Solutions to review

Answer key

Don't forget to check out Assignment 4 on your sheet for a nice overview of the kinds of problems you should be able to do.

2.3

In this lesson we talked about dividing polynomials. We used long division and synthetic division.

Long Division

x^3-3x^2-x+1+(1/x+3)
x+3/x^4+0x^3-10x^2-2x+4
-x^4+3x^3
-3x^3-9x^2
-x^2-2x
-x^2-3x
x+4
-x+3
1

Synthetic Division
This can only be used when the equation is linear and the leading term has a coefficient of one.

3/ 1 0 -10 -2 4
-3 9 3 -3

1 -3 -1 1 /1

1x^3-3x^2-x+1+(1/x+3)


We also talked about the Rational Rood Theorem

f(x)=px^n+...+q

factors of p/factors of q


Our homework was assignment 3
2.3 - 1-8, 11, 28-36 even, 39,45,48,63,96,102,103(a-f)

Chap. 2.2

Chapter 2.2 -Polynomial Functions of Higher Degree....check it out

-Assignment- 2.1 = 68(c-e), 73, 79, 80, & 85
2.2 = 1-8, 11, 22-36 even, 39, 45, 48, 63, 96, 102, 103(a-f)

-The Maximum number of intercepts for a Polynomial is the variables highest exponent.

extrema- relative minimums or maximums

Real Zeroes of Polynomial Functions-
1. x = a is a zero of the function
2. x = a is a solution of the polynomial equation f(x) = 0
3. (x-a) is a factor of the polynomial f(x)
4. (a,0) is an x-intercept of the graph of f

-Finding zeros of polynomial functions is closely related to factoring and finding x-intercepts

End Behavior
X -> 00 x -> -00
f(x) -> 00 f(x) -> 00

Example problem = 3x^5 - 7x^4 + 3x^3 - 2x + 1 = x -> 00 , f(x) -> 00

Intermediate Value Theorem - concerns the existence of real zeros of polynomial functions.
-theorem states that if (a, f(a)) and (b,f(b)) are two points on the graph of a polynomial functions such that f(a) can not = f(b), then for any number d between f(a) and f(b) there must be a number c between a and b such that f(c) = d


Helpful Video- http://www.youtube.com/watch?v=4TA5Zy8Hap8










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-Leading Coefficient Test- Test used to determine whether the graph of a polynomial eventually rises or falls

Chapter 2 Section 1 - Quadratic Functions

• A polynomial function is of the form:
• f(x)=AnX^n+An-1X^n-1+An-2X^n-2+...+A2X^2+A1X+A0
• Quadratic Formula
• http://www.youtube.com/watch?v=H_7lNT9oDzI
• The value of n must be a nonnegative integer (i.e., it must be a whole number; it is equal to zero or a positive integer)
• The coefficients are An, An-1, ... A1, A0. <-- These are real numbers
• The degree of the polynomial function is the highest value of n
• f(x)=X^2+5X-7 (second degree)
• g(x)=2X^3-8+/X[square root of X]-1 (not a polynomial)
• h(x)=X^1/2+X-6 (not a polynomial)
• i(x)=4 (zero degree)
• k(x)=2^x (exponential function)
• Degree               |Name               |Example          

                     0             Constant            f(x)=4
                     1             Linear                g(x)=5x-3
                     2             Quadratic          h(x)=3x^2-9x+8
                     3             Cubic                i(x)=x^3
                     4             Quartic              j(x)=x^4
                     5             Quintic              k(x)=x^5

• Zeros, X-intercepts, f(x)=0, roots <-- All mean the same thing
• Standard form: f(x)=ax^2+bx+c
• Vertex form: f(x)=a(x-h)^2+k
• a: stands for the shrinking or stretching of the parabola
• h: determines whether it moves left or right
• k: symbolizes if it goes up or down
• Completing the square
• f(x)=x^2-6x+5
• f(x)=(x^2-6x+__)                                            +5
• take the +5 and put it far away from the equation, it'll be used later
• to find the square use the equation ((b/2)^2)
• f(x)=(x-3)^2 --> f(x)=(x^2-6x+9)+5
• seeing as f(x) isn't a number, because you add 9 to one side, you still have to subtract it
• that's were the last digit comes in
• f(x)=(x^2-6x+9)+(-9+5)
• f(x)=(x-3)^2-4
• vertex: (3,-4)

Homework: 2.1-#1-8, 12, 17, 20, 29, 31, 35, 37, 41, 54, 61, 65, 67, 69

Section 1-5 - Inverses

The inverse of a function is defined as f^-1(x).

2 functions; F and G, are inverses iff (f(g(x)) = g(f(x))
In other words, when the input is the output, where there is no net change.

To find the inverse of a function:
Example:
f(x) = 2x+5

f^-1(x) = ?

You would perform the following:
the original function is:
f(x) = 2x+5
and since f(x) is equal to y...
y= 2x+5
then you switch x and y...
x= 2y+5
and then you solve for y...
x-5 = 2y +5 -5
x-5= 2y
(x-5) /2 = 2y /2
y= (x-5) /2
and so:
f^-1(x) = (x-5) /2

to validate, you would show that f(f^-1(x)) = f^-1(f(x))

so;

f(f^-1(x)) = f((x-5) /2)
= ((2x+5)-5) /2
= 2x /2
= x

f^-1(f(x)) = f^-1(2x+5)
= 2((x-5) /2) +5
= x

so the inverse is correct because both equations come out to x

One-to-one means that for every y value, there is only one x value

a function f is one to one iff f(a) = f(b) implies a=b

ex:
f(x) = 2x+5
f(a) =2a+5 f(b) =2b+5
-5 -5
f(a) =2a f(b)= 2b
/2 /2
a=2=b
a=b

so the function is one to one

one-to-one functions must pass the horizontal line test, which must be true to have an inverse that is a real function.

Compositions of Functions

Today we learned about composition of functions.

Another way of combining two functions is to form the composition of one with the other. For example, if f(x) = x^2 and g(x) = x + 1, the composition of f with g is:

f ( g( x ) )  =  f ( x + 1 )  =  ( x + 1 )^2



The composition of the function f with g is:

( f o g ) ( x )  =  f ( g ( x ) )

The domain of f o g is the set of all x in the domain of g such that g(x) is in the domain of f.


EXAMPLE:

f(x) = 2x+4      and       g(x) = x^2 - 7

(f o g) (x) = f (g(x))
                = 2(x^2 - 7) + 4

                = 2x^2 - 14 + 4
                = 2x^2 - 10             

Section 1-3 Shifting, Reflecting, and Stretching Graphs

Sunday, September 26, 2010

 

By: James Thomas


To understand how graphs can shift and reflect you must first know about the parent functions of the graph.

Parent Function: The most basic form of a equation for a graph.

Vertical and Horizontal Shifts

1) Vertical shift c units upward                    h(x)=f(x)+c
2) Vertical shift c units downward               h(x)=f(x)-c
3) Horizontal shift c units to the right          h(x)=f(x+c)
4) Horizontal shift c units to the right          h(x)=f(x-c)

Example:

              - g(x)=(x-2)²+3

Parent function = g(x)=x²
Horizontal Shift = -2....... Right Three (Translate the opposite of the sign)
Vertical Shift = +3...... Up Three





Reflecting Graphs

Graphs can also be reflected over the x-axis and y-axis depending on a (-) sign in the equation.

- Reflection over the x-axis                    h(x)=-f(x)
- Reflection over the y-axis                    h(x)=f(-x)

Example:



Nonrigid Transformations:

- Rigid Transformations- Horizontal and Vertical Shifts because they dont affect the graphs shape or structure, just the location.
- Nonrigid Transformations - Causes a distortion in the graph making it skinnier or fatter.

                      y=cf(x)
- Skinny Transformation = 0 < c < 1
- Fatter Transformation = c > 1


Example:

            - g(x) = 3x^{2 ,}           g(x)= 1/4x²