We began our in-class assignment by talking about why
x
3
=
–1
x
3
=–1 does have a solution, whereas
x
2
=
–1
x
2
=–1 does not. Let’s talk about the same thing graphically.
On the graph below, do a quick sketch of
y
=
x
3
y=
x
3
.
ppp
- a. Draw, on your graph, all the points on the curve where
y
=
1
y=1. How many are there?
- b. Draw, on your graph, all the points on the curve where
y
=
0
y=0. How many are there?
- c. Draw, on your graph, all the points on the curve where
y
=
–1
y=–1. How many are there?
On the graph below, do a quick sketch of
y
=
x
2
y=
x
2
.
ppp
- a. Draw, on your graph, all the points on the curve where
y
=
1
y=1. How many are there?
- b. Draw, on your graph, all the points on the curve where
y
=
0
y=0. How many are there?
- c. Draw, on your graph, all the points on the curve where
y
=
-1
y=-1. How many are there?
Based on your sketch in exercise #2…
- a. If
a
a is some number such that
a
>
0
a>0, how many solutions are there to the equation
x
2
=
a
x
2
=a?
- b. If
a
a is some number such that
a
=
0
a=0, how many solutions are there to the equation
x
2
=
a
x
2
=a?
- c. If
a
a is some number such that
a
<
0
a<0, how many solutions are there to the equation
x
2
=
a
x
2
=a?
- d. If
ii is defined by the equation
i
2
=
–1
i
2
=–1, where the heck is it on the graph?
OK, let’s get a bit more practice with
ii.
In class, we made a table of powers of ii, and found that there was a repeating pattern. Make that table again quickly below, to see the pattern.
Table 1
| i1i1 |
|
| i2i2 |
|
| i3i3 |
|
| i4i4 |
|
| i5i5 |
|
| i6i6 |
|
| i7i7 |
|
| i8i8 |
|
| i9i9 |
|
| i10i10 |
|
| i11i11 |
|
| i12i12 |
|
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.)
Table 2
| i-4i-4 |
|
| i-3i-3 |
|
| i-2i-2 |
|
| i-1i-1 |
|
| i0i0 |
|
Did it work? Let’s figure it out. What should
i
0
i
0
be, according to our general rules of exponents?
What should
i
–1
i
–1
be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?
What should
i
–2
i
–2
be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?
What should
i
–3
i
–3
be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?
Simplify the fraction
i4−3ii4−3i size 12{ { {i} over {4 - 3i} } } {}.
"This is the "main" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about them in […]"