Each group has three people. Designate one person as the “Leader” and one person as the “Recorder.” (These roles will rotate through all three people.) At any given time, the Leader is looking at a sheet with a list of “functions,” or formulas; the Recorder is looking at the answer sheet. Here’s how it works.
- One of the two players who is not the Leader says a number.
- The Leader does the formula (silently), comes up with another number, and says it.
- The Recorder writes down both numbers, in parentheses, separated by a comma. (Like a point.)
- Keep doing this until someone guesses the formula. (If someone guesses incorrectly, just keep going.)
- The Recorder now writes down the formula—not in words, but as an algebraic function.
- Then, move on to the next function.
Sound confusing? It’s actually pretty easy. Suppose the first formula was “Add five.” One player says “” and the Leader says “”. One player says “” and the Leader says “”. One player says “” and the Leader says “”. One player says “You’re adding five” and the Leader says “Correct.” At this point, the Recorder has written down the following:
1. Points:
Answer:
Sometimes there is no possible answer for a particular number. For instance, your function is “take the square root” and someone gives you “.” Well, you can’t take the square root of a negative number: is not in your domain, meaning the set of numbers you are allowed to work on. So you respond that “ is not in my domain.”
Leader, do not ever give away the answer!!! But everyone, feel free to ask the teacher if you need help.
The Function Game Leader’s Sheet
Only the leader should look at this sheet. Leader, use a separate sheet to cover up all the functions below the one you are doing right now. That way, when the roles rotate, you will only have seen the ones you’ve done.
- Double the number, then add six.
- Add three to the number, then double.
- Multiply the number by , then add three.
- Subtract one from the number. Then, compute one divided by your answer.
- Divide the number by two.
- No matter what number you are given, always answer “.”
- Square the number, then subtract four.
- If you are given a positive number, give the same number back. If you are given a negative number, multiply that number by .
- Cube the number.
- Add two to the number. Also, subtract two from the original number. Multiply these two answers.
- Take the square root of the number. Round up to the nearest integer.
- Add one to the number, then square.
- Square the number, then add .
- Cube the number. Then subtract the original number from that answer.
- Give back the lowest prime number that is greater than or equal to the number.
- If you are given an odd number, respond . If you are given an even number, respond . (Fractions are not in the domain of this function.)
The Function Game: Answer Sheet Recorder
1. Points:
Answer:
2. Points:
Answer:
3. Points:
Answer:
4. Points:
Answer:
5. Points:
Answer:
6. Points:
Answer:
7. Points:
Answer:
8. Points:
Answer:
9. Points:
Answer:
10. Points:
Answer:
11. Points:
Answer:
12. Points:
Answer:
13. Points:
Answer:
14. Points:
Answer:
15. Points:
Answer:
16. Points:
Answer:
Name:
Homework: The Function Game
1. Describe in words what a variable is, and what a function is.
There are seven functions below (numbered 2-8). For each function,
- Write the same function in algebraic notation.
- Generate three points from that function.
For instance, if the function were “Add five” the algebraic notation would be “”. The three points might be , , and .
2. Triple the number, then subtract six.
a. Algebraic notation:
b. Three points:
3. Return , no matter what.
a. Algebraic notation:
b. Three points:
4. Add one. Then take the square root of the result. Then, divide that result into two.
a. Algebraic notation:
b. Three points:
5. Add two to the original number. Subtract two from the original number. Then, multiply those two answers together.
a. Algebraic notation:
b. Three points:
6. Subtract two, then triple.
a. Algebraic notation:
b. Three points:
7. Square, then subtract four.
a. Algebraic notation:
b. Three points:
8. Add three. Then, multiply by four. Then, subtract twelve. Then, divide by the original number.
a. Algebraic notation:
b. Three points:
9. In some of the above cases, two functions always give the same answer, even though they are different functions. We say that these functions are “equal” to each other. For instance, the function “add three and then subtract five” is equal to the function “subtract two” because they always give the same answer. (Try it, if you don’t believe me!) We can write this as:
Note that this is not an equation you can solve for —it is a generalization which is true for all values. It is a way of indicating that if you do the calculation on the left, and the calculation on the right, they will always give you the same answer.
In the functions above, there are three such pairs of “equal” functions. Which ones are they? Write the algebraic equations that state their equalities (like my equation).
10. Of the following sets of numbers, there is one that could not possibly have been generated by any function whatsoever. Which set it is, and why? (No credit unless you explain why!)
a.
b.
c.
d.
e.
Name:
Homework: Functions in the Real World
Laura is selling doughnuts for each. Each customer fills a box with however many doughnuts he wants, and then brings the box to Laura to pay for them. Let represent the number of doughnuts in a box, and let represent the cost of the box (in cents).
a. If the box has doughnuts, how much does the box cost?
b. If , how much does the box cost? How many doughnuts does it have?
c. If a box has doughnuts, how much does it cost?
d. Write a function that gives the cost of a box, as a function of the number of doughnuts in the box.
2. Worth is doing a scientific study of graffiti in the downstairs boy’s room. On the first day of school, there is no graffiti. On the second day, there are two drawings. On the third day, there are four drawings. He forgets to check on the fourth day, but on the fifth day, there are eight drawings. Let represent the day, and represent the number of graffiti marks that day.
a. Fill in the following table, showing Worth’s four data points.
b. If this pattern keeps up, how many graffiti marks will there be on day
c. If this pattern keeps up, on what day will there be graffiti marks?
d. Write a function that gives the number of graffiti marks as a function of the day.
3. Each of the following is a set of points. Next to each one, write “yes” if that set of points could have been generated by a function, and “no” if it could not have been generated by a function. (You do not have to figure out what the function is. But you may want to try for fun—I didn’t just make up numbers randomly...)
a.
b.
c.
d.
e.
4.
a.
b.
c.
d.
e.
f.
g.
5. Make up a function that has something to do with movies.
a. Think of a scenario where there are two numbers, one of which depends on the other. Describe the scenario, clearly identifying the independent variable and the dependent variable.
b. Write the function that shows how the dependent variable depends on the independent variable.
c. Now, plug in an example number to show how it works.
Algebraic Generalizations
1. a. Pick a number: _____
b. Add three: _____
c. Subtract three from your answer in part (b): _____
d. What happened? ______________________________________________
e. Write an algebraic generalization to represent this rule. _________________
f. Is there any number for which this rule will not work? ___________________
2. a. Pick a number: _____
b. Subtract five: _____
c. Double your answer in part (b): _____
d. Add ten to your answer in part (c): _____
e. Divide your answer in part (d) by your original number (a): _____
f. Now, repeat that process for three different numbers. Record the number you started with (a) and the number you ended up with (e).
g. What happened? ______________________________________________
h. Write an algebraic generalization to represent this rule. _________________
i. Is there any number for which this rule will not work? ___________________
3. Here are the first six powers of two.
a. If I asked you for (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term? ________________
b. Write an algebraic generalization to represent this rule. _____________________
4. Now, we’re going to make that rule even more general. Suppose I want to multiply times . Well, means , and means . So we can write the whole thing out like this.
This shows that ()() = .
a. Using a similar drawing, demonstrate what ()() must be.
b. Now, write an algebraic generalization for this rule. ____________________
5. The following statements are true.
Write an algebraic generalization for this rule. __________________
6. Look at the following pairs of statements.
a. Based on these pairs, if I told you that , could you tell me (immediately, without a calculator) what is? ____________________________
b. Express this rule—the pattern in these numbers—in words.
c. Whew! That was complicated, wasn’t it? Good thing we have math. Write the algebraic generalization for this rule. _______________________
d. Try out this generalization with negative numbers, with zero, and with fractions. (Show your work below, trying all three of these cases separately.) Does it always work, or are there cases where it doesn’t?
Name: __________________
Homework: Algebraic Generalizations
In class, we talked about the following four pairs of statements.
You made an algebraic generalization about these statements: write that generalization again below.
Now, we are going to generalize it further. Let’s focus on the thing.
There are two numbers that are one away from ; these numbers are, of course, and . As we saw, is . It is one less than .
Now, suppose we look at the two numbers that are two away from Or three away? Or four away? We get a sequence like this (fill in all the missing numbers):
Do you see the pattern? What would you expect to be the next sentence in this sequence?
Write the algebraic generalization for this rule.
Does that generalization work when the “___away from ” is Is a fraction? Is a negative number? Test all three cases. (Show your work!)
Name: __________________
Homework: Graphing
The following graph shows the temperature throughout the month of March. Actually, I just made this graph up—the numbers do not actually reflect the temperature throughout the month of March. We’re just pretending, OK?
1. Give a weather report for the month of March, in words.
2. On what days was the temperature exactly
3. On what days was the temperature below freezing?
4. On what days was the temperature above freezing?
5. What is the domain of this graph?
6. During what time periods was the temperature going up?
7. During what time periods was the temperature going down?
8. Mary started a company selling French Fries over the Internet. For the first days, while she worked on the technology, she lost per day. Then she opened for business. People went wild over her French Fries! She made in one day, the day after that, and the day after that. The following day she was sued by an angry customer who discovered that Mary had been using genetically engineered potatoes. She lost in the lawsuit that day and closed up her business. Draw a graph showing Mary’s profits as a function of days.
9. Fill in the following table. Then, based on your table, draw graphs of the functions and .
Now describe in words what happened...
a. How did adding to the function change the graph?
b. How did subtracting from the function change the graph?
c. How did adding three to change the graph?
d. How did doubling the function change the graph?
e. How did multiplying the graph by change the graph?
f. By looking at your graphs, estimate the point of intersection of the graphs and . What does this point represent?
Name: __________________
Horizontal and Vertical Permutations
1.Standing at the edge of the Bottomless Pit of Despair, you kick a rock off the ledge and it falls into the pit. The height of the rock is given by the function , where is the time since you dropped the rock, and is the height of the rock.
a. Fill in the following table.
b. . What does that tell us about the rock?
c. All the other heights are negative: what does that tell us about the rock?
d. Graph the function . Be sure to carefully label your axes!
2. Another rock was dropped at the exact same time as the first rock; but instead of being kicked from the ground, it was dropped from your hand, up. So, as they fall, the second rock is always three feet higher than the first rock.
a. Fill in the following table for the second rock.
b. Graph the function for the new rock. Be sure to carefully label your axes!
c. How does this new function compare to the old one? That is, if you put them side by side, what change would you see?
d. The original function was . What is the new function?
(*make sure the function you write actually generates the points in your table!)
e. Does this represent a horizontal permutation or a vertical permutation?
f. Write a generalization based on this example, of the form: when you do such-and-such to a function, the graph changes in such-and-such a way.
3. A third rock was dropped from the exact same place as the first rock (kicked off the ledge), but it was dropped seconds later, and began its fall (at ) at that time.
a. Fill in the following table for the third rock.
b. Graph the function for the new rock. Be sure to carefully label your axes!
c. How does this new function compare to the original one? That is, if you put them side by side, what change would you see?
d. The original function was . What is the new function?
(*make sure the function you write actually generates the points in your table!)
e. Does this represent a horizontal permutation or a vertical permutation?
f. Write a generalization based on this example, of the form: when you do such-and-such to a function, the graph changes in such-and-such a way.
Name: __________________
Homework: Horizontal and Vertical Permutations
1. In a certain magical bank, your money doubles every year. So if you start with , your money is represented by the function , where is the time (in years) your money has been in the bank, and is the amount of money (in dollars) you have.
Don puts into the bank at the very beginning ().
Susan also puts into the bank when . However, she also has a secret stash of under her mattress at home. Of course, her stash doesn’t grow: so at any given time , she has the same amount of money that Don has, plus more.
Cheryl, like Don, starts with . But during the first year, she hides it under her mattress. After a year () she puts it into the bank, where it starts to accrue interest.
a. Fill in the following table to show how much money each person has.
Don | ||||
Susan | ||||
Cheryl |
b. Graph each person’s money as a function of time.
c. Below each graph, write the function that gives this person’s money as a function of time. Be sure your function correctly generates the points you gave above! (*For Cheryl, your function will not accurately represent her money between and , but it should accurately represent it thereafter.)
2. The function is defined on the domain as shown below.
a. What is (That is, what does this function give you if you give it a -)
b. What is
c. What is
d. The function has three zeros. What are they?
The function is defined by the equation: . That is to say, for any -value you put into , it first puts that value into , and then it subtracts from the answer.
e. What is
f. What is
g. What is
h. Draw next to the drawing above.
The function is defined by the equation: . That is to say, for any value you put into , it first adds to that value, and then it puts the new value into .
i. What is
j. What is
k. What is
l. Draw next to the drawing to the right.
m. Which of the two permutations above changed the domain of the function?
3. On your calculator, graph the function . Graph it in a window with going from to , and going from to .
a. Copy the graph below. Note the three zeros at and .
b. For what values is the function less than zero? (Or, to put it another way: solve the inequality .)
c. Construct a function that looks exactly like this function, but moved up . Graph your new function on the calculator (as , so you can see the two functions together). When you have a function that works, write your new function below.
d. Construct a function that looks exactly like the original function, but moved to the left. When you have a function that works, write your new function below.
e. Construct a function that looks exactly like the original function, but moved down and to the right. When you have a function that works, write your new function below.
Name: __________________
Sample Test: Functions I
1. Chris is years younger than his brother David. Let represent David’s age, and represent Chris’s age.
a. If Chris is fifteen years old, how old is David? ______
b. Write a function to show how to find David’s age, given Chris’s age. _____
2. Sally slips into a broom closet, waves her magic wand, and emerges as...the candy bar fairy! Flying through the window of the classroom, she gives every student two candy bars. Then five candy bars float through the air and land on the teacher’s desk. And, as quickly as she appeared, Sally is gone to do more good in the world.
Let represent the number of students in the class, and represent the total number of candy bars distributed. Two for each student, and five for the teacher.
a. Write a function to show how many candy bars Sally gave out, as a function of the number of students. _____
b. Use that function to answer the question: if there were students in the classroom, how many candy bars were distributed? First represent the question in functional notation—then answer it. __________
c. Now use the same function to answer the question: if Sally distributed candy bars, how many students were in the class? First represent the question in functional notation—then answer it. __________
3. The function is “Subtract three, then take the square root.”
a. Express this function algebraically, instead of in words: ________________
b. Give any three points that could be generated by this function: _____________________
c. What values are in the domain of this function? _____________________
4. The function is “Given any number, return .”
a. Express this function algebraically, instead of in words: _______________
b. Give any three points that could be generated by this function: _____________________
c. What values are in the domain of this function? _____________________
5.
a.
b.
c.
d.
e.
f.
6. Of the following sets of numbers, indicate which ones could possibly have been generated by a function. All I need is a “Yes” or “No”—you don’t have to tell me the function! (But, you may do so if you would like to...)
a.
b.
c.
d.
7. Make up a function involving music.
a. Write the scenario. Your description should clearly tell me—in words—how one value depends on another.
b. Name, and clearly describe, two variables. Indicate which is dependent and which is independent.
c. Write a function showing how the dependent variable depends on the independent variable. If you were explicit enough in parts (a) and (b), I should be able to predict your answer to part (c) before I read it.
d. Choose a sample number to show how your functio