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Algebra 2
EXPONENT RULES
11/10/08
Product of Powers
X3 ● X5 = X3+5 = X8
34 ● 35 = 34+5 = 39
Quotients of Powers
X8 = X8-3 = X3
X3
49 = 49-3 = 46
43
Power of a Power
(X4) 6 = X4●6 = X24
(53) 4 = 53●4 = 512 = 244140625
(XY) 6 = X6 Y6
(3X) 7 = 37 X7
Power of a Quotient
b
b8
7
( )8 =
( )4
a
a8
3
Rational Exponents
Simplify: get the terms down to less complicated.
Evaluate: solve, get the answer.
a0 =1 20 = 1
Any number to the zero power is equal to 1.
a =a
Negative exponents
X-3 = 1/X3
3-4 = 1/34
1
2=2
1
11/14/08
Operations with Functions
Function: Relationship between two variables.
Relation: For every X you will have a Y.
Domain (X) Range (Y)
Are these functions?
Domain Range
Domain Range
X
Y
X
Y
2
3
3
2
3
4
5
3
4
8
6
3
5
10
6
5
6
12
7
6
Function
Not a function b/c X repeats.
Vertical Line Test : Use to determine if you have a function or not.
If you draw a vertical line that crosses the graph at
two points then you do not have a function.
This isn’t a function.
This IS a function.
FUNCTION NOTATION: Helps you know the value of your function.
Y = 3X + 5
Y = 7X + 10
f(x) = 3X +5
g(X) = 7X + 10
f(6) = 3(6) +5 g(6) = 7(6) + 10
f(6) = 18 + 5 = 23
11/14/08
g)6) = 42 + 10 = 52
Addition & Subtraction
f(X) + g(X)
f(X) = 5X² -2X +3
Add vertically
g(X) = 4X²+ 7X -5
5X² -2X +3
+ 4X²+ 7X -5
9X² +2X -2
Subtract vertically
Note: By subtracting the – changes all signs.
5X² -2X +3
5X² -2X +3
-(4X²+ 7X -5) becomes
-4X² -7X + 5
X² -9X + 8
Example 1:
Add:
f(X) = 5X³ + 6X² -7X +12
g(X) = 7X³
+ 5
12X³ + 6X² -7X + 17
Subtract:
5X³ + 6X² -7X +12
- 7X³
- 5
- 2X³ +6X² -7X +7
Multiplication/Division
Example 1:
f(X) = 5X²
g(X) = 3X -1
(5X²)(3X-1) = (5●3)(X²●X1)(5X²● -1) = 15X³- 5X²
Example 2:
f(X)= 6X4 -3X³-2X -4
g(X) = 5X²+7X +8
(5X²+7X +8)( 6X4 -3X³-2X -4)
Take each term in the first function and
multiply everything in the second function.
5X²(6X4 )+5X²(-3X³)+5X²(-2X)+ 5X²(-4)= 30X6-15X5
7X(6X4 )+7X(-3X³)+7X(-2X)+7X(-4)
=
-10X³-20X²
42X5-21X4
8(6X4)+8(-3X³)+8(-2X)+8(-4) =
Answer:
-14X²-28X
48X4-24X³
-16X -32
6
5
4
30X +27X +27X -34X³-34X2 - 44X -32
Division
f ( x) 3 x 2  3x 3  2 x  4

 .5  .25  .4  .90  .25  .65
g ( x)
6 x 2  16 x  10
f ( x) 6 x 4  3x 3  2 x  4

 1.2 x 2  .43x 2  2 x  .5
2
g ( x)
5x  7 x  8
11/18/08
Solving Systems of Equations by Graphing or Substitution
y  x  3

 y  3 x  5
Y = mx + b
Y = dependent
X = independent
Remember m = slope
b = y-intercept
Y-intercept is where the line crosses the Y axis.
For the first equation, Y=x-3, the line should look like this:
0
-0.5 0
1
2
3
The line crosses the y-axis
4
at -3.
-1
-1.5
-2
-2.5
-3
-3.5
Series1
11/21/08
The solution of a system is where the two graphs
of the equations intersect.
This graph has only one solution.
It is CONSISTENT and INDEPENDENT.
These are parallel lines. THEY DO NOT
INTERSECT. No solution.
It is INCONSISTENT.
Two equations with the same line.
Multiple solutions.
It is CONSISTENT and DEPENDENT.
Example 1: Graph the system
x  y  5

 x  5 y  7
Make a table for both equations.
X+y = 5
x-5y = 7
X Y
X Y
5
2
3
2
1.4
4
4
1
4
2.2
3
X+Y=5
2
X-5Y=-7
1
0
The graph has one solution.
1
2
3
4
(3,2) It is consistent and independent.
Solving with substitution
x  y  5

 x  5 y  7
1)
First take the first equation and solve for X
X + Y =5
-Y
-Y
X = 5-Y Replace this value for X in second equation and solve for Y.
2) X-5Y=-7
5-Y -5Y =-7
5 – 6Y = -7
-5
-5
-6Y = -12
-6 -6
Y = 2 Take your value for Y, 2 and replace it in the first equation
and solve for x.
3) X + Y = 5
X+2=5
-2 -2
X=3
Solution of the system is (3,2)
Example 2:
2X + Y = 3
3X – 2Y = 8
1) 2X + Y =3
-2X
-2x
Y = -2X +3
2) 3X – 2(-2X + 3) = 8
3X + 4X – 6 = 8
7X -6 = 8
+ 6 +6
7X = 14
7
7
X=2
3) 2(2) + Y = 3
4+Y=3
-4
-4
Y = -1
Solution (2,-1)
11/25/08
2x+5y=15
To solve this you want to be able to cancel out one of the
-4x+7y=-13
variables. Find which one you could cancel by multiplying one
of the equations.
2(2x+5y)=15(2)
Multiply both sides by 2, now you can cancel out
4x+10y = 30
the x.
-4x+7y = -13
17y = 17
17
17
Y=1
Now put your value for y into the first equation and solve for X.
2x+5(1) = 15
2x +5 = 15
-5 -5
2x
= 10
2
2
X=5
Solution for this system is (5,1)
Example 1:
3x -7y =8
5x -6y =10