9.1 Homework: imaginary numbers

On the graph below, do a quick sketch of $y=x^3$ .

ppp

• A

  Draw, on your graph, all the points on the curve where $y=1$ . How many are there?

• B

  Draw, on your graph, all the points on the curve where $y=0$ . How many are there?

• C

  Draw, on your graph, all the points on the curve where $y=\mathrm{–1}$ . How many are there?

On the graph below, do a quick sketch of $y=x^2$ .

ppp

• A

  Draw, on your graph, all the points on the curve where $y=1$ . How many are there?

• B

  Draw, on your graph, all the points on the curve where $y=0$ . How many are there?

• C

  Draw, on your graph, all the points on the curve where $y=-1$ . How many are there?

Based on your sketch in exercise #2…

• A

  If $a$ is some number such that $a>0$ , how many solutions are there to the equation $x^2=a$ ?

• B

  If $a$ is some number such that $a=0$ , how many solutions are there to the equation $x^2=a$ ?

• C

  If $a$ is some number such that $a<0$ , how many solutions are there to the equation $x^2=a$ ?

• D

  If $i$ is defined by the equation $i^2=\mathrm{–1}$ , where the heck is it on the graph?

In class, we made a table of powers of $i$ , and found that there was a repeating pattern. Make that table again quickly below, to see the pattern.

Now let’s walk that table backward . Assuming the pattern keeps up as you back up, fill in the following table. (Start at the bottom.)

Did it work? Let’s figure it out. What should $i^0$ be, according to our general rules of exponents?

What should $i^{\mathrm{–1}}$ be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?

What should $i^{\mathrm{–2}}$ be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?

What should $i^{\mathrm{–3}}$ be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?

Simplify the fraction $\frac{i}{4-\mathrm{3i}}$ .