equation. The \begin{align*}f(x)\end{align*} is used to substitute for \begin{align*}y\end{align*}.
Exactly, let me try to explain this a little clearer. We know that the value of \begin{align*}y\end{align*} depends on the rest of the equation including whichever values we substitute for \begin{align*}x\end{align*}. Well, we can say that \begin{align*}y\end{align*} is a function of the rest of the equation. Therefore, we can say that the \begin{align*}f(x)\end{align*} is also dependent on the rest of the equation. The \begin{align*}f(x)\end{align*} is the same as \begin{align*}y\end{align*}.
Take a look at this situation.
\begin{align*}y=3x+1\end{align*}
To work with this equation, we have to create a table of values. Then we will know what the value of \begin{align*}y\end{align*} is based on the values that we substitute for \begin{align*}x\end{align*}.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 4 |
2 | 7 |
3 | 10 |
4 | 13 |
5 | 16 |
Now we have the values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. You can also notice that since we have these two values, we also have a set of ordered pairs that have been created in a table form.
Write each equation in function form.
Example A
\begin{align*}2x+y=7\end{align*}
Solution: \begin{align*}y=-2x+7\end{align*}
Example B
\begin{align*}-3x+y=18\end{align*}
Solution: \begin{align*}y=3x+18\end{align*}
Example C
\begin{align*}x+y=10\end{align*}
Solution: \begin{align*}y=-x+10\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
The problem can first be solved by writing an equation. To do this, we need to look at the given information.
Total cost \begin{align*}= y\end{align*}
\begin{align*}x = \end{align*} number of students and this can vary
$2.00 is the service fee
$5.00 per ticket
\begin{align*}y=5x+2\end{align*}
Now we can create our table of values based on the range of students in attendance. Uniqua and Tasha think that there will be anywhere from 22 to 18 students on the trip.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
22 | $112 |
21 | $107 |
20 | $102 |
19 | $97 |
18 | $92 |
Vocabulary
- Ordered pair
- the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values that can be found in a table or used to graph points or a line on the coordinate plane.
- Standard form
- the form of an equation \begin{align*}Ax+By=C\end{align*}
- Function form
- the form of an equation \begin{align*}y=mx+b\end{align*}
Guided Practice
Here is one for you to try on your own.
Rewrite this equation into function form.
\begin{align*}4x-y=-1\end{align*}
Solution
Here we have an equation in standard form. We will need to rewrite this equation into function form. To do this, we will move the negative one with the \begin{align*}4x\end{align*} and the \begin{align*}-y\end{align*} to the opposite side of the equals. We can do this by using inverse operations. Remember that an inverse operation is an opposite operation.
\begin{align*}4x-y+y &=-1+y \\ 4x &=-1+y \\ 4x+1 &=y \ or \ y=4x+1\end{align*}
Once we have an equation written in function form, we can use a table of values to figure out a set of ordered pairs.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 1 |
1 | 5 |
2 | 9 |
3 | 13 |
Now we have a set of ordered pairs for the equation \begin{align*}y=4x+1\end{align*}.
Video Review
Linear Equations in Standard Form
Practice
Directions: Find 4 solutions to the function \begin{align*}3x+y=24\end{align*}. Write your answers as ordered pairs.
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Directions: Find 4 solutions to the function \begin{align*}2x-y=9\end{align*}. Write your answers are ordered pairs.
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Directions: Write each equation in standard form.
- \begin{align*}y=2x+3\end{align*}
- \begin{align*}y=-4x+6\end{align*}
- \begin{align*}y=-2x-4\end{align*}
- \begin{align*}y=-5x+4\end{align*}
- \begin{align*}y=-3x-2\end{align*}
- \begin{align*}y=-4x-6\end{align*}
- \begin{align*}y=6x-1\end{align*}