12.1: Modeling Data with Functions
Name: __________________
Direct Variation
1. Suppose I make \begin{align*}\$6/\mathrm{hour}\end{align*}. Let \begin{align*}t\end{align*} represent the number of hours I work, and \begin{align*}m\end{align*} represent the money I make.
a. Make a table showing different \begin{align*}t\end{align*} values and their corresponding \begin{align*}m\end{align*} values. (\begin{align*}m\end{align*} is not how much money I make in that particular hour—it’s how much total money I have made, after working that many hours.)
time \begin{align*}(t)\end{align*} | money \begin{align*}(m)\end{align*} |
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b. Which is the dependent variable, and which is the independent variable?
c. Write the function.
d. Sketch a quick graph of what the function looks like.
e. In general: if I double the number of hours, what happens to the amount of money?
2. I am stacking bricks to make a wall. Each brick is \begin{align*}4\end{align*}" high. Let \begin{align*}b\end{align*} represent the number of bricks, and \begin{align*}h\end{align*} represent the height of the wall.
a. Make a table showing different \begin{align*}b\end{align*} values and their corresponding \begin{align*}h\;\mathrm{values}\end{align*}.
bricks \begin{align*}(b)\end{align*} | height \begin{align*}(h)\end{align*} |
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b. Which is the dependent variable, and which is the independent variable?
c. Write the function.
d. Sketch a quick graph of what the function looks like.
e. In general: if \begin{align*}I\end{align*} triple the number of bricks, what happens to the height?
3. The above two scenarios are examples of direct variation. If a variable \begin{align*}y\end{align*} “varies directly” with \begin{align*}x\end{align*}, then it can be written as a function \begin{align*}y=kx\end{align*}, where \begin{align*}k\end{align*} is called the constant of variation. (We also sometimes say that “\begin{align*}y\end{align*} is proportional to \begin{align*}x\end{align*},” where \begin{align*}k\end{align*} is called the constant of proportionality. Why do we say it two different ways? Because, as you’ve always suspected, we enjoy making your life difficult. Not “students in general,” but just you personally.) So, if \begin{align*}y\end{align*} varies directly with \begin{align*}x \ldots\end{align*}
a. What happens to \begin{align*}y\end{align*} if \begin{align*}x\end{align*} doubles? (Hint: You can find and prove the answer from the equation \begin{align*}y=kx\end{align*}.)
b. What happens to \begin{align*}y\end{align*} if \begin{align*}x\end{align*} is cut in half?
c. What does the graph \begin{align*}y(x)\end{align*} look like? What does \begin{align*}k\end{align*} represent in this graph?
4. Make up a word problem like numbers \begin{align*}(1)\end{align*} and \begin{align*}(2)\end{align*} above, on the subject of fast food. Your problem should not involve getting paid or stacking bricks. It should involve two variables that vary directly with each other. Make up the scenario, define the variables, and then do problems a-e exactly like my two problems.
Name: __________________
Homework Inverse Variation
1. An astronaut in space is performing an experiment with three balloons. The balloons are all different sizes, but they have the same amount of air in them. As you might expect, the balloons that are very small experience a great deal of air pressure (the air inside pushing out on the balloon); the balloons that are very large, experience very little air pressure. He measures the volumes and pressures and comes up with the following chart.
Volume \begin{align*}(V)\end{align*} | Pressure \begin{align*}(P)\end{align*} |
---|---|
\begin{align*}5\end{align*} | \begin{align*}270\end{align*} |
\begin{align*}10\end{align*} | \begin{align*}135\end{align*} |
\begin{align*}15\end{align*} | \begin{align*}90\end{align*} |
\begin{align*}20\end{align*} | \begin{align*}67\frac{1}{2}\end{align*} |
a. Which is the dependent variable, and which is the independent variable?
b. When the volume doubles, what happens to the pressure?
c. When the volume triples, what happens to the pressure?
d. Based on your answers to parts (a) - (c), what would you expect the pressure to be for a balloon with a volume of \begin{align*}30\end{align*}?
e. On the right of the table add a third column that represents the quantity \begin{align*}\mathrm{PV}\end{align*}: pressure times volume. Fill in all four values for this quantity. What do you notice about them?
f. Plot all four points on the graph paper, and fill in a sketch of what the graph looks like.
g. Write the function \begin{align*}P(V)\end{align*}. Make sure that it accurately gets you from the first column to the second in all four instances! (Part (e) is a clue to this.)
h. Graph your function \begin{align*}P(V)\end{align*} on your calculator, and copy the graph onto the graph paper. Does it match your graph in part \begin{align*}(f)\end{align*}?
2. The three little pigs have built three houses—made from straw, Lincoln Logs\begin{align*}^\circledR\end{align*}, and bricks, respectively. Each house is \begin{align*}20'\mathrm{high}\end{align*}. The pieces of straw are \begin{align*}\frac{1}{10}"\end{align*} thick; the Lincoln Logs\begin{align*}^\circledR\end{align*} are \begin{align*}1"\end{align*} thick; the bricks are \begin{align*}4"\end{align*} thick. Let \begin{align*}t\end{align*} be the thickness of the building blocks, and let \begin{align*}n\end{align*} be the number of such blocks required to build a house \begin{align*}20'\mathrm{high}\end{align*}. (Note: There are \begin{align*}12"\end{align*} in \begin{align*}1'\end{align*}. But you probably knew that ...)
a. Make a table showing different \begin{align*}t\end{align*} values and their corresponding \begin{align*}n\;\mathrm{values}\end{align*}.
Building Blocks | thickness \begin{align*}(t)\end{align*} | number \begin{align*}(n)\end{align*} |
---|---|---|
Straw | ||
Lincoln Logs\begin{align*}^\circledR\end{align*} | ||
Bricks |
b. Which is the dependent variable, and which is the independent variable?
c. When the thickness of the building blocks doubles, what happens to the number required?
(*Not sure? Pretend that the pig’s cousin used \begin{align*}8"\end{align*} logs, and his uncle used \begin{align*}16"\end{align*} logs. See what happens to the number required as you go up in this sequence...)
d. When the thickness of the building blocks is halved, what happens to the number required?
e. On the right of the table add a fourth column, that represents the quantity \begin{align*}tn\end{align*}: thickness times number. Fill in all three values for this quantity. What do you notice about them? What do they actually represent, in our problem?
f. Plot all three points on the graph paper, and fill in a sketch of what the graph looks like.
g. Write the function \begin{align*}n(t)\end{align*}.
h. Graph your function \begin{align*}n(t)\end{align*} on your calculator, and copy the graph onto the graph paper. Does it match your graph in part (f)?
3. The above two scenarios are examples of inverse variation. If a variable \begin{align*}y\end{align*}, “varies inversely” with \begin{align*}x\end{align*}, then it can be written as a function \begin{align*}y=\frac{k}{x}\end{align*}, where \begin{align*}k\end{align*} is called the constant of variation. So, if \begin{align*}y\end{align*} varies inversely with \begin{align*}x\end{align*}...
a. What happens to \begin{align*}y\end{align*} if \begin{align*}x\end{align*} doubles? (Hint: You can find and prove the answer from the equation \begin{align*}y=\frac{k}{x}\end{align*} )
b. What happens to \begin{align*}y\end{align*} if \begin{align*}x\end{align*} is cut in half?
c. What does the graph \begin{align*}y(x)\end{align*} look like? What happens to this graph when \begin{align*}k\end{align*} increases? (* You may want to try a few different ones on your calculator to see the effect \begin{align*}k\end{align*} has.)
4. Make up a word problem like numbers \begin{align*}(1)\end{align*} and \begin{align*}(2)\end{align*} above. Your problem should not involve pressure and volume, or building a house. It should involve two variables that vary inversely with each other. Make up the scenario, define the variables, and then do problems a-h exactly like my two with each other. Make up the scenario, define the variables, and then do problems a-h exactly like my two problems.
Name: __________________
Homework Direct and Inverse Variation
For questions \begin{align*}1-3\end{align*}, Please note that these numbers are meant to simulate real world data—that is to say, they are not necessarily exact! If it is “darn close to” direct or inverse variation, that’s good enough.
1. For the following set of data...
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}3\end{align*} | \begin{align*}5\end{align*} |
\begin{align*}6\end{align*} | \begin{align*}11\end{align*} |
\begin{align*}21\end{align*} | \begin{align*}34\end{align*} |
a. Does it represent direct variation, inverse variation, or neither?
b. If it is direct or inverse, what is the constant of variation?
c. If \begin{align*}x=30\end{align*}, what would \begin{align*}y\end{align*} be?
d. Sketch a quick graph of this relationship.
2. For the following set of data...
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}3\end{align*} | \begin{align*}18\end{align*} |
\begin{align*}4\end{align*} | \begin{align*}32\end{align*} |
\begin{align*}10\end{align*} | \begin{align*}200\end{align*} |
a. Does it represent direct variation, inverse variation, or neither?
b. If it is direct or inverse, what is the constant of variation
c. If \begin{align*}x=30\end{align*}, what would \begin{align*}y\end{align*} be?
d. Sketch a quick graph of this relationship.
3. For the following set of data\begin{align*}\ldots\end{align*}
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}3\end{align*} | \begin{align*}20\end{align*} |
\begin{align*}6\end{align*} | \begin{align*}10\end{align*} |
\begin{align*}21\end{align*} | \begin{align*}3\end{align*} |
a. Does it represent direct variation, inverse variation, or neither?
b. If it is direct or inverse, what is the constant of variation?
c. If \begin{align*}x=30\end{align*}, what would \begin{align*}y\end{align*} be?
d. Sketch a quick graph of this relationship.
4. In number \begin{align*}2\end{align*} above, as you (hopefully) saw, the relationship is neither direct nor inverse. However, the relationship can be expressed this way: \begin{align*}y\end{align*} is directly proportional to \begin{align*}x^2\end{align*}. Write the function that indicates this relationship. What is \begin{align*}k\end{align*}? one table is miss
5. In June, 2007, Poland argued for a change to the voting system in the European Union Council of Ministers. The Polish suggestion: each member’s voting strength should be directly proportional to the square root of his country’s population. This idea, traditionally known as Pensore’s Rule, is “almost sacred” among “people versed in the game theory of voting” according to one economist.
I swear I am not making this up.
Also in the category of “things I am not making up,” the following table of European Populations comes from Wikipedia.
\begin{align*}&\text {Germany} && 83,251,851\\ & \text{Italy} && 59,715,625\\ &\text{Poland} && 38,625,478\\ &\text{Luxemberg} && 48,569\end{align*}
a. Write an equation that represents Pensore’s Rule. Be sure to clearly label your variables.
b. Suppose that Pensore’s Rule was followed, and suppose that Poland voting strength was exactly \begin{align*}100\end{align*} (which I did actually make up, but of course it doesn’t matter). What would the voting strength of Germany, Italy, and Luxembourg be?
6. Write a “real world” word problem involving an inverse relationship, on the topic of movies. Identify the constant of variation. Write the function that shows how the dependent variable depends inversely upon the independent variable. Create a specific numerical question, and use your function to answer that question.
7. Joint Variation: The term “Joint Variation” is used to indicate that one variable varies directly as two different variables. This is illustrated in the following example.
Al is working as a waiter. When a group of people sit down at a table, he calculates his expected tip \begin{align*}(T)\end{align*} as follows: multiply the number of people \begin{align*}(N)\end{align*}, times the average meal cost \begin{align*}(C)\end{align*}, times \begin{align*}0.15\end{align*} (for a \begin{align*}15\%\end{align*} tip).
a. If the number of people at the table doubles, does Al’s expected tip double?
b. If the average cost per meal doubles, does Al’s expected tip double?
c. Write the function that expresses the dependent variable, \begin{align*}T\end{align*}, as a function of the two independent variables, \begin{align*}N\end{align*} and \begin{align*}C\end{align*}.
d. Write the general function that states “\begin{align*}z\end{align*} varies jointly as both \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.” Your function will have an unknown \begin{align*}k\end{align*} in it, a constant of variation.
8. Light Intensity: Stacy visits a tanning booth, where she spends several hours with lamps shining on her skin, thus giving her a beautiful copper-colored tan and a sharply increased risk of skin cancer. For reasons known only to herself, she considers this a good trade-off. Anyway, Stacy has a lot of time to just lie there and think, and she starts to consider the question: which bulb is shining on her skin with the most intensity? The answer is that the intensity I of a bulb varies directly with the strength \begin{align*}S\end{align*} of the bulb, and varies inversely with the square of the distance \begin{align*}d\end{align*} of the bulb from her skin.
a. Bulbs \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are the same distance away, but bulb \begin{align*}B\end{align*} is twice as strong as bulb \begin{align*}A\end{align*}. If bulb \begin{align*}A\end{align*} shines with an intensity of \begin{align*}5\end{align*}, what is the intensity of bulb \begin{align*}B\end{align*}?
b. Bulbs \begin{align*}A\end{align*} and \begin{align*}C\end{align*} are the same strength as each other, but bulb \begin{align*}A\end{align*} is twice as far away from Stacy as bulb \begin{align*}C\end{align*}. If bulb \begin{align*}A\end{align*} shines with an intensity of \begin{align*}5\end{align*}, what is the intensity of bulb \begin{align*}B\end{align*}?
c. Write a function to represent the statement “the intensity I of a bulb varies directly with the strength \begin{align*}S\end{align*} of the bulb, and varies inversely with the square of the distance d of the bulb from Stacy’s skin.” Your function will have an unknown \begin{align*}k\end{align*} in it, a constant of variation.
Name: _________________
Homework: Calculator Regression
1. Canadian Voters: The following table shows the percentage of Canadian voters who voted in the 1996 federal election.
\begin{align*}&\text{Age} && 20 && 30 && 40 && 50 && 60\\ &\% \text{voted} && 59 && 86 && 87 && 91 && 94\end{align*}
a. Enter these points on your calculator lists.
b. Set the Window on your calculator so that the \begin{align*}x-\end{align*}values go from to \begin{align*}60\end{align*}, and the \begin{align*}y-\end{align*}values go from to \begin{align*}100\end{align*}. Then view a graph of the points on your calculator. Do they increase steadily (like a line), or increase slower and slower (like a log), or increase more and more quickly (like a parabola or an exponent)?
c. Use the [STAT] function on your calculator to find an appropriate function to model this data. Write that function below.
d. Graph the function on your calculator. Does it match the points well? Are any of the points “outlyers?”
2. Height and Weight: A group of students record their height (in inches) and weight (in pounds). The results are on the table below.
\begin{align*}&\text{Height} && 68 && 74 && 66 && 68 && 72 && 69 && 65 && 71 && 69 && 72 && 71 && 64 && 65\\ &\text{weight} && 180 && 185 && 150 && 150 && 200 && 160 && 125 && 220 && 220 && 180 && 190 && 120 && 110\end{align*}
a. Enter these points on your calculator lists.
b. Set the Window on your calculator appropriately, and then view a graph of the points on your calculator. Do they increase steadily (like a line), or increase slower and slower (like a log), or increase more and more quickly (like a parabola or an exponent)?
c. Use the [STAT] function on your calculator to find an appropriate function to model this data. Write that function below.
d. Graph the function on your calculator. Does it match the points well? Are any of the points “outlyers?”
3. Gas Mileage: The table below shows the weight (in hundreds of pounds) and gas mileage (in miles per gallon) for a sample of domestic new cars.
\begin{align*}&\text{Weight} && 29 && 35 && 28 && 44 && 25 && 34 && 30 && 33 && 28 && 24\\ &\text{mileage} && 31 && 27 && 29 && 25 && 31 && 29 && 28 && 28 && 28 && 33\end{align*}
a. Enter these points on your calculator lists.
b. Set the Window on your calculator appropriately, and then view a graph of the points on your calculator. Do they decrease steadily (like a line), or decrease slower and slower (like a log), or decrease more and more quickly (like a parabola or an exponent)?
c. Use the [STAT] function on your calculator to find an appropriate function to model this data. Write that function below.
d. Graph the function on your calculator. Does it match the points well? Are any of the points “outlyers?”
4. TV and GPA: A graduate student named Angela Hershberger at Indiana University-South Bend did a study to find the relationship between TV watching and Grade Point Average among high school students. Angela interviewed \begin{align*}50\end{align*} high school students, turning each one into a data point, where the independent \begin{align*}(x)\end{align*} value was the number of hours of television watched per week, and the dependent \begin{align*}(y)\end{align*} value was the high school grade point average. (She also checked the types of television watched—eg news or sitcoms—and found that it made very little difference. Quantity, not quality, mattered.)
In a study that you can read all about at www.iusb.edu/~journal/2002/hershberger/hershberger.html, Angela found that her data could best be modeled by the linear function \begin{align*}y=–0.0288 x + 3.4397\end{align*}. Assuming that this line is a good fit for the data
a. What does the number \begin{align*}3.4397\end{align*} tell you? (Don’t tell me about lines and points: tell me about students, TV, and grades.)
b. What does the number \begin{align*}–0.0288\end{align*} tell you? (Same note.)
Name: _________________
Sample Test: Modeling Data with Functions
1. Three cars and an airplane are traveling to New York City. But they all go at different speeds, so they all take different amounts of time to make the \begin{align*}500-\end{align*}mile trip. Fill in the following chart.
Speed(s) - miles per hour | Time(t) - hours |
---|---|
\begin{align*}50\end{align*} | |
\begin{align*}75\end{align*} | |
\begin{align*}100\end{align*} | |
\begin{align*}500\end{align*} |
a. Is this an example of direct variation, inverse variation, or neither of the above?
b. Write the function \begin{align*}s(t)\end{align*}.
c. If this is one of our two types, what is the constant of variation?
2. There are a bunch of squares on the board, of different sizes.
s - length of teh side of a square | A - area of the square |
---|---|
\begin{align*}1\end{align*} | |
\begin{align*}2\end{align*} | |
\begin{align*}3\end{align*} | |
\begin{align*}4\end{align*} |
a. Is this an example of direct variation, inverse variation, or neither of the above?
b. Write the function \begin{align*}A(s)\end{align*}.
c. If this is one of our two types, what is the constant of variation?
3. Anna is planning a party. Of course, as at any good party, there will be a lot of on hand! \begin{align*}50\end{align*} Coke cans fit into one recycling bin. So, based on the amount of Coke she buys, Anna needs to make sure there are enough recycling bins.
c-Coke cans Anna buys | b - recycling bins she will need |
---|---|
\begin{align*}50\end{align*} | |
\begin{align*}100\end{align*} | |
\begin{align*}200\end{align*} | |
\begin{align*}400\end{align*} |
a. Is this an example of direct variation, inverse variation, or neither of the above?
b. Write the function \begin{align*}b(c)\end{align*}.
c. If this is one of our two types, what is the constant of variation?
4. Make up a word problem involving inverse variation, on the topic of skateboarding.
a. Write the scenario.
b. Label and identify the independent and dependent variables.
c. Show the function that relates the dependent to the independent variable. This function should (of course) be an inverse relationship, and it should be obvious from your scenario!
5. I found a Web site (this is true, really) that contains the following sentence:
[This process] introduces an additional truncation error [directly] proportional to the original error and inversely proportional to the gain \begin{align*}(g)\end{align*} and the truncation parameter \begin{align*}(q)\end{align*}.
I don’t know what most of that stuff means any more than you do. But if we use \begin{align*}T\end{align*} for the “additional truncation error” and E for the “original error,” write an equation that expresses this relationship.
6. Which of the following correctly expresses, in words, the relationship of the area of a circle to the radius?
A. The area is directly proportional to the radius
B. The area is directly proportional to the square of the radius
C. The area is inversely proportional to the radius
D. The area is inversely proportional to the square of the radius
7. Now, suppose we were to write the inverse of that function: that is, express the radius as a function of the area. Then we would write:
The radius of a circle is ___________ proportional to _____________________ the area.
8. Death by Cholera: In 1852, William Farr reported a strong association between low elevation and deaths from cholera. Some of his data are reported below.
\begin{align*}&\text{E:Elevation(ft)} && 10 && 30 && 50 && 70 && 90 && 100 && 350\\ &\text{C:Cholera morality}(\text{per} 10,000) && 102 && 65 && 34 && 27 && 22 && 17 && 8\end{align*}
a. Use your calculator to create the following models, and write the appropriate functions \begin{align*}C(E)\end{align*} in the blanks.
Linear: \begin{align*}C=\end{align*}
Quadratic: \begin{align*}C=\end{align*}
Lograthamic: \begin{align*}C=\end{align*}
exponential: \begin{align*}C=\end{align*}
b. Which model do you think is the best? Why?
c. Based on his very strong correlation, Farr concluded that bad air had settled into low-lying areas, causing outbreaks of cholera. We now know that air quality has nothing to do with causing cholera: the water-borne bacterial Vibrio cholera causes the disease. What might explain Farr’s results without justifying his conclusion?
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