Rational functions…..
Get three forms of the rational function
P(X)/Q(X)
Multiplied out (2x^2-8X+8)/(x^2-5x+6)
Factored (2(x-2)^2)/((x-2)(x-3))
Reduced (2(x-2))/ ((x-3))
1. Factor, if possible *
2. Domain
* in numerical order (123)
*
3. Reduce, if possible *
4. X- intercepts If any
5. Y- intercepts, if it exists
(numerator of the reduced form)
(multiplied out form)
6. Vertical Asymptotes (if any) X = (a number)
(denominator of the reduced form)
7. Horizontal asymptote (if it exists) Y = (a number) (multiplied out form)
8. Sign chart (the sign of the function) (look at the reduced form)
9. Graph it!
456&7 in any order
Asymptotes: A line that guides the shape of a graph
3 types:
Vertical asymptotes (a line the graph will not touch) will always be X= (a number) (it is the
changed sign of the factored denominator) Denominator OF THE REDUCED FORM is(X-
2)Vertical asymptotes are X=2, Denominator (reduced form) (X-3) the V. asymptote is X=3
D= (X+7) V asymptote is X=-7
Horizontal asymptote Will always be Y= (a number) Determine the degree of the numerator
and the degree of the denominator 1 if the degree of the numerator is larger the degree of the
denominator, there is no horizontal asymptote…… if the degree of the denominator is larger than
the degree of the numerator, then the horizontal asymptote is Y=0…….. if the degrees are equal,
then the horizontal asymptote is Y=(a/b) where a is the leading coefficient of the numerator, and
b is the leading coefficient of the denominator.
Degree 7 / degree 9 (9>7) or (Denominator > Numerator) means Y=0
degree 4 / degree 2 (4>2) or (numerator>denominator) means there is no horizontal asymptote
Degree 2 / degree 2 (2=2) or (numerator = denominator) meanslook at the co efficients and
divide them. Coefficient 2 / coefficient 3 means horizontal asymptote is Y = 2/3
Oblique asymptotes (not doing those)
Sign chart a number line. Mark the X-intercepts, and and the number(s) not in the domain.
We’ve been working with the horizontal asymptote, it describes the end behavior.
F(X)=(X^4-1)/(X^2-4) (step one, factor) ((X^2+1)(X+1)(X-1))/((X+2)(X-2))
Denominator cannot equal 0, so {xIx=/2,-2}
This form is also reduced (cannot be reduced.)
X-intercepts are…. 1,-1
Y-intercept is….. ¼
Vertical asymptote X=2, X=-2 (Places the line cannot touch)
Horizontal Asymptotes…. None numerator degree>denominator degree means there is not one
Sign chart (x-int. and non-domain X’s) + (-2) - (-1) + (1) - (2) +
Section 5.6 polynomial and rational inequalities.
Step 1 Make one side = 0 X^2-5x (</=) -6
X^2-5x+6 (</=) 0
Step 2 factor completely the other side (X-2)(X-3) (</=) 0
Step 3 create a sign chart and mark the numbers that are associated with all of the factors.
+++2---3+++
(we want the left side to be negative or equal to 0, the answer is the interval [2,3]
Example 2 (X-2)(X-3)>0 +++2---3+++ (-inf.,2)U(3,inf.)
Example 3
X^4 +x-16>X
X^4-16>0
(X^2-4)(X^2+4)>0
(X+2)(X-2)(X^2+4)>0
+++(-2)---(2)+++
(-inf.,-2)U(2,inf.)
Example4
X^4-3x^3 < 4X^2
X^4-3X^3-4x^2<0
X^2(X^2-3x-4)<0
X^2(X-4)(X+1)<0
+++(-1)---(0)---(4)+++
(-1,0)U(0,4)
RATIONAL INEQUALITIES
4/14/2014
8.1-8.2
Solving systems of equations. What a system looks like:
Two types of tickets: adults and kids. Two adult and two kids = $12 ….. three adults and four
kids = $19
HOW MUCH IS EACH TYPE OF TICKET?
Let x be the cost of an adult ticket
Let Y be the cost of a kids ticket
2x+2y=12
3x+4y=19
These are linear. Y=-x+6………Y=-3/4 x + 19/4+
Graph them both. The lines show the possible solutions. The intersection is the solution for the
whole system.
We are NOT going to use this as a solution technique fir the test.
8.1 systems of linear equations two types of slutions
-substitution
-elimination
eliminate one of the variables
SUBSTITUTION TECHNIQUE - Step 1: choose an equation & choose a variable, then solve for
that variable …….. x=-y+6
Step 2 in the OTHER equation, substitute the expression = to the variable that you solved for in
step 1 3(-y+6)+4y=19…… solve this one: Y=1
Step 3 in any original equation, plug in the value you just got from step 2 and solve for the other
variable.
2x+2(1)=12 then solve…… x=5
Answer the word problem with words. Answer the number problem with numbers.
Final number answer X=5 Y=1 (circled)
ELIMINATION: step 1: choose a variable. (we choose X) look at the variable in both equations.
Multiply 1 or both equations by a non-zero constant(s) so that the coefficients of the chosen
variable are either the same or opposites.
-3(2x+2y=12) becomes -6x-6y=-36
2(3x+4y=19) becomes 6x+8y=38
Step 2: either add or subtract these equations to create a new equation where the chosen variable
is eliminated.
Now add them together. (-6x-6y=-36)+(6x+8y=38) = 2y=2 then y=1
New example….
6x-8y=24….. becomes the same
-3x+4y=-15….(2) …. Becomes -6x+8y=-30
Using elimination
Add the new equations. 0=-6…… NO SOLUTION
NEW EXAMPLE
-6x-10y=-40
3x-5y=20…….(-2).. becomes -6x-10y=-40
They are THE SAME. A graph of the equation y=3/5X-4 is the answer (X is ANY real number)
X is all real numbers
8.2 matrix
A matrix is a rectangular array of numbers. For example.
2 1 2
0 3 3
-5 4 6
matrix size is 5X3
7 7 7
2 1 5
GET A GRAPHING CALCULATOR
A matrix can be used to represent a system of equations
Every row of the matrix corresponds to an equation in the system.
Every column corresponds to either a variable, or a constant.
(Baseball ticket price example) size will be 2X3
2 2 12
3 4 19
4/16/2014
Using the matrix process to solve systems of equations
8.2 Systems of linear equations: Matrices
Example of the matrix for the system
2x+2y=12
3x+4y=19
2
3
2
4
12
19
RREF: reduced Row Echelon Form
Step1: translate the system of equations to a matrix (shown above)
Use calculator, open the matrix function. (you’ll see “names” “Math” and “Edit”)
Step 2: go to “edit” hit enter. Enter the desired dimentions (2X3 for the example.)
Once done, hit “quit” (2nd function for mode) DO NOT CLEAR OR HIT ANYTHING ELSE
Step 3 Go back to the MATRIX menu, go right to “math”, and scroll down to rref (be careful
not to use “ref” instead)
Calculator should show rref(
Hit the matrix key again, chooses the matrix you made, and close the parenthases. You should
see rref( [A] )
Hit enter and the solution should be
1
0
Example 2
Solve using a matrix
0
1
5
1
2x+5y=17
3x+7y=-4
Make the matrix
2
3
Quit
Matrix - Math
5
7
- rref(
- matrix
17
-4
- Choose your matrix (I used [B])
Calc should say rref( [B] )
Hit enter, solution should show
1
0
0
1
That translates to X=-139, Y=59
-139
59
If there is a single solution, then it should look like
1 (the X)
0 (the X)
Which means X= A number
0 (the Y)
1 (the Y)
And Y= A number
A number
A number
Example of a no solution answer:
Number
Number
0
0
The bottom here says 0=2, which means there is no solution
Number
2
An example for infinitely many solutions
1
0
-2/3
0
2
0
Bottom row is 0=0, that means there are infinitely many solutions.
The first row translates to X+ (-2/3)Y=2 the correct answer to this is all of the below
(((Infinitely many solutions, X=(2/3)Y+2, Y is any real number)))
LAST CHAPTER 6.1 Composition
Notation (F (circle) G)(X) = F(G(X))
Imagine F as the midpoint in an assembly line
X -> [F] -> F(X)
In this chapter, we will be combining functions
(F(circle)G)(X) = F(G(X)) start with G(X), then work your way out to F(G(X))
For example. G(X) = X^2 and F(X) = 2x+1. So F(G(X)) is
2(X^2)+1
Be careful with the order of composition…… F(G(X)) is usually not equal to G(F(X))
F(G(X)) = 2X^2+2
G(F(X)) = 4x^2+4x+1
Always be careful of the order of the functions.
6.2 – Inverse functions
Back to basics…..
Y is a function of X
means
for every X value, there can be ONLY 1 Y value.
Horizontal line test: if no horizontal line cresses the graph of a function at more than one point,
then the function is 1 to 1
G(X)=3x-2 is a 1 to 1 function.
If F is a 1 to 1 function, then F-inverse is the inverse function. (it looks like F-1 but is not an
exponent)
An example of a function VS an inverse functions is
The function is you wrapping a gift, adding ribbon and bow (in that order)
The inverse function is your friend taking off the bow, then ribbon, then the wrapping (in that
order)
4/21/2014
If F is a 1 to 1 (1-1) function (where every X-value, there is only 1 Y-value AND VICEVERSA) the function is an inverse function (F inverse, seen as F-1 but is not an exponent)
4 steps to finding the inverse function……
1: replace the function notation (F(X)) with Y …. Example. F(X)=X3+8 becomes Y= X3+8
2: replaces all of the X’s with Y and all of the Y’s with X’s ….. Y= X3+8 becomes X=Y3+8
3: solve for Y ….. ∛(x-8) = Y
4: write the answer using the inverse notation…. F-1(X)= ∛(x-8)
verify the functions are inverse.
(H(circle)H-1)(X)=H(H-1(X))
H(X)= 2/(x-5)
H-1(X)=(2/X) + 5
Begin with (H(circle)H-1)(X)=H(H-1(X))
=H((2/x) + 5)
Side note, to verify the inverse of a function, graph the function, and any point on the graph
reversed will be a point on the inverse function. F(X)= 2x+8 intercepts are (-4,0),(0,8)
It’s inverse is (X/2)-4 and it’s intercepts are (0,-4),(8,0) … note the common differences
here.
6.3 Exponential functions
F(X) = Ax
(“A is a real number, A>0, A=/=1) or (0,1)U(1,inf.)
If a>1 then the graph would look like
If X is between 0 and 1 (0<a<1)then the graph looks like
Domain: (-inf,inf.)
Range( 0,inf.)
X-int. none
Y intercept = 1
Horizontal asymptote is Y=0
Reference points will be (0,1),(1,a),(-1,(1/a))
In an example of function conversions, a change up (Y-value) will change the Y-asymptote
G(X)= (1/3)x-3+2 the graph will move up 2 so the vertical asymptote will change from y=0 to
y=2
4/23/2014
6.4 -logarithmic functions (inverses of the exponential functions)
F(X)=logax (log base a of x)
Example.
F(x) = 2x
F(3) = 8
The inverse is f-1(x)= log2x
Rule of a : (a>0, a=/=1)
Log2(8)=3
Log10 10,000 = 4
(think of it as 104)
Two types of functions depending on a
(a>1) and (0<a<1)
Logarithmic stuff
Domain (0, inf.)
Range (-inf, inf.)
X-int 1
Y-int none
Vertical Asymptote X=0
Ref. points (1,0),(a,1),(1/a,-1)
Example1. F(x)=log6x Ref points are (1/6,-1),(1,0),(6,1)
Example2 g(x)=log7(x+2)+3 Ref poins are (1/7,-1),(1,0),(7,1) Asymptote X=0
Transformations: up 3. (1/7,2),(1,3),(7,4)
Right 2 new asymptote X=2. Ref points: (15/7,2),(3,3),(9,4)
Example 3 h(x)=-log1/4x
Asymptote x=0, points (1/4,1),(1,0),(4,-1)
Transformations: vertical reflection. Asymptote X=0, points (1/4,-1),(1,0),(4,1)