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Transcript
Mth 65
Module 3
Sections 3.1 through 3.3
Section 3.1 - Functions and the Quadratic Function
Which of the following are functions?
Which of the following relations are functions? Why? Identify the independent and dependent
variables. Find the domain and range.
x
y
-1
5
-2
6
Fall 2013
0
5
3
4
4
-1
r
t
0
-1
1
2
2
5
1
.7
3
5
4
-8
1
Mth 65
Module 3
Sections 3.1 through 3.3
Evaluate the function, f(x) = 3x2 – 6x + 1, to fill in the table below and make a graph by
hand.
x
-1
0
1
2
3
f(x)
The function above is called a _______________________________function.
The shape of the graph is called a ___________________________.
Each parabola has a _______________(maximum or minimum) and an axis of symmetry (always
a ______________________ line which passes through the vertex). State the vertex and the axis
of symmetry for the graph above. Vertex ___________ Axis of symmetry _____________
The general form of a quadratic function is
y = ax2 + bx + c, with a ≠ 0 where a, b, and c are constant coefficients.
Give a, b and c’s value for the function you graphed above.
a=
b=
c=
Explain why each function is quadratic or not quadratic? Identify a, b, and c for each quadratic.
g(x) = 2x2 -10
f(x) = 3x -1
h(x) = 15 + 2x – x2
Use your graphing calculator to complete the following table. Graph the function to determine
the vertex and the axis of symmetry.
g(x) = -x2 – 4
x
g(x)
Fall 2013
-3
-2
-1
0
1
2
3
2
Mth 65
Module 3
Sections 3.1 through 3.3
Graph the following quadratic function using a calculator and estimate its vertex. Tell whether
the vertex is a maximum or a minimum.
h(t) = 3(t – 5)2 – 2
y = 100 – x2
Section 3.2 – Properties of Exponents
Review: In the expression 43 , 4 is the ____________ and 3 is the exponent. The exponent
tells how many times to repeat the base as a _________________.
All of the properties of exponents are true for all real number a, b, c, and n.
1. Product Rule - When you multiply two numbers with the same _____________
keep the ______________ and add the ____________________
am · an = am+n
Examples
a3(2a4)
(-3m4n)(-2m3n2)
42 · 43
2. Raising Powers to Powers - Keep the ________and multiply the_____________
(am)n
Examples
(m2)4
(52)4
w2(w3)3
(23)2
3.
Raising Products to Powers - Take each __________ to that _____________.
(ab)m = ambm
Examples
(4a3)3
(-4a3b)(2ab)2
(4a)2
(2x3y2)3(-xy3)
Fall 2013
(-3y2)3(-4y4)
3
Mth 65
4.
Module 3
Sections 3.1 through 3.3
0
Zero Exponents - For any nonzero number a, a = 1, so
4 0 =_______and
100 = ________ , but 00 is ____________________ .
a0 = 1
Examples
(-6)0
-3y0
3x2(4x2y3)0
50
5.
Negative Exponents - Takes the reciprocal of a number to the _________ power,
but doesn’t change the sign of the coefficient
an 
1
an
x-3
and
1
 an
an
Examples
4x-2y3
1
4 2
7 x 3 y 1 z 0
14
x3
y 2
6.
(6x3)-2
Quotient Rule for Exponents - When you divide two numbers with the same
___________, subtract the exponents ( top exponent – bottom exponent).
am
 a mn
n
a
y5
y3
Fall 2013
Examples
4w5
2w
21x3
x 1 y 2
m5
m8
7 y3
3 y5
(2 x) 3
xy 2
4
Mth 65
7.
Module 3
Sections 3.1 through 3.3
Raising Quotients to Powers - Take each factor in the numerator and the denominator
to that __________________.
n
an
a
   n
b
b
Examples
 5 2 
 4 
5 
2
 2m 3 
 2 
 n 
2
 x2 
 
 2 
3
Scientific Notation
Scientific notation is used to express very large and very small numbers. A number is written in
scientific notation when it is in the form, n  10 y .
n always has ________ nonzero digit in front of the decimal point and y tells how many places
and the direction that you moved the point. Unnecessary zeros are omitted from n.
If the number is greater than one, then 1  n  10 and y will be an integer greater than 0.
Write each standard numeral in scientific notation.
760,000
3,040,000,000
Write each number given in scientific notation as a standard numeral.
4.1 x 106
5.02 x 108
If the number is less than negative one,
then 10  n  1 and y will be an integer greater than 0.
Write each standard numeral in scientific notation.
-958,000
-604,000,000
Write each number given in scientific notation as a standard numeral.
-6.3 x 105
Fall 2013
-3.007 x 109
5
Mth 65
Module 3
Sections 3.1 through 3.3
If the number is greater than ZERO but less than ONE,
then 1  n  10 and y will be an integer less than 0.
Write each standard numeral in scientific notation.
0.28
0.0000504
Write each number given in scientific notation as a standard numeral.
3.6 x 10-5
4.105 x 10-11
If the number is greater than negative one but less than zero,
then
10  n  1 and y will be an integer less than 0.
Write each standard numeral in scientific notation.
-0.00008
-0.000000000931
Write each number given in scientific notation as a standard numeral.
-1.8 x 10-6
-3.04 x 10-11
Number line summary for n x 10y
-1
0
1
Using a GRAPHING calculator to solve problems
To enter 5.3  108 into your calculator you will enter n’s value, hit a special key EE , and then enter
y’s value.
5.3  108 is entered 5.3 EE 8. The result should be 530,000,000.
Fall 2013
6
Mth 65
Module 3
Sections 3.1 through 3.3
Compute. Give your answer in scientific notation rounded to two decimal places. You will need
parentheses only if there is more than one factor in your denominator.
7.39 108
1.02 103
(4.2 108 )(3.37 104 )
Type 4.2 2nd EE 8 x 3.37 2nd EE 4 enter
= 1.4154 E 13 ≈
(4.26 109 )(1.5 102 )
3.3 103
(3.6 1010 )(1.5 102 )
(5.9 108 )(2.1103 )
Review
Simplify completely. Write your answer with positive exponents only.
 3m 5 
 3 
 n 
9 y6
3 y8
4
Compute. Give your answer in scientific notation rounded to two decimal places.
 2.67 10 1.110 
3 10  4.2 10 
4
2
3
4
Section 3.3 – Square Roots
When working with square roots, the symbol
For any positive real number a,
positive square root of a.
49 =
a b
is called a __________________sign.
implies
b 2  a and b is called the principal or
a is called the _________________________.
 49 =
 49 =
49 =
negative # is always a ________________________________ number.
Fall 2013
7
Mth 65
Module 3
Sections 3.1 through 3.3
Evaluate each of the following. If the result is not a real number explain why.
(a)
100 =
(b)
(c)  81 =
9 =
(d)
64 =
Multiplying Radicals
To multiply square roots we simply multiply the radicands.
If a  0 and b  0 , then
a
b ab
Multiply the radicands and evaluate.
(a)
6
6
(b)  5
(c)
20
  3   12 
Simplifying Radicals
A radical can be simplified by writing it as the product of two square roots.
If a  0 and b  0 , then
ab a
b . Example:
40  4 10 
There were other ways to factor 40 besides 4(10) but we chose 4(10) because 4 was the
largest square factor of 40.
Use your calculator to find approximate solutions for
40 and 2 10 .
Knowing the square numbers will help you simplify radicals
Square Numbers: 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64, 92 = 81,
102 = 100, 112 = 121, 122 = 144, 132 = 169, 142 = 196, 152 = 225, ...
Simplify each of the following square roots exactly. Verify that each result is equal to the
original square root with your calculator.
(a)
75
Fall 2013
(b)
98
(c)
96
8
Mth 65
Module 3
Sections 3.1 through 3.3
Sometimes multiplying radicals involves multiplying coefficients and radicands, and then
simplifying the square root.
3 10 7 5  3 7 10
5
2 12 5 6
Simplify. Give both the exact answers and the approximate answers.
62  3(2)(4)
4 63 3
Rational Numbers and Irrationals Numbers
A rational number is real number whose decimal expansion terminates or repeats. Any
number that can be written as a ratio is rational.
An irrational number is a real number whose decimal expansion does not terminate and does
not repeat. Irrational numbers cannot be written as ratios. (The square root of any non-perfect
square number is irrational.)
The square root of any negative radicand is a non-real number.
Identify each number as a rational real number, an irrational real number, or non-real number.
5
5
14,  ,1.287,  ,5 7,  27,54, 60
9
12
Rational real numbers
Fall 2013
Irrational real numbers
Non-real numbers
9