1.10 Homework: horizontal and vertical permutations ii

1. In a certain magical bank, your money doubles every year. So if you start with $1, your money is represented by the function $M=2^t$ , where $t$ is the time (in years) your money has been in the bank, and M is the amount of money (in dollars) you have.
   Don puts $1 into the bank at the very beginning ( $t=0$ ).
   Susan also puts $1 into the bank when $t=0$ . However, she also has a secret stash of $2 under her mattress at home. Of course, her $2 stash doesn’t grow: so at any given time t, she has the same amount of money that Don has, plus $2 more.
   Cheryl, like Don, starts with $1. But during the first year, she hides it under her mattress. After a year ( $t=1$ ) she puts it into the bank, where it starts to accrue interest.

1. Fill in the following table to show how much money each person has.

1. Graph each person’s money as a function of time.

1. 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 $t=0$ and $t=1$ , but it should accurately represent it thereafter.)

1. The function $y=f\left(x\right)$ is defined on the domain [–4,4] as shown below.

1. What is $f\left(\mathrm{–2}\right)$ ? (That is, what does this function give you if you give it a -2?)
2. What is $f\left(0\right)$ ?
3. What is $f\left(3\right)$ ?
4. The function has three zeros. What are they?

The function $g\left(x\right)$ is defined by the equation: $g\left(x\right)=f\left(x\right)\mathrm{–1}$ . That is to say, for any x -value you put into $g\left(x\right)$ , it first puts that value into $f\left(x\right)$ , and then it subtracts 1 from the answer.

1. What is $g\left(\mathrm{–2}\right)$ ?
2. What is $g\left(0\right)$ ?
3. What is $g\left(3\right)$ ?
4. Draw $y=g\left(x\right)$ next to the $f\left(x\right)$ drawing above.

The function $h\left(x\right)$ is defined by the equation: $h\left(x\right)=f(x+1)$ . That is to say, for any x -value you put into $h\left(x\right)$ , it first adds 1 to that value, and then it puts the new x -value into $f\left(x\right)$ .

1. What is $h\left(\mathrm{–3}\right)$ ?
2. What is $h\left(\mathrm{–1}\right)$ ?
3. What is $h\left(2\right)$ ?
4. Draw $y=h\left(x\right)$ next to the $f\left(x\right)$ drawing to the right.

1. Which of the two permutations above changed the domain of the function?

1. On your calculator, graph the function $Y1=x^3–13x–12$ . Graph it in a window with x going from –5 to 5, and y going from –30 to 30.

1. Copy the graph below. Note the three zeros at $x=\mathrm{–3}$ , $x=\mathrm{–1}$ , and $x=4$ .

1. For what x-values is the function less than zero? (Or, to put it another way: solve the inequality $x^3–13x–12<0$ .)
2. Construct a function that looks exactly like this function, but moved up 10. Graph your new function on the calculator (as Y2, so you can see the two functions together). When you have a function that works, write your new function below.
3. Construct a function that looks exactly like the original function, but moved 2 units to the left. When you have a function that works, write your new function below.
4. Construct a function that looks exactly like the original function, but moved down 3 and 1 unit to the right. When you have a function that works, write your new function below.