Download Algebra II Chapter 5 Notes

Algebra II Chapter 5 Notes
Day 1: 5.1 Use Properties of Exponents
1. Product of powers: When the bases are the_____– add the exponents
Ex.1: 53 51  531  52  25
2. Power of a power: When have one base with ____ powers - _________ the exponents.
Ex.2: (32 )3  32 3  36  729
3. Power of a product: When two bases with _____ exponent - ______ the bases and raise
to power of the exponent.
Ex.3: 24 34  (2 3)4  64  1296
4. Negative exponent: When have a base with a ________ exponent- _____ the base and
raise to positive power. DO NOT make the answer negative.
2
1
1
Ex.4: 7 2    
49
7
5. Zero exponent: Anything raised to the _____ power is 1.
Ex.5: (89)0  1
6. Quotient of powers: When the bases are the same in a division problem -_______ the
exponents.
63
1
Ex.6: 2  63( 2)  63 2  61 
6
6
Try these:
1.  2  
3
2.
 4   4  
2
3
2
4.   
3
4 6 7
5. b b b 
3
 r 2 
3.  3   =
 s 
16m4 n 5
6.

2m1n 5
Scientific Notation: Use properties of exponents
Ex. 1:  8.5 X 107 1.2 X 103  
Ex. 2:
1.1X 103

5.5 X 108
1
Day 26:5.2 Evaluate and graph polynomial functions
Polynomial function: f ( x)  ax n  bx n1  cx n2  ...  mx1  n
Where: a  0 and the exponents are all _____ numbers, and the coefficients are all _____ numbers.
Leading coefficient: The coefficient of the term with the highest exponent.
Degree: The degree is the value of the highest exponent.
Type
Constant
Linear
Quadratic
Common Polynomial Functions
Standard Form
f ( x)  a
f ( x)  ax  b
Example
f ( x)  14
f ( x)  5 x  7
f ( x)  ax 2  bx1  c
f ( x)  2 x 2  x  9
3
Cubic
f ( x)  ax3  bx3  cx1  d
f ( x)  x 3  x 2  3x
4
Quartic
f ( x)  ax 4  bx3  cx 2  dx1  e
f ( x)  x 4  2 x  1
Degree
0
1
2
Decide whether the function is a polynomial function. If so write in standard form, state its degree,
type, and leading coefficient.
1
1. h( x)  x 4  x 2  3
2. g ( x)  7 x  3   x 2
4
3. f ( x)  5x 2  3x 1  x
4. k ( x)  x  2x  0.6 x5
Evaluate by direct substitution: Plug in value given for x into function wherever x is.
Use direct substitution to evaluate f ( x)  2 x 4  5 x3  4 x  8 when x=3.
f (3)  2(3)4  5(3)3  4(3)  8 =
Try: #1: f ( x)  x 4  2 x3  3x 2  7; x  2
#2: g ( x)  x3  5x 2  6 x  1; x  4
2
Synthetic substitution: How to evaluate a function more easily than direct substitution
Use synthetic substitution to evaluate f ( x)  2 x 4  5 x3  4 x  8 when x=3.
Step 1: Write coefficients in descending order using zeros for terms missing.
3
2
-5
0 -4 8
Step 2: Bring down first coefficient. Multiply that by x-value. Write the product under the second
coefficient. Repeat until done.
Step 3: Final sum is value of f(x) when x = 3.
Try these:
#1:
f ( x)  x 4  2 x3  3x 2  7; x  2
#2:
g ( x)  x3  5x 2  6 x  1; x  4
End Behavior : Behavior of the graph of a function as x approaches positive infinity ( ,  ) or
negative infinity ( ,  ) .It is determined by the function’s ________ and the sign of the
___________ coefficient.
End Behavior of Graphs of Polynomial Functions


Odd-degree polynomial functions have graphs with opposite behavior at each end.
Even-degree polynomial functions have graphs with the same behavior at each end.
The Leading Coefficient Test
The coefficient on the term with the largest exponent on the variable is called the leading coefficient.
 If the largest exponent is an odd number:
If the leading coefficient is
positive, the graph falls to
the left and rises to the right.
Rises
Left
If the leading coefficient is
negative, the graph rises to
the left and falls to the right.
Rises
Right
Falls
Left
Falls
Right
Odd Degree; positive leading
coefficient
Odd Degree; negative leading
coefficient
3
The Leading Coefficient Test (continued)
The coefficient on the term with the largest exponent on the variable is called the leading coefficient.
 If the largest exponent is an even number:
If the leading coefficient is
positive, the graph rises to
the left and rises to the right.
If the leading coefficient is
negative, the graph falls to
the left and falls to the right.
Falls
Right
Rises
Right
Rises
Left
Falls
Left
Even Degree; negative leading
coefficient
Even Degree; positive leading
coefficient
Graphing polynomial functions: Make a table of values and plot the points. Connect the points
into a smooth curve and check end behaviors.
Graph f ( x)   x3  x 2  3x  3
x -3 -2 -1 0
1
2
y
3
4