3.7 Graphs of Real- World Situations

Transcription

1 3.7 Graphs of Real- World Situations In this lesson ou will describe graphs using the words increasing, decreasing, linear, and nonlinear match graphs with descriptions of real- world situations learn about continuous and discrete functions use intervals of the domain to help ou describe a function s behavior Like pictures, graphs communicate a lot of information. So ou need to be able to draw and make sense of graphs. In Unit 2, ou learned to interpret dotplots, histograms, and boplots based on one quantit. In this lesson ou ll look at graphs that show how two real- world quantities are related, and ou ll practice interpreting and describing graphs. Investigation 1: Interpreting Graphs EX 1: This graph shows the relationship between time and the depth of water in a leak swimming pool. Depth (ft) Time (hrs) a. What is the initial depth of the water? b. For what time interval(s) is the water level decreasing? What accounts for the decrease(s)? c. For what time interval(s) is the water level increasing? What accounts for the increase(s)? d. Is the pool ever empt? How can ou tell? In this eample, the depth of the water is a function of time. That is, the depth depends on how much time has passed. So, in this case, depth is called the dependent variable. Time is the independent variable. When ou draw a graph, put the independent variable on the - ais and put the dependent variable on the - ais. On the graph of this function, ou can see the domain values that are possible for the independent variable in this real- world contet. This is called the practical domain. The practical domain in this eample is the set of all instants of time from 0 to 16 hours. We can epress this as 0 16, where is the independent variable representing time. Adapted from Discovering Algebra: An Investigative Approach b Murdock, Kamischke, and Kamischke

2 You can also see the values that are possible for the dependent variable. In this eample the range is the set of all numbers between 1 ft and about 3.3 ft. We can epress this as 1 3.3, where is the dependent variable representing the depth of the water in feet. Notice that the lowest value for the range (1 ft) does not have to be the starting value when is zero (2 ft). The relationship between the independent and dependent variable and the dependent variable is not alwas a cause and effect relationship. In man situations, time is the independent variable. It is the independent variable in graphs such as population growth or car depreciation and in several relationships of the form (time, distance). But time does not cause a population to grow or a walker s distance from a given point to change. People do that. The values of the range depend on the values of the domain. If ou know the value of the independent variable, ou can determine the corresponding value of the dependent variable. You do this ever time ou locate a point on the graph of a function. EX 2: This graph shows the volume of air in a balloon as it changes over time. a. What is the independent variable? How is it measured? b. What is the dependent variable? How is it measured? c. For what intervals is the volume increasing? What accounts for the increases? d. For what intervals is the volume decreasing? What accounts for the decreases? \ e. For what intervals is the volume constant? What accounts for this? f. What is happening for the first 2 seconds? Adapted from Discovering Algebra: An Investigative Approach b Murdock, Kamischke, and Kamischke

3 Investigation 2: Matching Up a. The graphs below show increasing functions, meaning that as the - values increase, the - values also increase. In Graph A, the function values increase at a constant rate. In Graph B, the values increase slowl at first and then more quickl. In Graph C, the function switches from one constant rate of increase to another. Graph A Graph B Graph C b. The graphs below show decreasing functions, meaning that as the - values increase, the - values decrease. In Graph D, the function values decrease at a constant rate. In Graph E, the values decrease quickl at first and then more slowl. In Graph F, the function switches from one constant rate of decrease to another. Graph D Graph E Graph F c. The graphs below show functions that have both increasing and decreasing intervals. In Graph G, the function values decrease at a constant rate at first and then increase at a constant rate. In Graph H, the values increase slowl at first and then more quickl and then begin to decrease quickl at first and then more slowl. In Graph I, the function oscillates between two values. Graph G Graph H Graph I Adapted from Discovering Algebra: An Investigative Approach b Murdock, Kamischke, and Kamischke

4 Read the description of each situation below. Identif the independent and dependent variables. Then decide which of the graphs above match the situation. a. White Tiger Population A small group of endangered white tigers are brought to a special reserve. The group of tigers reproduces slowl at first, and then as more and more tigers mature, the population grows more quickl. Matching Graph: b. Temperature of Hot Tea Grandma pours a cup of hot tea into a tea cup. The temperature at first is ver hot, but cools off quickl as the cup sits on the table. As the temperature of the tea approaches room temperature, it cools off more slowl. Matching Graph: c. Number of Dalight Hours over a Year s Time In Januar, the beginning of the ear, we are in the middle of winter and the number of dalight hours is at its lowest point. Then the number of dalight hours increases slowl at first through the rest of winter and earl spring. As summer approaches, the number of dalight hours increases more quickl, then levels off and reaches a maimum value, then decreases quickl, and then decreases more slowl into fall and earl winter. Matching Graph: d. Height of a Person Above Ground Who is Riding a Ferris Wheel When a girl gets on a Ferris wheel, she is 10 feet above ground. As the Ferris wheel turns, she gets higher and higher until she reaches the top. Then she starts to descend until she reaches the bottom and starts going up again. Adapted from Discovering Algebra: An Investigative Approach b Murdock, Kamischke, and Kamischke

5 Matching Graph: e. Make Your Own Stor! Choose one of the graphs that ou have not matched et (or sketch our own) and create a real- world situation that would match the graph. Describe the situation below and identif the independent and dependent variables. Indicate which graph ou chose Graph: Investigation 3: Discrete vs. Continuous Functions that have smooth graphs, with no breaks in the domain or range, are called continuous functions. Functions that are not continuous often involve quantities such as people, cars, or stories of a building that are counted or measured in whole numbers. Such functions are called discrete functions. Below are some eamples of discrete functions. Match each description with its most likel graph. Then label the aes with the appropriate quantities. a) the amount of product sold vs. advertising budget b) the amount of a radioactive substance over time c) the height of an elevator relative to floor number d) the population of a cit over time e) the number of students who help decorate for the homecoming dance vs. the time it takes to decorate Graph 1 Graph 2 Graph 3 Graph 4 Graph 5 Adapted from Discovering Algebra: An Investigative Approach b Murdock, Kamischke, and Kamischke

6 Sort the following ke terms into two groups. Then draw lines connecting pairs of terms that go together (one from each group). dependent, distance, horizontal ais, independent, input, output, time, vertical ais,, Domain Range Adapted from Discovering Algebra: An Investigative Approach b Murdock, Kamischke, and Kamischke