# Input-Output Tables for Function Rules

## Use function rules to complete patterns in tables.

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Graphs of Linear Functions from Tables

Bonita will be celebrating her sixteenth birthday next month. Her parents would like to give her a surprise party at the local pool. To rent the pool for a private party costs $100 plus$55.00 for each hour the pool is rented. Write a linear function to represent the cost of the pool party and list five prices from which her parents can choose.

### Graphing Linear Functions from Tables

One way to graph a linear function is to first create a table of points that work with the function and so must be on the graph of the function. A linear function will always result in a graph that is a straight line.

To create a table, substitute values for \begin{align*}x\end{align*} into the function (you can choose values for \begin{align*}x\end{align*}) and use the function to calculate the corresponding value for \begin{align*}y\end{align*}. Each pair of values is one point on the graph. It is easier to create the table if you first solve the equation for \begin{align*}y\end{align*}.

You can also use a graphing calculator to create a table of values and graph of the function.

#### Let's complete the table of values for the following linear function:

\begin{align*}3x+2y=-6\end{align*}.

Before completing the table of values, solve the given function in terms of ‘\begin{align*}y\end{align*}’. This step is not necessary, but it does simplify the calculations.

\begin{align*}& \qquad \ 3x+2y=-6\\ &3x-3x+2y=-3x-6\\ &\qquad \qquad \ \ 2y=-3x-6\\ &\qquad \qquad \ \frac{2y}{2}=\frac{-3x}{2}-\frac{6}{2}\\ &\qquad \qquad \ \ \boxed{y=\frac{-3x}{2}-3}\end{align*}

Now pick a few values for \begin{align*}x\end{align*} and substitute them into the equation to find the corresponding value for \begin{align*}y\end{align*}. Here, pick \begin{align*}x=-4\end{align*}, \begin{align*}x=2\end{align*}, \begin{align*}x=0\end{align*}, and \begin{align*}x=6\end{align*}.

\begin{align*}& y=\frac{-3x}{2}-3 && y=\frac{12}{2}-3 && y=\frac{-3x}{2}-3 && y=0-3\\ & y=\frac{-3({\color{red}-4})}{2}-3 && y=6-3 && y=\frac{-3({\color{red}0})}{2}-3 && \boxed{y=-3}\\ & && \boxed{y=3}\\ & y=\frac{-3x}{2}-3 && y=\frac{-6}{2}-3 && y=\frac{-3x}{2}-3 && y=\frac{-18}{2}-3\\ & y=\frac{-3({\color{red}2})}{2}-3 && y=-3-3 && y=\frac{-3({\color{red}6})}{2}-3 && y=-9-3\\ & && \boxed{y=-6} && && \boxed{y=-12}\end{align*}

Here is the table that shows the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} pairs.

\begin{align*}y=-\frac{3}{2}x-3\end{align*}
\begin{align*}X\end{align*} \begin{align*}Y\end{align*}
\begin{align*}{\color{red}-4}\end{align*} \begin{align*}3\end{align*}
\begin{align*}{\color{red}0}\end{align*} \begin{align*}-3\end{align*}
\begin{align*}{\color{red}2}\end{align*} \begin{align*}-6\end{align*}
\begin{align*}{\color{red}6}\end{align*} \begin{align*}-12\end{align*}

#### Now, let's use a calculator to create a table of values for the following linear function:

\begin{align*}f(x)=-\frac{1}{2}x+4\end{align*}

When the table is set up, you choose the beginning number as well as the pattern for the numbers in the table. In this table, the beginning value for ‘\begin{align*}x\end{align*}’ was –2 and the difference between each number was +2.

#### Finally, let's complete the table of values for \begin{align*}x-2y=4\end{align*}, and use those values to graph the following function:

First solve the equation for \begin{align*}y\end{align*}.

\begin{align*}x-2y&=4 \\ -2y&=-x+4\\ \frac{-2y}{-2}&=\frac{-x}{-2}+\frac{4}{-2}\\ y&=\frac{1}{2}x-2\end{align*}

Next choose values for \begin{align*}x\end{align*} to help make the table. Remember that you can choose any values for \begin{align*}x\end{align*}.

\begin{align*}& \ y=\frac{1}{2}x-2 && \ y=\frac{1}{2}x-2 && \ y=\frac{1}{2}x-2 && \ y=\frac{1}{2}x-2\\ & \ y=\frac{1}{2}({\color{red}-4})-2 && \ y=\frac{1}{2}({\color{red}0})-2 && \ y=\frac{1}{2}({\color{red}2})-2 && \ y=\frac{1}{2}({\color{red}6})-2\\ & \ y=-2-2 && \ y=0-2 && \ y=1-2 && \ y=3-2\\ & \boxed{y=-4} && \boxed{y=-2} && \boxed{y=-1} && \boxed{y=1}\end{align*}

Next make the table.

\begin{align*}y=\frac{1}{2}x-2\end{align*}
\begin{align*}X\end{align*} \begin{align*}Y\end{align*}
\begin{align*}{\color{red}-4}\end{align*} \begin{align*}-4\end{align*}
\begin{align*}{\color{red}0}\end{align*} \begin{align*}-2\end{align*}
\begin{align*}{\color{red}2}\end{align*} \begin{align*}-1\end{align*}
\begin{align*}{\color{red}6}\end{align*} \begin{align*}1\end{align*}

Last, plot the points from the table and connect to make the graph. You connect the points because there are more than just the four points in the table that work with the function and appear on the graph.

### Examples

#### Example 1

Earlier, you were told that Bonita's parents would like to throw her a surprise birthday party at the pool and we want to write a linear function to represent the cost of the pool party and list five prices from which her parents can choose.

There is a $100 fee that must be paid to rent the pool. In addition, Bonita’s parents will also have to pay$55.00 for each hour the pool is rented. Therefore, the linear function to represent this situation is \begin{align*}y=55x+100\end{align*} where ‘\begin{align*}y\end{align*}’ represents the cost in dollars and ‘\begin{align*}x\end{align*}’ represents the time, in hours, that the pool is rented.

\begin{align*}y=55x+100\end{align*} - To determine five options for her parents, replace ‘\begin{align*}x\end{align*}’ with the values 1 to 5 and calculate the cost for each of these hours.

\begin{align*}& \ y=55x+100 && \ y=55x+100 && \ y=55x+100 \\ & \ y=55(1)+100 && \ y=55(2)+100 && \ y=55(3)+100\\ & \boxed{y=\155} && \boxed{y=\210} && \boxed{y=\265}\\\\& y=55x+100 && \ y=55x+100 \\&y=55(4)+100 && \ y=55(5)+100\\&\boxed{y=\320} && \boxed{y=\375} \end{align*}

These results can now be represented in a table of values:

\begin{align*}& X(hours) \qquad 1 \qquad \quad \ \ 2 \qquad \quad \ 3 \qquad \quad \ 4 \qquad \quad \ \ 5\\ & Y(Cost) \qquad \155 \qquad \210 \qquad \265 \qquad \320 \qquad \375\end{align*}

The values in the table represent the coordinates of points that are located on the graph of \begin{align*}y=55x+100\end{align*}.

\begin{align*}(1,155);(2,210);(3,265);(4,320);(5,375)\end{align*}

Bonita’s parents can use the table of values and/or the graph to make their decision.

#### Example 2

Complete the following table of values for the linear function \begin{align*}3x-2y=-12\end{align*}

\begin{align*}3x-2y=-12\end{align*}
\begin{align*}X\end{align*} \begin{align*}Y\end{align*}
\begin{align*}{\color{red}-6}\end{align*}
\begin{align*}{\color{red}-4}\end{align*}
\begin{align*}{\color{red}0}\end{align*}
\begin{align*}{\color{red}6}\end{align*}

Solve the equation for ‘\begin{align*}y\end{align*}’ to get \begin{align*}y=\frac{3}{2}x+6\end{align*}.

Substitute the given values for ‘\begin{align*}x\end{align*}’ into the function.

\begin{align*}& \ y=\frac{3}{2}x+6 && \ y=\frac{3}{2}x+6 && \ y=\frac{3}{2}x+6 && \ y=\frac{3}{2}x+6\\ & \ y=\frac{3}{2}({\color{red}-6})+6 && \ y=\frac{3}{2}({\color{red}-4})+6 && \ y=\frac{3}{2}({\color{red}0})+6 && \ y=\frac{3}{2}({\color{red}6})+6\\ & \ y=-9+6 && \ y=-6+6 && \ y=0+6 && \ y=9+6\\ & \boxed{y=-3} && \boxed{y=0} && \boxed{y=6} && \boxed{y=15}\end{align*}

Complete the table.

\begin{align*}3x-2y=-12\end{align*}
\begin{align*}X\end{align*} \begin{align*}Y\end{align*}
\begin{align*}{\color{red}-6}\end{align*} \begin{align*}-3\end{align*}
\begin{align*}{\color{red}-4}\end{align*}
\begin{align*}{\color{red}0}\end{align*} \begin{align*}6\end{align*}
\begin{align*}{\color{red}6}\end{align*} \begin{align*}15\end{align*}

#### Example 3

Use technology to complete a table of values for the linear function \begin{align*}2x-y=-8\end{align*}. Use the coordinates to draw the graph.

To enter the function into the calculator, it must be in the form \begin{align*}y= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}. Solve the function for ‘\begin{align*}y\end{align*}’ to get \begin{align*}y=2x+8\end{align*}. Then, use your calculator.

The graph can also be done using technology. The table can be used to set the window.

#### Example 4

A local telephone company charges a monthly fee of $25.00 plus$0.09 per minute for calls within the United States. If Sam talks for 200 minutes in one month, calculate the cost of his telephone bill.

The function \begin{align*}y=0.09x+25\end{align*} represents the word problem.

\begin{align*}y &=0.09(200)+25 && \text{Substitute the time of} \ 200 \ \text{minutes for the variable} \ x.\\ y&=\43.00\end{align*}

The cost of Sam’s telephone bill is \$43.00.

### Review

Solve each of the following linear functions for ‘\begin{align*}y\end{align*}’.

1. \begin{align*}2x-3y=18\end{align*}
2. \begin{align*}4x-2y=10\end{align*}
3. \begin{align*}3x-y=8\end{align*}
4. \begin{align*}5x+3y=-12\end{align*}
5. \begin{align*}3x-2y-2=0\end{align*}

For each of the following linear functions, create a table of values that contains four coordinates.

1. \begin{align*}y=-4x+5\end{align*}
2. \begin{align*}5x+3y=15\end{align*}
3. \begin{align*}4x-3y=6\end{align*}
4. \begin{align*}2x-2y+2=0\end{align*}
5. \begin{align*}2x-3y=9\end{align*}

For each of the linear functions, complete the table of values and use the values to draw the graph.

1. \begin{align*}y=-2x+1\end{align*}

\begin{align*}& x \qquad -3 \qquad 0 \qquad 1 \qquad 5\\ & y\end{align*}

1. \begin{align*}x=2y-3\end{align*}

\begin{align*}& x \qquad -4 \qquad 0 \qquad 2 \qquad 6\\ & y\end{align*}

1. \begin{align*}3x+2y=8\end{align*}

\begin{align*}& x \qquad -6 \qquad -2 \qquad 0 \qquad 4\\ & y\end{align*}

1. \begin{align*}4(y-1)=12x-7\end{align*}

\begin{align*}& x \qquad -2 \qquad 0 \qquad 3 \qquad 7\\ & y\end{align*}

1. \begin{align*}\frac{1}{2}x+\frac{1}{3}y=6\end{align*}

\begin{align*}& x \qquad 0 \qquad 4 \qquad 6 \qquad 10\\ & y\end{align*}

Using technology, create a table of values for each of the following linear functions. Using technology, graph each of the linear functions.

1. \begin{align*}y=-2x+3\end{align*}
2. \begin{align*}y=-\frac{1}{2}x-3\end{align*}
3. \begin{align*}y=\frac{4}{3}x-2\end{align*}

To see the Review answers, open this PDF file and look for section 3.4.

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### Vocabulary Language: English

TermDefinition
Linear Function A linear function is a relation between two variables that produces a straight line when graphed.

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