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# 7.1: Line Graphs

Difficulty Level: Basic Created by: CK-12
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You're scanning some photographs for a customer. The customer is charged a 25 set-up fee for the scanner and then 0.55 per scan. How much would the customer get charged for 8 scans? What about for 15 scans? ### Watch This First watch this video to learn about line graphs. Then watch this video to see some examples. Watch this video for more help. ### Guidance Before you continue to explore the concept of representing data graphically, it is very important to understand the meaning of some basic terms that will often be used in this concept. The first such definition is that of a variable. In statistics, a variable is simply a characteristic that is being studied. This characteristic assumes different values for different elements, or members, of the population, whether it is the entire population or a sample. The value of the variable is referred to as an observation, or a measurement. A collection of these observations of the variable is a data set. Variables can be quantitative or qualitative. A quantitative variable is one that can be measured numerically. Some examples of a quantitative variable are wages, prices, weights, numbers of vehicles, and numbers of goals. All of these examples can be expressed numerically. A quantitative variable can be classified as discrete or continuous. A discrete variable is one whose values are all countable and does not include any values between 2 consecutive values of a data set. An example of a discrete variable is the number of goals scored by a team during a hockey game. A continuous variable is one that can assume any countable value, as well as all the values between 2 consecutive numbers of a data set. An example of a continuous variable is the number of gallons of gasoline used during a trip to the beach. A qualitative variable is one that cannot be measured numerically but can be placed in a category. Some examples of a qualitative variable are months of the year, hair color, color of cars, a person’s status, and favorite vacation spots. The following flow chart should help you to better understand the above terms. Variables can also be classified as dependent or independent. When there is a linear relationship between 2 variables, the values of one variable depend upon the values of the other variable. In a linear relation, the values of \begin{align*}y\end{align*} depend upon the values of \begin{align*}x\end{align*}. Therefore, the dependent variable is represented by the values that are plotted on the \begin{align*}y\end{align*}-axis, and the independent variable is represented by the values that are plotted on the \begin{align*}x\end{align*}-axis. Linear graphs are important in statistics when several data sets are used to represent information about a single topic. An example would be data sets that represent different plans available for cell phone users. These data sets can be plotted on the same grid. The resulting graph will show intersection points for the plans. These intersection points indicate a coordinate where 2 plans are equal. An observer can easily interpret the graph to decide which plan is best, and when. If the observer is trying to choose a plan to use, the choice can be made easier by seeing a graphical representation of the data. #### Example A Select the best descriptions for the following variables and indicate your selections by marking an ‘\begin{align*}x\end{align*}’ in the appropriate boxes. Variable Quantitative Qualitative Discrete Continuous Number of members in a family A person’s marital status Length of a person’s arm Color of cars Number of errors on a math test The variables can be described as follows: Variable Quantitative Qualitative Discrete Continuous Number of members in a family \begin{align*}x\end{align*} \begin{align*}x\end{align*} A person’s marital status \begin{align*}x\end{align*} Length of a person’s arm \begin{align*}x\end{align*} \begin{align*}x\end{align*} Color of cars \begin{align*}x\end{align*} Number of errors on a math test \begin{align*}x\end{align*} \begin{align*}x\end{align*} #### Example B Sally works at the local ballpark stadium selling lemonade. She is paid15.00 each time she works, plus 0.75 for each glass of lemonade she sells. Create a table of values to represent Sally’s earnings if she sells 8 glasses of lemonade. Use this table of values to represent her earnings on a graph. The first step is to write an equation to represent her earnings and then to use this equation to create a table of values. \begin{align*}y=0.75x+15\end{align*}, where \begin{align*}y\end{align*} represents her earnings and \begin{align*}x\end{align*} represents the number of glasses of lemonade she sells. Number of Glasses of Lemonade Earnings 015.00
1 $15.75 2$16.50
3 $17.25 4$18.00
5 $18.75 6$19.50
7 $20.25 8$21.00

The dependent variable is the money earned, and the independent variable is the number of glasses of lemonade sold. Therefore, money is on the \begin{align*}y\end{align*}-axis, and the number of glasses of lemonade is on the \begin{align*}x\end{align*}-axis.

### Vocabulary

Continuous data is data for which the plotted points can be joined, while discrete data is data for which the plotted points cannot be joined. A data set is a collection of observations of a variable. A variable is a characteristic that is being studied, with a continuous variable being a variable that can assume all values between 2 consecutive values of a data set, and a discrete variable being a variable that can only assume values that can be counted. Another way of classifying variables is by distinguishing between a qualitative variable, which is a variable that can be placed into specific categories according to some defined characteristic, and a quantitative variable, which is a variable that is numerical in nature and that can be ordered. When data is plotted on a coordinate grid, the independent variable is the variable represented by the values that are plotted on the \begin{align*}x\end{align*}-axis, and the dependent variable is the variable represented by the values that are plotted on the \begin{align*}y\end{align*}-axis.

### Guided Practice

The local arena is trying to attract as many participants as possible to attend the community’s “Skate for Scoliosis” event. Participants pay a fee of $10.00 for registering, and, in addition, the arena will donate$3.00 for each hour a participant skates, up to a maximum of 6 hours. Create a table of values and draw a graph to represent a participant who skates for the entire 6 hours. How much money can a participant raise for the community if he/she skates for the maximum length of time?

The equation for this scenario is \begin{align*}y=3x+10\end{align*}, where \begin{align*}y\end{align*} represents the money made by the participant, and \begin{align*}x\end{align*} represents the number of hours the participant skates.

Numbers of Hours Skating Money Earned
0 $10.00 1$13.00
2 $16.00 3$19.00
4 $22.00 5$25.00
6 28.00 The dependent variable is the money made, and the independent variable is the number of hours the participant skated. Therefore, money is on the \begin{align*}y\end{align*}-axis, and time is on the \begin{align*}x\end{align*}-axis as shown below: A participant who skates for the entire 6 hours can make28.00 for the "Skate for Scoliosis" event. The points are joined, because the fractions and decimals between 2 consecutive points are meaningful for this problem. A participant could skate for 30 minutes, and the arena would pay that skater \$1.50 for the time skating. The data is continuous.

### Practice

1. What term is used to describe a data set in which all points between 2 consecutive points are meaningful?
1. discrete data
2. continuous data
3. random data
4. fractional data
2. What type of variable is represented by the number of pets owned by families?
1. qualitative
2. quantitative
3. independent
4. continuous
3. What type of data, when plotted on a graph, does not have the points joined?
1. discrete data
2. continuous data
3. random data
4. independent data
4. Select the best descriptions for the following variables and indicate your selections by marking an ‘\begin{align*}x\end{align*}’ in the appropriate boxes.
Variable Quantitative Qualitative Discrete Continuous
Men’s favorite TV shows
Salaries of baseball players
Number of children in a family
Favorite color of cars
Number of hours worked weekly

You are selling your motorcycle, and you decide to advertise it on the Internet on Walton’s Web Ads. He has 3 plans from which you may choose. The plans are shown on the following graph. Use the graph and explain when it is best to use each plan.

1. When would it be best to use Plan A?
2. When would it be best to use Plan B?
3. When would it be best to use Plan C?
4. What is the dependent variable in the following relationship? The time it takes to run the 100 yard dash and the fitness level of the runner.
1. fitness level
2. time
3. length of the track
4. age of the runner
5. If the relationship in question 8 were graphed on a coordinate grid, what variable would be on the x-axis?
6. If the relationship in question 8 were graphed on a coordinate grid, what variable would be on the y-axis?

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### Vocabulary Language: English

TermDefinition
bar chart A bar chart is a graphic display of categorical variables that uses bars to represent the frequency of the count in each category.
broken line graph A broken line graph is a graph that is used to show changes over time. A line is used to join the values but the line has no defined slope.
continuous variables A continuous variable is a variable that takes on any value within the limits of the variable.
data set A collection of these observations of the variable is a data set.
dependent variable The dependent variable is the output variable in an equation or function, commonly represented by $y$ or $f(x)$.
discrete random variables Discrete random variables represent the number of distinct values that can be counted of an event.
independent variable The independent variable is the input variable in an equation or function, commonly represented by $x$.
qualitative variable A qualitative variable is one that cannot be measured numerically but can be placed in a category.
quantitative variable A quantitative variable is a variable that takes on numerical values that represent a measurable quantity. Examples of quantitative variables are the height of students or the population of a city.
variable In statistics, a variable is simply a characteristic that is being studied.

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