Functions and Relations
Relation: a correspondence between two sets –
We can write as ordered pairs of the form (x,y)
Function: a relation in which each x has only one value of y associated with it.
Domain: set of all permissible value of x
Range: set of all permissible values of y ( obtained by using permissible values of x)
sets:
a)
b)
1
2
3
2
1
-1
-2
4
4
1
Relation, Function, both, neither ?
Domain:
_____________________
Relation, Function, both, neither ?
Domain:
_____________________________
Range:
Range:
_____________________
c)
_____________________________
d)
1
2
3
4
2
5
10
17
1
2
4
8
0
9
7
Relation, Function, both, neither ?
Domain:
_____________________
Relation, Function, both, neither ?
Domain:
_____________________________
Range:
Range:
_____________________
_____________________________
Graphs:
Vertical line test:
construct vertical line – if each line crosses graph only once, then we have a function
if more than one crossing point, then it is just a relation
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101
y=4x
y=|x|
a)
b)
c) y2 = x2 + 1
d) y2 = x+1
Equations:
a) y = 2x – 3
b) x2 + y2 = 4
c) y = log b x
d) xy = 3
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Graph each of the following.
a) 2x = y – 4
b) x2 = y – 2x + 1
c) f(x) = 3x
d) g(x) = log4 x
Which of these relations are also functions ?
a)
2x
y = ----------------x2 + 5x - 14
b) | y | = 2x
c) x2 + 2x + y2 = 4
d) y + 4= 0
Find the domain of
a)
y = 2x – 1
c) y =
•
b) y =
x 2 − 2x − 3
x+2
x+2
x−3
d) y =
•
x+2
103
4. Find the range of
a) y = x2 + 2
b) y = x + 2
c) circle of radius 2 with center at (2, -1 ).
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Linear Equations
An equation of the form ax + by = c is called a linear equation
ex. 2x – 3y = 4,
4x = 6x + 1 ,
x = - 2y + 4,
ex. 0x + 2y = 4
x = - 3,
y= 4
ex. 3x + 0y = 9
ex. 3x – y = 6
There are three types of lines.
horizontal, vertical, and slant lines.
A horizontal line has slope zero and because it crosses the y –axis, its equation is of the form y = b
A vertical line has an undefined slope and because it crosses the x-axis its equation is of the form x = a
Slant lines have a slope and are of the type y = mx + b
We can find the slope by using m =
y 2 − y1
rise Δy
=
or by writing an equation in the form y = mx + b
=
run Δx
x 2 − x1
ex. Find the slope of
a) y = 3 Î _________
b) x = - 3 Î _____
c) 2x – y = 3 Î ______
ex. Find the equation of the line that
a) is horizontal and passes through ( 4, 7 )
b) is vertical and passes through the point ( -1, 5).
c) passes through (-1, 4) and has slope 2.
d) is parallel to 2x + 3y = 1 and passes through ( -1, 4)
e) passes through ( 4, -1) and ( 3, 0 )
f) is perpendicular to 2x – 3y = 4 and passes through the point (2, -3 )
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Other Material: Use of Quadratic Equations
x2 + 12x – 64 = 0
Find the sum of the roots ( solutions ) . ______________
product of the roots . _____________
What about
x2 – 2x - 123 = 0
sum = _____________
product = _____________
Now try,
21x2 + 4x - 32 = 0 ==> sum = _____________
product = ______________
Notation:
if f(x) = x2 + 2x – 1,
then f(0) = ______________
•
f(2) = _____________ and f(h) = ___________
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Quadratic Functions
Quadratic Functions: The graph of a function of the form
f(x) = ax2 + bx + c or y = ax2 + bx + c is
a parabola that opens up if a > 0 , opens downward if a < 0
with vertex V ( - b/2a, f(-b/2a) )
We can find the x-intercept, the y-intercept, and a couple of points to get an idea of the graph of the function.
ex. Sketch the graph of f(x) = - 2x2 – 4x + 1
ex. Sketch the graph of g(x) = 4x2 + 2x
ex. What is the maximum value of h(x) = - 2x2 + 3 ? and where does it occur ?
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ex.
The sum of two numbers is 28. Find the two numbers so that their product is a maximum.
ex. find the minimum ( maximum) :
f(x) = ½ x2 + 4x
ex. profit: P = 16x - 0.1x2 - 100
a) at what level of production is the profit at its maximum ?
b) What is the maximum profit ?
ex. I have 150 feet of fencing. what should the dimensions of my rectangular yard be if the area enclosed is as large as
possible.
ex. 42/290: f(x) = 104.5x2 - 1501.5x + 6016 → models the death rate per year per 100,000 males, f(x) , for US men who
average x hours of sleep each night. How many hours of sleep, to the nearest tenth of an hour, corresponds to
the minimum death rate ? What is this minimum death rate, to the nearest whole number ?
ex. 57/291: f(x) = - 0.018x2 + 1.93x - 25.34 describes the miles per gallon, f(x), of a Ford Taurus
driven at x miles per hour. Suppose that you own a Ford Taurus. describe how you can use
this function to save money.
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Functional Notation:
Let f(x) = x2
and g(x) = 2x + 4
Find
a) f + g:
b) fg :
c) f/g :
d) composition of functions --f o g : ( f (g(x) ) ) =
2. Find each of the four values above for x = 1
a) (f + g )(1) = __________
b) (fg)(1) = _________________
c) ( f/g) ( 1 ) = __________
•
d) ( f o g ) ( 1 ) = ______________
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Logarithms and Exponential Functions
Exponential:
We write y = ax or f(x) = ax
ex. let f(x) = 2x ,
find f(0) = ________, f(1) = _______, f( 2) = ________ f( -1 ) = ________
f( -2) = ______
x
f(x)
0
1
2
-1
-2
-3
We get a graph for this function - is it really a function ? _____
This idea would work with any exponential function of the form f(x) = ax.
What is the graph of y = 2x ?
x
0
f(x)
1
2
-1
-2
-3
What about y = - 2x
x
f(x)
•
0
1
2
-1
•
-2
-3
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Exponential Functions:
General equation: f(x) = ax
Graph
Some examples: y = 12 x,
g(x) = 2 x + 3
Graphs of
y = a-x
x-intercept: ____________
y = - ax
y = - a –x
y-intercept = ____________
What about equations like y = 3 + 2x, what is the y-intercept ? the x-intercept ?
49/383
52/383
55/383
Also, find ( 1 + 1/m ) m as m gets larger and larger ( as m → ∞ ) . ( 1 + 1/m)m → _______
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Logarithms
Logarithm: We write y = logb x or f(x) = logb x
We say “the logarithm of x base b” to mean there is an exponent y so that by = x.
ex. Log 5 125 = y → 5y = 125 → y = ? _________
log 64 8 = y → 64 y = 8 → y = ? _________
ex. let f(x) = log 2 x find
f( 0 ) = __________ f(1) = _________ f(2) = ______________
f( 4) = ________
f( ½ ) = __________
x
f(x)
0
f( 1/8 ) = _______ What about f( - 2) = ? _________
1
2
½
¼
-1
This is the general graph for y = log b x
Examples:
. Find x so that 128 = 2x , x = _________
What about 345 = 2x, x = _____________
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Properties of Exponents and Logarithms:
1) Domain
of y = ax ==>
2) Range
of y = ax ==>
of y = log b x ==>
of y = log b x ==>
3) x-intercept
of y = ax
of y = log b x
Other Properties of Exponents and Logarithms.
1. log b xy = log bx + log b y
2. log b (x/y) = log b x - log b y
3. log b (xk ) = k log b x
Other properties
4. log b 1 = 0
5. log b b = 1
6. log b 0 = undefined
7. log b ( x) = undefined if x < 0
IF b = 10 ,
we write log 10 x = log x and call it the common logarithm
If b = e ( where is the irrational number e ),
we write log e x = ln x ---- and call it the natural logarithm
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Domain: Find the domain of y = log b ( x + 2 )
Find the domain of y = ( x2 – 2x – 3 )
examples:
81/396
84/396
Note: log x2 = 2 log x so do they represent the same thing ? In other words look at their graphs
y = log x2
and
y = 2 log x
ex. Find the solution of
log2 x -
log2 (x - 2 ) = 1
what about
log2 x -
log2 (x + 2 ) = 1 ?
log 2 x + log 2 (x – 3 ) = 2
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Other examples
ex. Find the domain of
a) y = log 3 ( 2x – 1 ) → __________________________________________
2 x + 3 → ___________________________________________
b) f(x) =
c) g(x) = log 2 ( x 2 - 2x – 8 ) → ____________________________________
d) h(x) =
x 2 − 2 x → __________________________________________
ex. Find x if
a) 2
log 2 7
= x → x = _______________________
b) log2 x + log2 (x+1 ) = 1 → x = ________________________
c) log4 165 = x → __________________
d) log x - log (2x – 1 ) = 0 → ________________
e) If log b 16 = 0.21, then find
•
logb 2 = __________
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Chapter 6. Solving Polynomial Equations
Long hand division –
ex. 12 ÷ 5 = ___________
ex. Suppose you had 17 apples that were to be evenly divided by five individuals. How much should
each one get so that nothing remains ?
ex. Find ( x2 - 4 ) ÷ ( x + 2 ) = _________________
ex. Find ( x2 + 3x - 4 ) ÷ ( x – 1 ) = _______________
ex. Find ( x2 + 2 ) ÷ ( x + 2 ) = ______________________
The remainder Thm.
Let P(x) be a polynomial with real coefficients. The remainder of P(x) ÷ ( x – r ) is the same as P(r).
ex. Find the remainder of
(x2 + 3x - 4 ) ÷ ( x – 1 ) Î _______________
( x2 + 2 ) ÷ ( x + 2 ) Î __________________
What happens when the remainder is zero ? ___________________________________
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The Factor Thm and its converse.
If (x – r) is a factor of the polynomial P(x), then r is a root of P(x) = 0
ex. x2 – 4x – 5:
we can see that x – 5 is a factor and what are the solutions of x2 – 4x – 5 = 0 ? _________
another factor ? _________
If r is a root (zero, solution of ) of P(x) = 0 , then x – r is a factor of the polynomial P(x) = 0
ex. when we solve the equation x2 – 4x = 0 we get x = ____________
find the factors.
______________
Use of the Remainder and Factor Theorems.
1) Is ( x + 1 ) a factor of ( x4 - 5x - 4 ) ? _____________________
2) Is ( x – 2 ) a factor of 3x3 - 9x – 6 ? ____________________________
3) is x = 3 a solution of the equation x3 – 6x – 9 = 0 ? Can you find all of the factors of x3 – 6x – 9 ?
4) Factor x3 + 2x + 1 by using the fact that x = -1 is a solution of the equation x3 + 2x + 1 = 0
Sometimes finding the remainder is not sufficient. Finding the quotient may be useful and in that case
the remainder thm. is not sufficient. We can use long-hand division or Synthetic Division.
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Synthetic Division
shorthand way of dividing two polynomials where the divisor is of the form x – r.
ex. (x2 + 2 ) ÷ ( x + 2 ) = ____________
ex. Find ( x4 - 5x - 4 ) ÷ ( x + 1 ) = __________
ex.
Find all of the roots ( solutions ) of x3 + 2x + 1 = 0
ex.
Find all zeros of the polynomial P(x) = x3 + 1.
A polynomial P(x) can always be written in the form anxn + an-1xn-1 + … + a2x2 + a1x + a0
example: 3x4 + x2 – 2x + 7 Î __________________
4x3 + 2x – 3 Î _________________
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The degree of a polynomial provides information as to the number of roots (solutions) the polynomial
equation will have. We can use the factor thm to arrive at the following conclusion.
Fundamental Theorem of Algebra
Let P(x) be a polynomial with real coefficients and of degree n. Then P(x) has n roots which
1) may or may not be distinctive
2) may or may not be real
ex. x2 + 4 = 0 has how many roots ? __________ and they are both ? __________
ex. 4x2 - 9 = 0
ex. x2 - 4x + 4 = 0
____________
________________
_____________
__________________
Now we find all of the roots of the equation x3 + 1 = 0 . there are _______ roots and they are
____________
Descartes’ Rule of signs:
can be used to reduce the number of possibilities(roots).
If the original polynomial P(x) has no sign variations, then it has no positive roots
If P( - x ) has no sign variations , then P(x) has no negative roots.
By itself Descartes’ Rule of signs is not very helpful but when used with the following thm. , it is useful in finding roots of
some polynomial equations.
ex. x4 + 3x2 + 2 = 0 Î Find all of the roots. How many of them are positive ? ___________
How many are negative ? ______________
ex. What about x5 - 2x - 3 = 0 → positive ? ______________
•
negative ? __________
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119
Conjugate Pairs Thm.
Let P(x) be a polynomial with real coefficients. If a + bi
is
____________.
is a solution (root) of P(x) = 0, then so
ex. x2 + 9 = 0 Î _______________
ex. x4 + 5x2 + 4 = 0 Î _____________________
Quadratic Pairs:
Let P(x) be a polynomial with rational coefficients. If a +
perfect square, then so is
_______________
is a solution of P(x) = 0 , b not a
b
ex. x2 + 6x - 5 = 0
ex. x2 - 3 = 0
Rational roots:
Let P(x) be a polynomial with rational coefficients. If r is a rational solution of the equation
P(x) = 0,
then r can be written in the form r = p/q, where p is a factor of the constant term and q is a factor
of the leading coefficient of P(x).
ex. x3 - 4x + 3 = 0
c=3
and leading coefficient is 1
ex. 2x4 + 3x2 - 5 = 0
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Additional Examples
Synthetic Division the remainder thm is useful but it does not provide a quotient.
(x2 + 2x + 1 ) ÷ ( x – 1 ) = ___________________
(x3 + x + 2 ) ÷ ( x + 2 ) = _________________
Find P( 4 ) if P(x) = 2x3 – 2x + 1 _________________
Is 4 a solution of P(x) = 0 ? Why or why not ?
Is x – 4 a factor of P(x) ? Why or why not ?
Find ( 2x3 - 2x + 1 ) ÷ ( x – 4 ) = ____________________
Is x + y factor of x3 + 2xy2 – y3 ?
Find all of the roots of x3 - 2x + 1 = 0 if x = 1 is known to be a solution.
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121
Find all of the zeros of the polynomial x3 - 2x2 + x – 2 = 0
Find all of the zeros of x4 - 2x3 + 5x2 - 8x + 4 = 0
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122
Binomial Expansion
( a + b)n = _______________
ex. ( a + b)0 = _____________
ex. ( 2x + y)1 = ____________
ex. ( x – y )2 = ______________________________
ex ( x + 2y )3 = _______________________________
ex. ( x + y)5 = _________________________________
In general we patterns that allow us to find specific terms of an expansion without having to find all of the terms.
ex. Find the first two terms of the expansion of
( 2x – 1/x )6 = ____________________________
ex. Find the last two terms of the expansion of ( x + 1/x)7 = __________________________
ex. How many terms are in the expansion of ( 3x + 2y )12 ? _______________________
ex. We can find any term along the way, say the 7th term of ( x2 + x)12 → ______________________
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Inequalities in two variables ( on the plane )
Find the solution of the following inequalities.
x + 2y > 4
2x – y < 2
x<4
Find the solution of the following system of inequalities.
x + 2y > 4
2x – y < 2
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124
2x – y < 2
x<4
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125
System of Equations
Substitution :
2x – 3y = 6
x + 5y = 3
1) decide which variable in what equation to solve for:
2) Solve for that variable in that equation:
3) Substitute in the other equation:
4) Go back and use equation from 2 to obtain the remaining part of your solution:
→ solution: (x,y) =
Another example:
3x – 6y = 3
2x + 4y = 2
•
1)
2)
3)
4)
•
126
Elimination:
x + 4y = -2
3x – 2y = 8
1) decide which variable to eliminate:
2) Get the LCM of the coefficients of the chosen variable:
3) ( Add-subtract) to eliminate variable and create a new equation without variable.
4) Solve for remaining variable.
5) Back substitute into original equations ( any one of them ) to solve for remaining variable .
Solution: (x, y )
Another example:
3x – 12y = 1
2x - 8y = 3
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127
Additional examples:
page 452:
1)
5)
17)
19)
29)
43)
47)
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128
System in three variables:
Reduce to a system in two variable by eliminating one variable and creating two new equations in
only two variables. Solve the new system of two equations and two variables.
x + 2y + 3z = 7
2x – y – 4z = -1
x + 2y – z = 5
1) eliminate ____
a) use equations: ___ and __
b) use equations: ___ and _____
2) Solve the new system
3) Final Solution:
Another example:
x + 2y – z = 5
x+ y
=3
x z = 2
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More Examples on page 481
2)
8)
14)
20)
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130
Variation: direct and inverse
We say that y varies directly as x if
there exists a constant k so that y = kx.
We say that y varies inversely as x if
there exists a constant k so that y = k/x.
ex. 32/364
ex. 34/ 364
ex. 38/364
ex. 39/364
ex. 45/ 364
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Matrices
General Notation:
1) rectangular array of numbers with rows and columns
We normally use capital letters to name the matrices.
A = [3] ,
B = [3 1 / 5] ,
2
3
4
5⎤
⎡1
⎢6
7
8
9
0 ⎥⎥
⎢
D=
, E=
⎢ − 1 10 − 2 − 3 − 4⎥
⎢
⎥
⎣− 5 − 6 − 7 − 8 − 9 ⎦
⎡3 − 2⎤
C= ⎢
⎥,
⎣5 7 ⎦
⎡4⎤
⎢7⎥
⎣⎦
2) Dimension of a matrix: m x n
We use the number of rows and columns to describe the matrix.
A is a ___________ matrix
C: __________
B is a ____________ matrix
D: _____________
E: ____________
3) elements of a matrix: aij
Look at matrix C: we can label the elements of C as follows:
Look at matrix E: we can label the elements of E as follows:
Look at matrix D: find each of the following entries (elements)
d13 = _________
d32 = _________
d42 = __________
d25 =__________
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Special Types of Matrices:
Zero Matrices:
All entries are zero
2x2 zero matrix
1x5 zero matrix
4x3 zero matrix
Square Matrix: A matrix that has the same number of rows as columns
⎡1 0 0 ⎤
⎡1 2⎤
A = [5] , B = ⎢
C = ⎢⎢0 1 0⎥⎥
⎥
⎣3 4⎦
⎢⎣0 0 1⎥⎦
Identity Matrices: diagonal entries – a11, a22, a33,.... are all = 1 while all other entries = 0
[1] ,
⎡1 0⎤
⎢0 1 ⎥ ,
⎣
⎦
⎡1 0 0⎤
⎢0 1 0 ⎥ ,
⎥
⎢
⎢⎣0 0 1⎥⎦
⎡1
⎢0
⎢
⎢0
⎢
⎣0
0
1
0
0
0
0
1
0
0⎤
0⎥⎥
, ....
0⎥
⎥
1⎦
Addition: add corresponding entries so that you end up with a matrix that resembles the original
two in size- this can
only occur if the original matrices are identical in size .
A + B is defined if A : m x n matrix, then B must also be m x n matrix.
6
[3 − 2] + ⎡⎢ ⎤⎥ = _________
⎣ − 2⎦
⎡1 − 2⎤ ⎡4 − 3⎤ ⎡ ___
⎢2 − 1⎥ + ⎢0 − 3⎥ = ⎢ ___
⎥ ⎢
⎥ ⎢
⎢
⎢⎣0 4 ⎥⎦ ⎢⎣1 2 ⎥⎦ ⎢⎣ ___
___ ⎤
___ ⎥⎥
___ ⎥⎦
Subtraction: if treat matrices as real numbers, we can use addition.
Let - A represent the opposite of matrix A. Then B – A = B + ( -A).
⎡2 − 3⎤ ⎡ 3 − 2⎤
⎡ ___
⎢4 0 ⎥ - ⎢− 1 1 ⎥ = ⎢ ___
⎣
⎦ ⎣
⎦
⎣
___ ⎤
___ ⎥⎦
⎡4⎤ ⎡2⎤
⎡ ___ ⎤
⎢− 2⎥ - ⎢− 2⎥ = ⎢ ___ ⎥
⎥
⎢ ⎥ ⎢ ⎥
⎢
⎢⎣ 0 ⎥⎦ ⎢⎣ 3 ⎥⎦
⎢⎣ ___ ⎥⎦
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There are two types of products of matrices –
multiplication by a scalar (nonmatrix – real number)
multiplication of two matrices
Scalar Multiplication: easy product - distributive law
⎡ 3 ⎤ ⎡ ___ ⎤
a) 4 ⎢ ⎥ = ⎢
⎥
⎣− 2⎦ ⎣ ___ ⎦
⎡ 2 1⎤ ⎡ ___
c) - 2 ⎢
⎥ = ⎢
⎣− 2 1⎦ ⎣ ___
b) - 2 [2 − 3 1 0] = [___
___ `___ ___ ]
___ ⎤
___ ⎥⎦
Some Simple products of Two matrices:
If we multiply matrix A by B( in that order), then the number of columns of A must be the same
as the number of rows of A. If A is an m x p matrix, then B must be a p x n matrix
ex.
[2
ex.
⎡4⎤
[1 − 2 3] • ⎢⎢ 0 ⎥⎥ =
⎢⎣− 3⎥⎦
⎡ 1 ⎤ ⎡2 − 1⎤
⎢ − 2 ⎥ • ⎢1 0 ⎥ = ?
⎣ ⎦ ⎣
⎦
− 1] • [1 − 2] = ?
ex.
⎡4⎤
⎢−1⎥ • [2 − 3] =
⎣ ⎦
In the two examples above, what do you get if you change the order of the matrices ?
⎡− 1 2 ⎤
ex. [1 2 3] • ⎢⎢ 0 − 2⎥⎥ =
⎢⎣ 4
1 ⎥⎦
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134
General Product of Matrices
⎡1 2⎤ ⎡ 1 − 2 3⎤
ex. ⎢
⎥ • ⎢
⎥ =
⎣3 4⎦ ⎣− 2 2 4⎦
⎡2 − 3⎤
⎡ 1 − 2 3⎤ ⎢
ex. ⎢
• ⎢2 3 ⎥⎥ =
⎥
⎣ − 2 2 4⎦ ⎢ 4 1 ⎥
⎦
⎣
ex.
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135
Sequences
Factorials:
Def. n! = n(n-1)(n-2) • • • (2) (1)
ex. 4 ! = 4(3)(2)(1) = 24
ex. 6 ! = ______________
ex. 100 ! = ______________
We define 1 ! = 1 and 0! = ______
Find 5 ! = ________
4 ! / ( 5 ! - 7 ! ) = ____________
240! / 241 ! = ______
Sequences:
a1, a2, a3, …
a correspondence between the set of natural numbers and a second set ( we can list the numbers in a list, 1st, 2nd, 3rd, … )
We can have a finite sequence; there is a beginning term and an ending term
a1, a2, a3,… an Î here an represents the last term and n represents the number of terms in the
sequence.
We can have an infinite sequence;
a1, a2, a3, …, an,… Î here an represents a general term of the sequence, the 3rd , the 10th, …
1, 4, 7, _____, ______
12, 5, - 2, ________, ________
1, 3, 4, 7, 11, ______, _________
2, 6, 10, 18, 34, ________, _________
2, - 4, 8, _________, ___________
16, 4, 1, ________, __________ , ________
-2, 0, 2, 0, -2, 0, 2, __________, _________
1, ½, 1/3, ________, ________, _________
2, ½, 3, 1/3, ______, _________,
½, 2/3, ¾, 4/5, ______, ________
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2, x + 4, x2 + 6x + 8, ....
x, 3x – 1, 5x – 2, ....
There are several ways to describe a sequence.
By its position ( the value of n). If an represents the fifth term, then n = 5, its position.
ex. if an = 3n + 1
ex. an = ( -1)n - 1 ,
then a1 = ______, a3 = __________
a1 = ________, a2 = __________
a25 = _________
a3 = ________, a20 = __________
By using preceding terms in the sequence, an represents the current term in question,
while an-1 represents the preceding term, an-2 represents the term right before the preceding term,…
ex. an = ( an-1 ) 2 ,
a1 = - 2,
ex. an = 2 - an-1,
a1 = 3,
a2 = __________, a3 = __________ , a20 = ________
a2 = _________ a3 = ___________,
a5 = __________
Summation of a sequence:
Suppose you wanted to find the sum the first five terms of the sequence defined by
an = 2n, we can easily list the five terms and find their sum.
We can also write
Σ an or Σ 2n to represent the sum.
Find each of the following sums:
1)
2)
3)
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Three types of sequences and progressions.
Arithmetic Progressions (AP ): need 1st term (a1) and common difference ( d )
Geometric Progressions (GP ) : need 1st term (a1 ) and the common ratio ( r )
Harmonic Progressions (HP):
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Test III Review – April 18, 2002
Old Material:
A. Sets: complex numbers, real, rational, irrational, integers, whole numbers, natural numbers,
1. Use the set { 2 – 3i, - 4, 0, - 2/3, 0.2222…, \/5, none of these } to find an example of
a) a complex number: ________________
b) an irrational number: __________
b) a natural # _____________
B. Properties of Real numbers:
2) Give me an example of
a) the associative law of addition _____________
b) the commutative law of multi. _________
c) the distributive law : __________________
3) Simplify each of the following
a) - 40 = ___________
b) 00 = ________
c) 4 – 2( 4) = __________
C. Quadratic equations:
4) Write Down the quadratic formula:
5) Find the solution of
a) x2 + 4x = 0 ___________
b) 4x2 + 25 = 0
D. Exponents.
Use the rules of exponents to simplify.
a) ( 4x2y)4 = ___________
b) ( - 4x-2)( 3x2y-3 ) = ____________________
c) ( 4x-2y-3 ) / ( 2x2 y – 4 ) = _____________
E. Radicals.
________
a) \/ 16x4y16
3
b) \/ 8x6y10
________
New Material:
A. Find the solution to each of the following equations.
1) | 2 – x | = 4 Î _______________
b) | 1 + 3x | = -2 Î _______________
3) | 2 + 5x | = 0
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B. Find the solution of the following inequalities.
4) | 1 + x | ≤ 4 Î ______________________
5 ) | 2 – 3x | > 2 Î _________________
6) | 2 + x | ≥ - 4 Î ___________________
7) | 3 + 2x | < - 2 Î __________________
8)
x / ( x + 2 ) > 0 Î ______________
9) ( x – 2 ) / (x + 1)(x + 2 ) ≤ 0 Î _______
10) x( x + 2) ( x – 2 ) ≤ 0 Î ________________________
11) 1/x > 2 Î ______________________
C. Relations and Functions.
Which of these are Relations and which are relations that are also Functions ?
sets, graphs, equations, and may also include lines, parabolas, absolute values,
logarithms, exponents,
1) Relation or Function
a) x- 3 - y = 0
b) y – x2 = 4
c) y = 4x
d) y = | x + 2 |
c) y = log2 x
d) y = x2 – 1
2) Domain: Find the domain of
a) y = x / x2 – 4 b) y = x / x 2 + 9
3) Range: Find the range of
a) y = 3x – 4
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b) y = 4x
c) y = x2 – 1
d) y = | x + 2 |
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4) Sketch the graph of : include absolute value, parabola, lines, logarithms, exponents
c) y = 2x
a) y - 2x = 4
b) y = log4 x
5) Find the x and y – intercepts of
a) y = x2 – 4x – 12 Î ___________
d) y = x2 + 4x
y = | x + 1 | Î _________________
D. Lines –
1) Find the slope of each of the following
a) a vertical lineÎ __________
b) a line parallel to the x-axis Î _________
c) a line passing through the points ( 4, -2 ) and ( - 4, - 3 ) Î ____________
d) a line perpendicular, parallel to 3y – 4x = 4 Î _________
_____________
2) Sketch the graph of the line
a)with slope 2 passing through ( 3, - 2 )
b) x = 3
3) What is the equation of a line that is parallel to the y – axis and passes through the point ( 3, 0 )
What is the equation of the line that has slope 0 and passes through the point ( -4, 4 ) ?
E. Quadratic Functions: Parabolas: vertex, opens, graphs , max., min, word problems
1) Let f(x) = 3x – 1
and g(x) = x + 5 Find
a) f + g : ________________
b) fg : ____________________
c) f o g:__________________
2) Use the functions from above to find
a) f( - 3)= ________
b) fg ( - 1)= _________ c) fog ( - 1 ) = ______________
3) Word Problems: pages -- _____________
___________________
F. Roots of a polynomial equation.
1. Roots and degree of a polynomial
a) What is the degree of each of the following polynomial ? _____________
b) How many solutions does the following equation have ? ______________
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c) How many positive roots does x4 + 3x2 – x – 2 = 0 have ? ____________
d) If P(x) can be factored as x(x + 2 )( x- 3) (x + 5)(x2 + 1 ), then
how many solutions does the P(x) = 0 have ? ___________
what are they ? ______________________________
2. Remainder Thm. , Factor Thm and its converse
a) Find the remainder of (x3 + 2x + 4 ) ÷ ( x + 2 ) Î ________________
b) If x = 3 , x = -2 are the only solutions of a quadratic polynomial P(x), then find the
polynomial P(x).
c) See question 1d above – previous page
d) Is (x + 1 ) a factor of (x12 - 3x2 + 2x + 4) ? Show work.
e) Is x = -2 a solution of x4 - 8x2 + 3x + 6 = 0 ? Show work
3. Synthetic division.
a) Use synthetic division to find the remainder of (x3 + 5x - 1 ) ÷ ( x - 2 ). _____________
b) Use synthetic division to determine if x = -3 is a solution of x4 - 9x2 - x - 3 = 0
c) Use synthetic division to find ( x3 + 4x – 5 ) ÷ ( x + 1 ) = _______________________
4. Other Theorems:
a) If P(x) = 0 has the following numbers as solutions: 3, 4 – i, 2 + \/ 3, then what must the
degree of the polynomial P(x) be ?
b) If 2 + 3i is a root, then list one other root ---_______________
c) If 3 - \/ 5 is a root, then list one other root --- ______________________
5. Rational Root Thm.
a) Given 3x4 + x + 2 = 0 , what are the only rational numbers that should be tested to find all of
the rational roots.
b) If x = 2 is a solution of x3 - 5x + 2 = 0, then find the other solutions.
d) Find all of the solutions of the polynomial
a solution
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x4 - 2x3 + x2 – 8x - 12 = 0 , assume that x = 3 is
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Name _________________________- Math 1302 – Short QZ, April 4, 2002
1. Which of these has to do with values of x ?
Domain,
Range
2. Which of these is always true ?
A function is always a relation
A relation is always a function
3. True or False.
______________ a. A parabola that opens upward has a domain = set of all real numbers
_______________ b. The graph of x = 2 is a vertical line
_______________ c. To find the slope of a line , you can use m = ( y2 - y1 ) / ( x2 - x1 )
_______________ d. horizontal lines always have slope o
________________ e. the graph of a quadratic functions y = Ax2 + Bx + C is always a parabola.
4. Give me a rough sketch of the 2x – y = 4 by finding the x and the y intercepts.
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5. Use x = - B/ 2A to find the vertex of the parabola f(x) = 4x2 - 2x + 2Î ______________
Use the vertex to sketch the graph of f(x) .
Name ______________________________ Math 1302 - Long Quiz – April 9, 2002
1. Match the graph with the most appropriate equation.
y = 2x – 1
a)
y = x2 + 1
_________________
2. Given f(x) = 2x – 1 and
y=|x|
d) ______________________
g(x) = 2 - x , h(x) = | x | find
a) f( 0 ) = _____________
c)
y = log2 x
b) _____________________
c) __________________
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y = 2x
b) h( h ) = ________ if h represents a whole number.
g( f( - 2 ) ) = __________
3. Use functions f and g from #2 to find
a) f + g: ____________________
4. Which of these is not a functions ? y = x2
b) fg: ________________________
y=|x|
y=3
ALL are
5. What is the domain of y = 2x / ( x + 4 )? __________________
6. What is the range of the parabola y = x2 – 4x + 4,
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- B/2A = 2 Æ ________________________
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7. Divide 127 by 11 and write answer in fractional form(no decimals )
127/ 11 = _________________
Name ____________________________ Math 1302 – Long Quiz – April 2, 2002
1. State the quadratic formula.
2. Sketch the graph of 2x – y = 4
3. Which of these represents a parabola
2x = y
or
y = -x2
or
neither one
4. What is the y-intercept of the curve y = 2x – 4 ? ___________________
What is the x-intercept ?
5. Is y = | x | a function or just a relation ? ______________________
6. What is the domain of the function f(x) = 3x ? ___________________________
7. What is the range of the function f(x) = x2 + 2 ? __________________________
4x
8. What is the domain of f(x) = -------------- ? __________________________________
x + 2
Name _________________________________ Math 1302 – Short QZ - April 11, 2002
1. Find the remainder of ( x4 + 1 ) ÷ ( x – 1 ) Î _______________
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What is the remainder of ( 3x2 + 5x - 2 ) ÷ ( x + 1 ) Î _______________
2. If P(x) is a polynomial that has the following prime factorization ( it factors as )
P(x) = x ( x + 1 ) ( x – 2 ), then find all solutions of P(x) = 0 .
_________________________________
3. If P(x) represents a quadratic polynomial with x = 2 and x = -3 as its only solution, then find
the polynomial P(x).
P(x) = ________________________________
4. What are the x-intercepts of the polynomial represented by the following graph ?
5. Use long-hand division to find (x2 + 5x + 3 ) ÷ ( x + 2 ) = _________________________
Quiz - Math 1302 – Name ___________________________________________________________________
1. Find 124 ÷ 8 = __________
2. Find ( x2 + 1 ) ÷ (x + 1 ) = ____________
3. What is P(-1) if P(x) = 2x3 - x + 1 ?
______________
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4. What is the remainder of (x4 + x – 2 ) ÷ ( x + 1 ) ?
___________
5. I f P(x) = x( x + 1)(x + 2 ) (x – 3 ), what are the
solutions of P(x) = 0 ?
_________________________
6. If P(x ) represents a polynomial of degree 2 (quadratic
polynomial ) and 2, -4 are solutions of P(x) = 0, then
what is
P(x) ? _________________________
7. How many solutions (roots ) does
x5 + 4x3 + 2x2 - 4 = 0 have ? __________
8. If
- 6, 2 - 3i and - 4i
are roots (solutions ) of
P(x ) = 0 , then can you tell me other values that are
also solutions ?
9. How many positive roots does 3x4 + 4x + 2 = 0
have ? _________
__
10. If - 2 + \/ 5 is a root of P(x) = 0 , then give me
another real number that is also a solution .
11. What are the only rational numbers that could
possibly be roots of x3 + 2x + 3 = 0 ?
Name _________________________ Math 1302 – Qz - June 26, 2001
1. Find the remainder of ( x4 - 2x + 4 ) ÷ ( x + 2 ) ==> _________________
2. Find (x3 + 2x - 3 ) ÷ ( x – 1 ) = ___________________________________________
3. If x = 2, x = 3 are solutions of a quadratic equation, then what is the quadratic equation. ______________________
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4. How many solutions does x5 + 3x2 + 4x + 2 = 0 have ? _____________
How many of them are positive ? ____________
5. If 2 + 3i , - 4, 3/2 ,
and \/ 5 are solutions of a polynomial equation, then what is the smallest degree possible of that
polynomial ? __________
6. If x = 1 is a solution of x3 + x2 - 2 = 0 , then find the other solutions. _________________
7. Use synthetic division to tell me the value of P ( -1 ) if P( x ) = x3 + x2 - 2x + 2 ? ___________
Name __________________________ Math 1302 - Short QZ -- April 18, 2002
1. Give me an example of quadratic polynomial.
2. Show me that x = 1 is a solution of x3 + x2 - 2 = 0 by using synthetic division.
3. List all possible values of p/q in reduced form that should be tested to find the solutions of
a) x3 + 5 = 0 Î ____________________
b) 2x4 - 2x - 3 = 0 Î ___________________
4. If x = 1 is a solution of 2x3 + x – 3 = 0, then find the other solutions.
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Name ________________________ Math 1302 – Long QZ, April 25, 2002
1. Find each of the following factorials.
0 ! = ___________
1 ! = ________________
4 ! = ___________
2. Find without the use of a calculator.
a)
100 ! / 99 ! = _________________
b)
201 ! / 200 ! = ________________
3. An infinite sequence is a sequence that has how many terms ? _______________________
4. Find the next term in each sequence.
a)
1, 5, 9, 13 , ________
b) - 2, 2, 6, _________
c) 4, 2, 1, __________
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