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3.4: Graphs of Linear Functions from Tables

Difficulty Level: At Grade Created by: CK-12

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.

Guidance

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

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

You can also use a graphing calculator to create a table of values and graph of the function. This will be explored in Example B.

Example A

Complete the table of values for the linear function $3x+2y=-6$ .

Solution:

Before completing the table of values, solve the given function in terms of ‘ $y$ ’. This step is not necessary, but it does simplify the calculations.

$& \qquad \ 3x+2y=-6\\&3x-3x+2y=-3x-6\\&\qquad \qquad \ \ 2y=-3x-6\\&\qquad \qquad \ \frac{2y}{2}=\frac{-3x}{2}-\frac{6}{2}\\&\qquad \qquad \ \ \boxed{y=\frac{-3x}{2}-3}$

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

$y=-\frac{3}{2}x-3$
$X$ $Y$
${\color{red}-4}$ $3$
${\color{red}0}$ $-3$
${\color{red}2}$ $-6$
${\color{red}6}$ $-12$

Example B

Use technology to create a table of values for the linear function $f(x)=-\frac{1}{2}x+4$ .

Solution:

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 ‘ $x$ ’ is -2 and the difference between each number is +2. The table is consecutive, even numbers. When consecutive numbers are used as the input numbers ( $x$ -values), there is a definite pattern in the output numbers ( $y$ -values). This will be expanded upon in a later lesson.

Example C

Complete the table of values for $x-2y=4$ , and use those values to graph the function.

Solution:

$x-2y=4 && x-x-2y=-x+4 && -2y=-x+4 && \frac{-2y}{-2}=\frac{-x}{-2}+\frac{4}{-2} && \boxed{y=\frac{1}{2}x-2}$

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

$y=\frac{1}{2}x-2$
$X$ $Y$
${\color{red}-4}$ $-4$
${\color{red}0}$ $-2$
${\color{red}2}$ $-1$
${\color{red}6}$ $1$

Concept Problem Revisited

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.

The cost of renting the pool is $100. This amount is a 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 $y=55x+100$ where ‘ $y$ ’ represents the cost in dollars and ‘ $x$ ’ represents the time, in hours, that the pool is rented.

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

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

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

$& X(hours) \qquad 1 \qquad \quad \ \ 2 \qquad \quad \ 3 \qquad \quad \ 4 \qquad \quad \ \ 5\\& Y(Cost) \qquad \155 \qquad \210 \qquad \265 \qquad \320 \qquad \375$

The values in the table represent the coordinates of points that are located on the graph of $y=55x+100$ .

$(1,155);(2,210);(3,265);(4,320);(5,375)$

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

Vocabulary

Linear Function
The linear function is a relation between two variables, usually $x$ and $y$ , in which each value of the independent variable $(x)$ is mapped to one and only one value of the dependent variable $(y)$ .

Guided Practice

1. Complete the following table of values for the linear function $3x-2y=-12$

$3x-2y=-12$
$X$ $Y$
${\color{red}-6}$
${\color{red}-4}$
${\color{red}0}$
${\color{red}6}$

2. Use technology to complete a table of values for the linear function $2x-y=-8$ and use the coordinates to draw the graph.

3. 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.

1. $3x-2y=-12$ Solve the equation in terms of the variable ‘ $y$ ’.

$3x-3x-2y=-3x-12 && -2y=-3x-12 && \frac{-2y}{-2}=\frac{-3x}{-2}-\frac{12}{-2}$

$\boxed{y=\frac{3}{2}x+6}$

Substitute the given values for ‘ $x$ ’ into the function.

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

$3x-2y=-12$
$X$ $Y$
${\color{red}-6}$ $-3$
${\color{red}-4}$ $0$
${\color{red}0}$ $6$
${\color{red}6}$ $15$

2. $2x-y=-8$ To enter the function into the calculator, it must be in the form $y= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$ . Solve the function in terms of the letter ‘ $y$ ’.

$&2x-y=-8 && 2x-2x-y=-2x-8 && -y=-2x-8 && \frac{-y}{-1}=\frac{-2x}{-1}\frac{-8}{-1}\\& \boxed{y=2x+8}$

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

3. $y=.09x+25$ Write a linear function to represent the word problem.

$y &=.09(200)+25 && \text{Substitute the time of} \ 200 \ \text{minutes for the variable} \ `x'.\\y&=\43.00$

Date Created:

Dec 19, 2012

Apr 29, 2014
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