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## Algebra, Done!

I just finished all of algebra!

The last thing I did in algebra was conic sections.

In conics we learned how to graph ellipses, hyperbolas, circles, and parabolas.

This a line:

y = x + 5

Parabola:

y = -3x2 – 6x – 1

Circle:

(x – 2)2 + (y + 3)2 = 9

Ellipse:

(x2/6) + (y2/16) = 1

Hyperbola:

x2 – y2 = 39

A parabola is U-shaped. But it can open up or down.

An ellipse is an oval.

Hyperbola is like two parabolas.

You can tell what shape an equation is just by looking at it.

If one variable either ,x or y, is squared it is a parabola.

If both variables are squared then the graph might be a circle, an ellipse, or a hyperbola.

If the coefficients of x2 and y2 are the same sign AND equal, the equation represents a circle.

If the coefficients of x2 and y2 are the same sign BUT different, you have an ellipse.

If the coefficients of x2 and y2 have different signs(one positive and one negative), then the graph will be a hyperbola.

I did this with my dad:

Now I’m exited to start calculus and advanced geometry.

## Exponents

Today I was doing exponents.

These are their properties:

xa · xb = xa+b – Multiplying Powers

(xa)b = xab – Raising a Power to a Power

x-n =1/xn – Negative Exponents

Examples:

x3 · x7 = x3+7 = x10

(x52)2 = (x52)2 = x104

x-4 = 1/x4

32 · 33 = 35 = 243

(23)2 = 26 = 64

4-2 = 1/42 = 1/16

By | March 20th, 2011|algebra, math|0 Comments

## Imaginary Numbers

Yesterday I learned about i. We use it to describe imaginary numbers. This is the definition of i: i =  -1  Now if that is true, then the following is true: i2 =i· i = -1 i3 = i2 · i = –i i4 = i3 · i = 1 i5 = i4 · i = i i6 = i4 · i2 = -1 i7 = i4 · i3 = i3 = –i i8 = i4 · i4 = 1 i9 = i8 · i = i i10 = i8 · i2 = -1 As you can see, every 4th power of i = 1. So you can just divide i‘s exponent by 4 and the remainder will tell you the answer {1,i,-1,or –i}.

These are the questions I did yesterday. A complex number has two parts – a real part and an imaginary part. For example: 3 + 6i I’ll write another post on them later.

By | March 17th, 2011|algebra, math|1 Comment