given zero, find other zeros. parabola writing equations given zeros inequalities write equation...
TRANSCRIPT
Given zero, find other zeros.
Parabola
Writing Equations given zeros
Inequalities
Write Equation Given a Sketch
Word Problem
Intermediate Value Theorem \ Bounds
Rational Root Theorem
Polynomial Information
Complex Numbers
Rational Functions\Asymptotes
Click on buttons to go to a topic.
Not all topics are covered here, but this should help you study.
Click home buttons on the bottom right of each page to come back to this screen.
If there is an error or a question, please notify me by e-mail or AIM.
Given zeros, find other zeros.
i
xxxx
21 is zero one given that
3073f(x) of zeros theFind 234
30713 1 21 i
1) Use synthetic to find the depressed equation given a zero.
2) If first zero was complex, use the conjugate to find the depressed equation again.
3) Reduce until you reach a quadratic, then factor or use quadratic formula to find the other zeros.
1
i21i22
i
iii
ii
26
4422
)22)(21(2
i26
i25
i
iii
ii
121
41025
)25)(21(2
i121
i126
30
2412126
)126)(21(2iii
ii
30
0
i21
1i21
1i21
6i126
01 1 6
xx20 )2)(3(0 xx
23 xx ix 21
)21)(21)(2)(3()(
))21())(21()(2)(3()(
ixixxxxf
ixixxxxf
FormFactored
Parabolas
542)( 2 xxxf
1) Write the equation of the formula in vertex form using completing the square!
2) State the vertex and the axis of symmetry
3) Which way does it open and why?
4) State the intercepts
5) Describe the transformation and graph, state all key points
6) State the range
7) State the intervals of increase and decrease
5)2(2)( 2 xxxf1
2
2
12
2
2
1 2
7)1(2)( 2 xxf
You only factor the x2 and x term
Remember to balance the equation.
khxaxf 2)()(
hxsymmetryofAxis
khVertex
:
),(:
Remember the ‘x =‘ for axis of symmetry!
1:
)7,1(:
xsymmetryofAxis
Vertex
02
,2
)(0
)(0
abecausedownopensThis
frownNegativeDownOpensa
smilepositiveUpOpensa
1:
)7,1(
xAos
V
downOpens
)5,0(
int
5)0(
5)0(4-2(0)f(0)
form) (standardintercept -y2
y
f
x – intercept: y = 0
y – intercept: x = 0
2
141
4
1424
4
564
)2(2
)5)(2(4)4()4(
Formula) Quadraticor (Factor intercept -x
2
x
x
x
0,
2
141
(0,5)
units 7 Shift Up
axis-Reflect x
2StretchVertical
unit1leftShift
Remember, range is y-values, and include the vertex.
,7](-
Intervals of increase and decrease, use x-values for interval notation.
Use parenthesis.
),1(:
)1,(:
Dec
Inc
Word Problem
:find sold, units is x and revenue isr where
603
1-f(x)
:function the
by given isequation revenue theIf
2 xx
revenue. maximum find and revenue,
maximumfor sold bemust unitsmany How
Explain. possible?even thisIs
$10,000. of revenue aearn tosold bemust
unitsmany how show oequation tan upSet
square. thecompletenot Do
for these. 2
,2a
b- Usevertex.
by the foundboth are revenuemax
and revenuemax earn tosold Units
a
bf
90
31-
2
60-
2700
54002700
)90(60)90(3
1 2
revenuein
soldunits
2700$
90
xx 603
110000 2
values.possible of range in thenot
isIt $10,000.earn toimpossible
sit' $2700, is revenuemax if No,
n.informatio of pieces various
other find tohow and for those represents
vertex what theknow and book, the
in at thoseLook also. problem fence aor
objects falling a be couldit mind,in Keep
Writing equations given zeros
2i3 :zero
2 :tymultiplici 2 :zero
:ninformatio with this
polynomial a ofequation possible a Write
zero a also is conjugate the
thatmeans also zerocomplex a And
repeats. zero
themeansty multiplici Remember,
))23())(23()(2)(2()( ixixxxxf
)136)(44()( 22 xxxxxf
)23)(23)(2)(2()( ixixxxxf
52764110)( 234 xxxxxf
136
46269323
)23)(23(
2
22
xx
iiixixixxx
ixix
x -3 -2i
x x2 -3x -2ix
-3 -3x 9 6i
2i 2ix -6i -4i2
x2 -4x 4
x2 x4 -4x3 4x2
-6x -6x3 24x2 -24x
13 13x2 -52x 52Distribute Work Box Work
52764110
5224452244136
)136)(44(
234
223234
22
xxxx
xxxxxxxx
xxxx
Clear
Clear
Intermediate value theorem, bounds.
Intermediate value theorem: Given a continuous function in the interval [a,b], if f(a) and f(b) are of different signs, then there is at least one zero between a and b.
]6,5[)
]5,4[)
]4,3[)
35)(
Explain. exist? zero a does
intervalin what ,continuous is f(x)Given
2
c
b
a
xxxf
f(3) = -9
f(4) = -7
f(5) = -3
f(6) = 3
There is a zero in the interval [5,6] because there is a sign change, and by intermediate value theorem, a zero must exist in that interval.
Regarding bounds, just understand the application of the formula and what it means. Remember, in using the formula, you don’t use the leading coefficient.
Bounds. Let f denote a polynomial function whose leading coefficient is 1. A bound M on the zeros of f
is the smaller of the two numbers:
Max{1, |a0| + |a1| + .. + |an-1|}, 1 + Max{|a0|, |a1|, .. |an-1|}
4
1
6
1
2
1)( 23 xxxxf
1
12
11,1
4
1
6
1
2
1,1
Max
Max
2
32
11
4
1,
6
1,
2
11
Max
11 andbyboundedisx
Inequalities 1) Move Polynomial so that f is on the left side, and zero is on the right side. Write as a single quotient (Common denominator)
2) Determine the numbers where f equals zero or is undefined.
3) Use those values to separate the real number line. (open and closed)
4) Select a number in each interval and evaluate.
1) If f(x) > 0, all x’s in interval are greater than zero.
2) If f(x) < 0, all x’s in interval are less than zero.
1
2
1
4
xx
1
2
1
4
xx0
1
1
x
x
1
1
x
x
0)1)(1(
62
0)1)(1(
2244
0)1)(1(
)1(2)1(4
xx
x
xx
xx
xx
xx
1,1
3
xxatUndefined
xatZero
-3 -1 1
Closed, equals to and it’s a zero.
Open, even though it’s equals to, it’s undefined there, so x can’t equal 1 or -1.
-4 -2 0 2
3
10)2(;6)0(
3
2)2(;
15
2)4(
ff
ff
– – + +
We want greater than or equal to zero, so use the POSITIVE intervals and use open and closed circles to decide [ ] or ( )
[ -3, -1) U (1, ∞)
Write Equation given a sketch Remember:
Cross is odd multiplicity.
Touch is even multiplicity.
-4 -2 1 4
)4)(1()2)(4()( 2 xxxxxf
22 )3)(1)(1()3()( xxxxxf
Rational Root Theorem 1) If all coefficients are integers, list all possible combinations. Remember, p are all the factors of the constant, q are all the factors of the leading coefficient.
2) Test each possible zero until you find a factor. A factor HAS A REMAINDER OF ZERO!
3) Repeat the process until you get a quadratic or a factorable equation.
4) Find other zeros by quadratic formula or factoring.
Note: If all coefficients are not integers, you cannot use rational root theorem. In this case, it will most likely be a calculator problem where you will estimate and use those zeros.
Note 2: It’s possible for a zero to repeat, remember that.
10133)( 234 xxxxxf
10,5,2,1 q
p
1 1 -3 -1 13 -101 -2 -3 10
1 -2 -3 10 0
-1 1 -2 -3 10-1 3 0
1 -3 0 10
2 1 -2 -3 102 0 -6
1 0 -3 4
-2 1 -2 -3 10-2 8 -10
1 -4 5 0
0542 xx
)1(2
)5)(1(4)4()4( 2 x
i
ix
22
24
2
44
Polynomial Information Degree, end behavior, parent function.
X and Y intercepts
Cross or Touch at x-intercepts.
Determine max\min with calculator.
Draw Graph by Hand
Range
Intervals of increase and decrease
)2()1()( 2 xxxfDegree: 3
Parent function y = x3
End behavior follows the parent function y = x3
Remember, parent function matches up with the degree.
x-intercept, y = 0. y-intercept, x = 0
x-intercept 0 = (x-1)2(x+2)
(1,0) (-2,0)
y-intercept y = (0-1)2(0+2)
(0, 2)
Even multiplicity, Touch. Odd multiplicity, Cross
Touch at x = 1
Cross at x = -2
Max (-1, 4)
Min (1, 0)
Note, you can have more than one max and min. Also round to hundredths if necessary.
Max (-1, 4)
Min\x-int (1, 0)
x-int (-1, 0)
y-int (0, 2)
Notice how it touches at x = 1 and crosses at x = -2
Range, lowest y-value to highest y-value. If necessary, look at the y-values of the min and max to help determine range, and use either [ or ] to include that y-value.
(-∞, ∞)
Look at the x-values of the max and mins to help determine intervals. Use parenthesis.
Increase: (- ∞,-1) U (1, ∞)
Decrease: (-1, 1)
Rational Functions\Asymptotes
34
32)(
2
23
xx
xxxxf
)1)(3(
)1)(3()(
xx
xxxxf
1) Domain
2) Reduce Equation
3) Intercepts
4) Even\Odd
5) Holes & Vertical Asymptotes
6) Horizontal or Oblique Asymptotes.
7) Sketch with key points.
Domain: Find where the denominator equals zero before you reduce. It is possible you may have to use quadratic formula or have radicals in the denominator.
(-∞, 1) U (1 , 3) U (3, ∞)
)3(
)3()(
x
xxxR
y – intercept x = 0
x – intercept y = 0
)3(0
)3(
)3(0
int
xx
x
xx
x
0)0(
)30(
)30(0)0(
int
f
f
y
(-3, 0) (0, 0) (0, 0)34
32
3)(4)(
)(3)(2)()(
2
23
2
23
xx
xxx
xx
xxxxf
34
32
34
32)(
2
23
2
23
xx
xxx
xx
xxxxf
f(x) = f(-x) even
f(-x) = -f(x) odd
Not even
Not odd
Neither
Where the factor cancels out, there is a hole there. To find the y-value of the hole, plug in the x into the reduced equation.
After you reduce, the vertical asymptote is where the denominator equals zero. Remember to put ‘x =‘
23)-(1
3)1(1 R(1)
R.equation reduced theinto Plug
,coordinate-y thefind To 1. at x hole
a is thereout, cancels 1- xBecause
)2,1( Hole
at asymptote verticalSo 3. at x zero
equalsr denominato reducing,After
3: xVA
When the degree on bottom is bigger, it is proper, horizontal asymptote is y = 0
When the degree on top is bigger, it is improper, use long division. It will be either horizontal or oblique. REMEMBER PLACEHOLDERS!
Don’t forget ‘y =‘
6
006
34
03234
2
23
232
x
xx
xxx
xxxxx
- (- | – )
Oblique
xy 6
Complex Numbers
i
i
24
32
)2)(24(2 iii
The important thing to remember is to put these numbers back into standard form: a + bi
i2 = -1
i
i
24
24
20
162416
612482
2
ii
iii
i5
4
10
1
)2448(1 2iii
10
)10(1