Pre­Calc 7 and 8 7.1
May 15, 2014
day1
TOPICS
• NUMBER OF SOLUTIONS IN A SYSTEM
• SOLVING LINEAR SYSTEMS
• SOLVING NON-LINEAR SYSTEMS
1
Pre­Calc 7 and 8 May 15, 2014
solution
Methods to solve systems:
Substitution, Elimination, Graphing, Matrices
2
Pre­Calc 7 and 8 May 15, 2014
Solutions to linear systems
What does this look like algebraically?
3
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #1
2
4
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #2
5
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #3
6
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #4
7
Pre­Calc 7 and 8 May 15, 2014
How many solutions could the following pairs of equations have?
1) Line and Parabola
2) Parabola and Parabola
3) Line and Circle
4) Parabola and Circle
5) Circle and Circle
8
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #5
Dividing out factors will cause you to miss answers.
THIS IS A
BIG
2x = x
NO NO!
8x = x2 + 6x
2
2x = x2
x x
2 = x
9
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.1 p.526
#1­22 mod 3
23­26 all
Be able to do both elimination and substitution method.
EXAMPLE #1
EXAMPLE #2
EXAMPLE #3
EXAMPLE #4
EXAMPLE #5
Write these examples from class today in your steno notebook.
10
Pre­Calc 7 and 8 May 15, 2014
11
Pre­Calc 7 and 8 7.1
May 15, 2014
day2
TOPICS
APPLICATIONS OF SYSTEMS
7.2 TOPICS
Introduction
to Matrices
No we are not watching the movie!
12
Pre­Calc 7 and 8 May 15, 2014
We will look at applications of systems in the following:
• Geometry
• Velocity and Wind
• Mixtures
13
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #1 (GEOMETRY)
Find the dimensions of a rectangular garden that has
perimeter 100 ft and area 300 ft2
14
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #2 (VELOCITY and WIND)
An airplane flying with the wind from Los Angeles to New
York takes 3.75 hr. Flying against the wind, the airplane
takes 4.4 hr for the return trip. If the air distance from
Los Angeles and New York is 2500 mi and the airplane speed
and wind speed are constant, find the airplane speed and the
wind speed.
15
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #3 (MIXTURE)
Suppose the owner of a candy store mixes two types of candies.
She decides to create a 20-pound mixture of raspberry-flavored
gumdrops and cherry-flavored jelly beans. The gumdrops sell for
$0.95 per pound and the jelly beans sell for $1.20 per pound. She
plans to sell the mix for $1.10 per pound. How many pounds of each
candy should she use in her mix?
16
Pre­Calc 7 and 8 May 15, 2014
Matrix intro will include:
• Order • elements
• matrix algebra • matrix multiplication
17
Pre­Calc 7 and 8 May 15, 2014
Matrix Order: M x N
Matrix Elements: ai,j
18
Pre­Calc 7 and 8 May 15, 2014
Matrix Algebra
19
Pre­Calc 7 and 8 May 15, 2014
Matrix Multiplication
Order matters!
Matrix Size Matters!
[ 4 x 2] [ 2 x 3]
# of columns of first must MATCH
# of rows of second
[ 4 x 2] [ 2 x 3]
The size of the resulting matrix is the # of rows of the first and the # of columns of the second
20
Pre­Calc 7 and 8 May 15, 2014
Checking to see if you understand...
Are BOTH AB and BA possible?
Find AB and BA, if possible
21
Pre­Calc 7 and 8 May 15, 2014
Matrix Multiplication
22
Pre­Calc 7 and 8 May 15, 2014
EXAMPLES #1­3
23
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.1 p.526
#48, 49, 50, 51, 53,
54, 57, 58, 61­64
3 examples from class today should have been written down.
Assignment 7.2 p.540
#1­22
3 examples from class today should have been written down.
24
Pre­Calc 7 and 8 May 15, 2014
TODAY
Scientific Calculators ONLY ­ No graphing calculators for quiz tomorrow.
1) Get out your Steno notebooks and show your teacher you have copied down:
Examples #1­5 (7.1 Monday)
Example #1­3 (7.1 Tuesday)
Examples #1­3 (7.2 Tuesday)
2) Then begin doing all the problems, WITHOUT help from anyone or your notes.
Mark questions you are unsure of how to do.
3) Check your answers with the answers key on the side board when you are done.
What should you do about questions you don't know how to do?
25
Pre­Calc 7 and 8 May 15, 2014
Can you do all of these?
Solve each system (do at least one with substitution)
1) x + 6y = 12
-5x - 12y = -24
2) 18x + 8y = 10
-9x - 3y = -15
3) -5x - 2y = 4
-10x - 4y = 8
4) A boat traveled 165 miles downstream and back. The trip downstream took 11
hours and the trip back took 15. What is the speed of the boat? Of the current?
5) Susan and Tim are selling pies for a fundraiser. Susan sold 4 cherry pies and 2
apple pies for $64. Tim sold 5 cherry pies and 6 apple pies for $129. What is the
cost of each pie?
ANSWERS
1) (0, 2) 2) (5, ­10) 3) All solutions 4) 13mph and 2mph 5) cherry $9, apple $14
26
Pre­Calc 7 and 8 May 15, 2014
Can you do all of these?
Find AB.
Add =
What is A21?
Multiply
=
27
Pre­Calc 7 and 8 May 15, 2014
PRODUCT and PRICE EXAMPLE
Company A rents copiers for a monthly charge of $200 plus
10 cents per copy. Company B rents copiers for a monthly
charge of $400 plus 5 cents per copy. What is the number of
copies above which Company A's charges are the higher of
the two?
28
Pre­Calc 7 and 8 May 15, 2014
29
Pre­Calc 7 and 8 May 15, 2014
7.3 TOPICS
Solving systems of three variable equations.
30
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #1
ANSWER
(15, 10)
31
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #2
(3, ­2, 1)
ANSWER
Try to combine
equations that will
cause variables to be
eliminated.
32
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #3
With a partner, try to solve this system.
­3x + 2y ­ 6z = 6
5x + 7y ­ 5z = 6
x + 4y ­ 2z = 8
( ­2, 3, 1)
ANSWER
33
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #4
3x - 3y + 6z = 20
x - 3y + 10z = 40
-x + 3y - 5z = 30
( 18, 39.3, 14)
ANSWER
34
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.3 p.552
#1­7
You should have _______ examples from today.
35
Pre­Calc 7 and 8 May 15, 2014
36
Pre­Calc 7 and 8 May 15, 2014
7.2 (cont.) TOPICS
• Determinants of a 2x2 Matrix
• Inverses of Matrices
7.3 TOPICS cont.
Systems and Matrices...you're gonna LOVE this!!
Applications of Systems of Equations
37
Pre­Calc 7 and 8 May 15, 2014
Determinant of the following matrix:
(4)(7)­(2)(5)=18
Ok...why do we need determinants?
Being able to find the determinant of a square matrix is a quick way to let us know if there is an inverse to the matrix... ...which is used for???
38
Pre­Calc 7 and 8 May 15, 2014
Inverses
Sw
Sw itc
itc h e
h lem
SI
GN ent
S s a
of a
el nd
em d
en .
ts
c an
d d.
39
Pre­Calc 7 and 8 May 15, 2014
Ok...why do we need inverse matrices?
We can now write and solve MATRIX EQUATIONS...YEAH!!
40
Pre­Calc 7 and 8 May 15, 2014
Same example as Monday!
(15, 10)
ANSWER
Write as a Matrix Equation AX=B
First, make sure the system is all in SAME order!
Second, use coefficients of variables for matrix A.
[
[ = A
2 ­3
2 ­2
Third, write the variable matrix, called X.
[
[ = X
Fourth, write the resulting matrix, called B.
0 [
[ = B
X
Y
10
41
Pre­Calc 7 and 8 May 15, 2014
This is a system of equations.
[
[ = A
2 ­3
2 ­2
[
X[= 0 [
[ [ Y [ 10
This is a matrix EQUATION!
2 ­3
2 ­2
[
[ = X
0 [
[ = B
X
Y
10
AX=B
To solve, multiply both sides by the inverse of A.
Remember, order matters in matrix multiplication.
A­1AX = A­1B
X = A­1B
This can be done on your calculator!
42
Pre­Calc 7 and 8 Matrix Menu
Enter the coefficients
for A matrix.
May 15, 2014
Go to EDIT
Select a Matrix and change the SIZE
Exit to Home screen and go back into
matrix edit menu.
Put in B matrix.
On home screen, use Matrix button to call up names of matrices. Use the x­1 key to enter the inverse power.
X
Hit Enter
Y
43
Pre­Calc 7 and 8 EXAMPLE #1
Write the Matrix equation.
THEN solve on your calculator.
May 15, 2014
(3, ­2, 1)
ANSWER
44
Pre­Calc 7 and 8 EXAMPLE #2
May 15, 2014
APPLICATIONS
Define Variables
Find relationships between the variables so you can write equations.
At the Pittsburgh zoo, children ride a train for 25 cents, adults
pay $1.00 and senior citizens 75 cents. On a given day, 1,400
paid a total of $740 for the rides. There were 250 more
children riders than all other riders. Find the number of
children, adult and senior citizen riders.
45
Pre­Calc 7 and 8 EXAMPLE #3
May 15, 2014
APPLICATIONS
Define Variables
Find relationships between the variables so you can write equations.
Matthew has 74 coins consisting of nickels, dimes and
quarters in his coin box. The total value of the coins is $8.85.
If the number of nickels and quarters is four more than the
number of dimes, find out how many of each coin Matthew has
in his coin box.
46
Pre­Calc 7 and 8 EXAMPLE #4
May 15, 2014
APPLICATIONS
Define Variables
Find relationships between the variables so you can write equations.
Monica receives an $80,000 inheritance. She invests part of it in CDs
earning 6.7%, part in bonds earning 9.3% and the remainder in a growth
fund earning 15.6%. She invests three times as much in the growth fund
as in the other two combined. How much does she have in each investment
if she receives $10,843 interest the first year?
47
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.3 day 2 p.552
#45­53 odd, 74, 76, 79
You should have ________ examples from today.
48
Pre­Calc 7 and 8 May 15, 2014
POP QUIZ
You will be given one system of equations to solve with a scientific calculator. Show all your work. Keep it organized.
49
Pre­Calc 7 and 8 May 15, 2014
50
Pre­Calc 7 and 8 May 15, 2014
7.4 TOPICS
Partial Fractions
51
Pre­Calc 7 and 8 May 15, 2014
How do you ADD or SUBTRACT fractions?
1
1
+
b
a
=
b + a
ab
Add the following fractions together.
52
Pre­Calc 7 and 8 May 15, 2014
Partial Fractions and Decomposition of Fractions
Distinct Linear Factors
Repeated Linear Factors
px + q
A + B
= (x ­ a)(x ­ b)
(x ­ a) (x ­ b) Then: px + q = A(x ­ b) + B(x ­ a)
Solve TWICE, using the root from each factor as the value for X.
If you have more parts, then you will have more coefficients to solve for. 53
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #1
54
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #2
55
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #3
3x ­ 4
x2 ­ 2x
56
Pre­Calc 7 and 8 May 15, 2014
EXAMPLE #4
4
x2 ­ 1
57
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.4 p.563
#5, 6, 13, 15, 21, 22 ­­ fractions
You should have 3 examples from today.
58
Pre­Calc 7 and 8 May 15, 2014
TODAY
May use Graphing Calculators for quiz tomorrow.
1)
Get out your Steno notebooks and show your teacher you have copied down:
7.3 Monday: 2 examples
7.3 Tuesday: 2 examples
7.4 Wednesda: 3 Examples
2) Then begin doing all the problems, WITHOUT help from anyone or your notes.
Mark questions you are unsure of how to do.
3) Check your answers with the answers key on the side board when you are done.
What should you do about questions you don't know how to do?
59
Pre­Calc 7 and 8 May 15, 2014
7.5 TOPICS
• Inequalities
• Graphing inequalities
• Systems of inequalities
60
Pre­Calc 7 and 8 May 15, 2014
Inequalities
Equalities use EQUAL signs
3x + 2 = 17
Answers are exact values. x = 5
Inequalities use signs of inequalities:
>
greater than
<
less than
≤
less than or equal to
≥
greater than or equal to
Answers are a RANGE of VALUES
61
Pre­Calc 7 and 8 May 15, 2014
Solving and Graphing Inequalities
Other Equations
Linear equations
Solve inequality for y.
Remember: y = mx +b
Plot b on the y axis
Use m to find a second point
Know the 12 basic functions
y = x2 is a parabola
x2 + y2 = r2 is a circle
y = √x is the square root graph
OR
Plug a 0 in for x and solve for y. Plot point (0, y).
Plug a 0 in for y and solve for x.
Plot point (x, 0). Graphing using x & y intercepts
Connect the points with EITHER a solid line or a dashed line.
≤ and ≥
< and >
Find a test point , usually (0,0) provided the graph doesn't go through it, and plug in the values for x and y.
If statement is true, shade side WITH the test point.
If statement is false, shade side WITHOUT the test point.
62
Pre­Calc 7 and 8 May 15, 2014
Examples
63
Pre­Calc 7 and 8 May 15, 2014
Example
y > x2 + 3
64
Pre­Calc 7 and 8 May 15, 2014
Systems of inequalities ­ graph each and shade. Section with double shading is the solution set
65
Pre­Calc 7 and 8 May 15, 2014
Example
2x + y ≤ 10
2x + 3y ≤ 14
x≥0
y≥0
66
Pre­Calc 7 and 8 May 15, 2014
Example
3x + 8y ≥ 240
9x + 4y ≥ 360
x ≥ 60
y≥0
67
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.5 day 1 p.571
#1­34 mod 3
Steno Notebooks: 3 examples
*One single system
*One double system
*One system of 4 equations
68
Pre­Calc 7 and 8 May 15, 2014
Which region matches each set?
B
C
Set 1
A
D
Set 2
Set 3
Set 4
69
Pre­Calc 7 and 8 May 15, 2014
70
Pre­Calc 7 and 8 May 15, 2014
7.5 TOPICS continued
Linear Programming
71
Pre­Calc 7 and 8 May 15, 2014
Steps to Linear Programming
1) Write the equation that is to be Maximized or Minimized.
2) Write the equations/inequalities that are the constraints on the system.
3) Graph the system and find the feasible region. 4) Find the coordinates of all corner points.
5) Put the coordinates of the corner points into the equation to be max/min and find the set that answers the question.
72
Pre­Calc 7 and 8 May 15, 2014
A small TV manufacturing company produces flat screen and portable TVs using 3 different machines, A, B and C. The table below shows how many hours are required on each machine per day in order to produce a flat screen or portable TV.
Machine
Flat screen
Portable
Hours
A
1 hour
2 hours
16
B
1 hour
1 hour 9
C
4 hours
1 hour
24
Steps to Linear Programming
Write the equation that is to be Maximized.
Write the equations that are the constraints on the system.
Graph the system and find the feasible region. Find all corner points.
Suppose that the company makes a $60 profit on each flat screen and a $40 profit on each portable. How many of each type of TV should be produced each day to maximize profit?
73
Pre­Calc 7 and 8 May 15, 2014
2
2
60X + 40Y = Profit
74
Pre­Calc 7 and 8 May 15, 2014
60X + 40Y = Profit
Corner Points by intersections of lines:
(0,0)
(0,8)
(6,0)
(2,7)
(4,5)
Substitute them into the profit equation and which is the max??
(0,0)
(0,8)
(6,0)
(2,7)
(4,5)
75
Pre­Calc 7 and 8 May 15, 2014
76
Pre­Calc 7 and 8 May 15, 2014
Cost = 1.7X + 1.2Y
77
Pre­Calc 7 and 8 May 15, 2014
78
Pre­Calc 7 and 8 May 15, 2014
79
Pre­Calc 7 and 8 May 15, 2014
this shows 90 long sleeve and 165 short sleeve gives max profit!!
80
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.5 day 2 p. 571
#38, 39, 40, 45, 46
Steno Notebooks: 2 examples
81
Pre­Calc 7 and 8 May 15, 2014
Use an algebraic method to solve the systems of
inequalities
1.
5x - 3y > 1
3x + 4y ≤ 18
2.
x - 3y < 6
y > -x2 - 2x +2
Find the minimum and maximum, if they exist, of the
objective function f, subject to the constraints.
3. Objective function: f = 5x + 2y
Constraints:
2x + y ≥ 12
4x + 3y ≥ 30
x + 2y ≥ 10
x ≥ 0, y ≥ 0
82
Pre­Calc 7 and 8 May 15, 2014
P
I
s
K
a
r
S xt
E
**Add in as extra examples/review/warm ups if needed for the remaining slides
83
Pre­Calc 7 and 8 May 15, 2014
Every day Rhonda Miller needs a dietary supplement of 4 mg of vitamin A, 11 mg of vitamin B and 100 mg of vitamin C. Either of two brands of vitamin pills can be used: Brand X at $0.06 a pill or Brand Y at $0.08 a pill. A Brand X pill supplies 2 mg of vitamin A, 3 mg of vitamin B and 25 mg of vitamin C. Likewise, a Brand Y pills supplied 1,4, and 50 mg of vitamin A, B and C, respectively. How many pills of each brand should she take each day in order to satisfy the minimum daily need most economically?
P
I
s
K
a
r
S xt
E
84
Pre­Calc 7 and 8 May 15, 2014
x + 2y + z = ­1
P
I
x ­ 3y + 2z = 1
s
K
a
2x ­ 3y + z = 5
r
S xt
E
85
Pre­Calc 7 and 8 May 15, 2014
P
I
s
K
a
r
S xt
E
.06x + .08y = C
(3, .5)
(0, 4)
(1, 2)
(4, 0)
86
Pre­Calc 7 and 8 May 15, 2014
WARM­UP ­­ 87
Pre­Calc 7 and 8 May 15, 2014
88
Pre­Calc 7 and 8 May 15, 2014
WARM­UP ~ aka Practice Quiz
1.
Solve the system of inequalities.
2x + y ≤ 80
x + 2y ≤ 80
x ≥ 0, y ≥ 0
2.
A carpentry shop makes dinner tables and coffee tables. Each week the
shop must complete at least 9 dinner tables and 13 coffee tables to be
shipped to furniture stores. The shop can produce at most 30 dinner
tables and coffee tables combined each week. If the shop sells dinner
tables for $120 and coffee tables for $150, how many of each should be
produced for a maximum weekly income?
3.
Mrs. Klumb's farm contains 240 acres available for planting corn and oats.
Profit per acre for corn is $40 and for oats is $32. The total number of
hours available for labor is 320. Each acre of corn uses 2 hours of labor
while each acre of oats requires 1 hour. How should the land be divided to
maximize profit?
89
Pre­Calc 7 and 8 May 15, 2014
90
Pre­Calc 7 and 8 May 15, 2014
8.1 TOPICS
Conic Sections - Parabolas
Vertex on axis
91
Pre­Calc 7 and 8 May 15, 2014
92
Pre­Calc 7 and 8 May 15, 2014
CONIC SECTIONS
Ellipse
Parabola
Hyperbola
93
Pre­Calc 7 and 8 May 15, 2014
94
Pre­Calc 7 and 8 May 15, 2014
The vertex lies halfway between the
directrix and the focus
What does the mean?
•
•
•
Focus to Vertex is the same
distance as Directrix to Vertex
FV = DV
Focus to any point is the same as
Directix to any point
FP = DP
That means we are going to use the
distance formula
95
Pre­Calc 7 and 8 May 15, 2014
What is the distance formula?
Well,if you can't remember it, it is just the
Pythagorean Formula
F (0, p) and P (x, y) and D (x, -p)
Basic formula for a parabola that has a vertex at (0, 0)
96
Pre­Calc 7 and 8 May 15, 2014
DETAILS of a parabola with vertex at (0, 0).
97
Pre­Calc 7 and 8 May 15, 2014
98
Pre­Calc 7 and 8 May 15, 2014
99
Pre­Calc 7 and 8 May 15, 2014
example
Find the focus point and directrix and graph the parabolas: General form of a parabola:
What do you know about this parabola
that would help you graph it?
100
Pre­Calc 7 and 8 May 15, 2014
example
Find the focus point and directrix and graph the parabolas: General form of a parabola:
What do you know about this parabola
that would help you graph it?
101
Pre­Calc 7 and 8 May 15, 2014
example
Write the equation for the parabolas with the following characteristics:
Focus (­4, 0), directrix x = 4
Vertex (0, 0), opens up, focal width 12
102
Pre­Calc 7 and 8 May 15, 2014
More examples for finding equations with (0,0) center
103
Pre­Calc 7 and 8 May 15, 2014
Assignment 8.1 day 1 p.587
#7­10 all, #11­19 odds
104
Pre­Calc 7 and 8 May 15, 2014
105
Pre­Calc 7 and 8 May 15, 2014
8.1 TOPICS
Parabolas with vertex
not at origin
106
Pre­Calc 7 and 8 May 15, 2014
What will the only difference for parabolas that
have a vertex at places other than (0, 0)?
107
Pre­Calc 7 and 8 May 15, 2014
DETAILS of a parabola with vertex at (h, k).
108
Pre­Calc 7 and 8 May 15, 2014
Example
Find the equation of the parabola with focus (­5, 3) and vertex (­5, 6).
Sketch the parabola to help you out!
109
Pre­Calc 7 and 8 May 15, 2014
Example
Find the equation of the parabola with focus (1, 3) and directrix x = ­3
Sketch the parabola to help you out!
110
Pre­Calc 7 and 8 May 15, 2014
Example
Find the vertex, focus and directrix and graph the parabola
y = 2x2 ­ 8x + 1
111
Pre­Calc 7 and 8 May 15, 2014
Example
Find the focus, vertex and directrix of the parabola
x2 ­ 8x ­ y + 5 = 0
112
Pre­Calc 7 and 8 May 15, 2014
Example
Find the focus, vertex and directrix
2y2 ­ 4x + 6y ­10 = 0
113
Pre­Calc 7 and 8 May 15, 2014
Assignment 8.1 day 2 p.587
#1­6 all, 21­29 odd, 49­56 all
114
Pre­Calc 7 and 8 May 15, 2014
Practice Questions for Quiz
1.
Solve the system of inequalities.
7x + 3y ≤ 210
3x + 7y ≤ 210
x + y ≥ 30
2.
Ted is about to take a history test consisting of matching questions worth
10
points each and essay questions worth 25 points each. He is required to do at least
3 matching questions, but time restricts doing more than
12. He must do at least 4
essays, but time restricts doing more than 15.
If Ted is required to answer at most
total 20 questions, how many of each should he answer to get the maximum score?
3.
Find the equation of the parabola in standard form that satisfies the
conditions.
a.
Vertex (0, 0)
b.
Focus (0, -3)
Focus (5, 0)
Directrix y = 3
given
115
Pre­Calc 7 and 8 May 15, 2014
116
Pre­Calc 7 and 8 May 15, 2014
117
Pre­Calc 7 and 8 May 15, 2014
Review of 8.1
1)
What is the equations for parabolas and when do you know which one to use?
2) Find the equation of the parabola with focus (­2, 1) and vertex (­2, 7).
3)
Find the focus, vertex and directrix
y2 ­ 4x + 16y ­8 = 0
118
Pre­Calc 7 and 8 May 15, 2014
8.2 TOPICS
Ellipses with center at origin
119
Pre­Calc 7 and 8 May 15, 2014
(longer) major
120
Pre­Calc 7 and 8 May 15, 2014
121
Pre­Calc 7 and 8 May 15, 2014
122
Pre­Calc 7 and 8 May 15, 2014
123
Pre­Calc 7 and 8 May 15, 2014
124
Pre­Calc 7 and 8 May 15, 2014
(0, B)
(x, y)
(A, 0)
(­A, 0)
(C, 0)
(­C, 0)
(0,­B)
125
Pre­Calc 7 and 8 May 15, 2014
(0,a)
(0, c)
(b, 0)
(0, ­b)
(0, ­c)
(0,­a)
126
Pre­Calc 7 and 8 May 15, 2014
Example
Find the equation of the ellipse with center at the origin, one vertex at (0, 5) and one focus at (0, 2).
Focus is at (0, 2) so c = 2
Vertex is at (0, 5) so a = 5
b2 = a2 ­ c2
so 52 ­ 22 = 21
127
Pre­Calc 7 and 8 May 15, 2014
Example
Find an equation of the ellipse with center at the origin, vertex at (4, 0) and minor axis 4 units long. Also, Find the coordinates of the foci.
128
Pre­Calc 7 and 8 May 15, 2014
Example
Sketch the graph of 4x2 + y2 = 64. State what the vertices are and what the foci are.
129
Pre­Calc 7 and 8 May 15, 2014
Questions to Practice for Wednesday's Quiz
Find an equation in vertex form for the parabola that satisfies the given conditions
2) Vertex (­4, 3), Focus (­4, 1)
1) Focus (5, 6), directrix x = ­2
2)
Given the parabola y = 3x2 ­ 12x + 17
a) Write the equation of the parabola in vertex form
b)
Identify the vertex, the focus and the directrix of the parabola
130
Pre­Calc 7 and 8 May 15, 2014
Assignment 8.2 day 1 p. 599
#1­6, 11­14, 21­28
131
Pre­Calc 7 and 8 May 15, 2014
132
Pre­Calc 7 and 8 May 15, 2014
Example
133
Pre­Calc 7 and 8 May 15, 2014
8.2 TOPICS
Ellipses with center moved from origin
134
Pre­Calc 7 and 8 May 15, 2014
Remember...(h, k) for circles and parabolas
135
Pre­Calc 7 and 8 May 15, 2014
Example
Sketch the given ellipse. Label the foci.
136
Pre­Calc 7 and 8 May 15, 2014
Example
Find the center, vertices, and foci of the ellipse:
137
Pre­Calc 7 and 8 May 15, 2014
Eccentricity comes from the adjective eccentric which
means off-center. The larger the e value, the more off
center the foci are so the more elongated the ellipse is.
The closer to 0 the e value is, the more circular the
ellipse.
138
Pre­Calc 7 and 8 May 15, 2014
Example
Write the equation for the ellipse.
Find the eccentricity.
Sketch the ellipse. Label the foci.
3x2 + 5y2 ­ 12x + 30y + 42 = 0
139
Pre­Calc 7 and 8 May 15, 2014
Example
Write the equation for the ellipse.
Find the eccentricity.
Sketch the ellipse. Label the foci.
2x2 + 3y2 +8x ­12y +2 = 0
140
Pre­Calc 7 and 8 May 15, 2014
Example
Find the vertices and the foci of the ellipse:
141
Pre­Calc 7 and 8 May 15, 2014
Example
142
Pre­Calc 7 and 8 May 15, 2014
Assignment 8.2 day 2 p.599
#15, 16, 31­37, 45, 47, 49
50, 53, 54, 56, 61, 62, 65­70
143
Pre­Calc 7 and 8 May 15, 2014
Questions to Practice for Wednesday's Quiz
Find an equation in vertex form for the parabola that satisfies the given conditions
2) Vertex (5, 2), Focus (­3, 2)
1) Focus (­3, 4), directrix y = 2
2)
Given the parabola y = 5x2 + 10x + 3
a) Write the equation of the parabola in vertex form
b)
Identify the vertex, the focus and the directrix of the parabola
3)
An ellipse has vertex (0, 5), focus (0, 3) and center at the origin. Find an equation of the ellipse.
4)
An ellipse has vertex at (­17, 0), focus at (­8, 0) and center at the origin. Find an equation of the ellipse.
144
Pre­Calc 7 and 8 May 15, 2014
Questions to Practice for Wednesday's Quiz
Find an equation in vertex form for the parabola that satisfies the given conditions
1) Focus (­3, 4), directrix y = 2
2) Vertex (5, 2), Focus (­3, 2)
2)
Given the parabola y = 5x2 + 10x + 3
a) Write the equation of the parabola in vertex form
b)
Identify the vertex, the focus and the directrix of the parabola
3)
An ellipse has vertex (0, 5), focus (0, 3) and center at the origin. Find an equation of the ellipse.
4)
An ellipse has vertex at (­17, 0), focus at (­8, 0) and center at the origin. Find an equation of the ellipse.
145
Pre­Calc 7 and 8 May 15, 2014
What do you remember from BEFORE Spring Break?????
146
Pre­Calc 7 and 8 May 15, 2014
solve systems of equations
solve with matrices
graph inequalities
graph and shade and solve linear programming problems
decomposition of fractions
parabola equation
ellipse equation
147
Pre­Calc 7 and 8 May 15, 2014
148
Pre­Calc 7 and 8 May 15, 2014
149
Pre­Calc 7 and 8 May 15, 2014
150
Pre­Calc 7 and 8 May 15, 2014
8.3 TOPICS
• Hyperbolas
151
Pre­Calc 7 and 8 May 15, 2014
152
Pre­Calc 7 and 8 May 15, 2014
153
Pre­Calc 7 and 8 May 15, 2014
Conjugate Axis - axis not crossed by the
hyperbola
Focal Axis or Transverse Axis
axis that is crossed by the
hyperbola
154
Pre­Calc 7 and 8 May 15, 2014
c2 = a2 + b2
Slope of Asymptotes = Δy/Δx
155
Pre­Calc 7 and 8 May 15, 2014
Slope of Asymptotes = Δy/Δx
156
Pre­Calc 7 and 8 May 15, 2014
Put it all together.
Which way does it face? How do you know?
What/where is the a?
What/where is the b?
What is the relation to c? Where is the c?
What is the conjugate axis? What does it's length tell us?
What are the formulas for the asymptotes?
157
Pre­Calc 7 and 8 May 15, 2014
Example
Find the vertices and foci of the hyperbola and sketch the hyperbola.
158
Pre­Calc 7 and 8 May 15, 2014
Example
159
Pre­Calc 7 and 8 May 15, 2014
Example
160
Pre­Calc 7 and 8 May 15, 2014
Example
Find an equation of the hyperbola in standard form that satisfies the given conditions
The slope of one asymptote is 3/4 and the transverse axis length is 16
161
Pre­Calc 7 and 8 May 15, 2014
Assignment 8.3 day 1 p.609
#1­6, 11­14, 23­26
162
Pre­Calc 7 and 8 May 15, 2014
163
Pre­Calc 7 and 8 May 15, 2014
8.3 TOPICS cont.
hyperbolas - that are not
centered at 0,0
What is hyperbola formula?
What is new formula?
What should you find first?
164
Pre­Calc 7 and 8 May 15, 2014
165
Pre­Calc 7 and 8 May 15, 2014
166
Pre­Calc 7 and 8 May 15, 2014
167
Pre­Calc 7 and 8 May 15, 2014
Put it all together.
Which way does it face? How do you know?
What/where is the a?
What/where is the b?
What is the relation to c? Where is the c?
What is the conjugate axis? What does it's length tell us?
What are the formulas for the asymptotes?
168
Pre­Calc 7 and 8 May 15, 2014
Example
169
Pre­Calc 7 and 8 May 15, 2014
170
Pre­Calc 7 and 8 May 15, 2014
Example
Find the equation of the hyperbola
Transverse axis endpoints (2, 3) and (2, ­1), conjugate axis length 6
171
Pre­Calc 7 and 8 May 15, 2014
Find the equation of the hyperbola
Transverse axis endpoints (2, 3) and (2, ­1), conjugate axis length 6
172
Pre­Calc 7 and 8 May 15, 2014
Example
Find the equation of the hyberbola
Transverse axis endpoints (­1, 3) and (5, 3), slope of one asymptote is 4/3
173
Pre­Calc 7 and 8 May 15, 2014
Example
Find the equation of the hyberbola
Transverse axis endpoints (­1, 3) and (5, 3), slope of one asymptote is 4/3
174
Pre­Calc 7 and 8 May 15, 2014
And again...
175
Pre­Calc 7 and 8 May 15, 2014
Assignment 8.3 day 2 p.609
#32­42 even, 47­52, 65­68
176
Pre­Calc 7 and 8 May 15, 2014
Practice for Friday's Quiz
Use the information provided to write the standard form of each ellipse
1. Foci: (2, ­1), (­6, ­1)
Endpoints of major axis: (3, ­1), (­7, ­1)
2.
Foci: (­2, 7), (­10, 7)
Endpoints of major axis: (­1, 7), (­11, 7)
3.
Endpoints of major axis: (20, ­2), (­4, ­2)
Endpoints of minor axis: (8, 2), (8, ­6)
4.
Vertices: (9, ­2), (­1, ­2)
Foci: (7, ­2), (1, ­2)
5.
4x2 + 25y2 + 48x ­ 150y ­ 31 = 0
6.
64x2 + 9y2 ­ 384x + 108y + 324 = 0
Identify the vertices and foci of each hyperbola then sketch a graph
7.
8.
177
Pre­Calc 7 and 8 May 15, 2014
Practice Answers for Friday's Quiz
1. +
3.
+
5.
7.
+
Vertices Foci
2.
+
4.
+
6.
8.
+
Vertices
Foci
178
Pre­Calc 7 and 8 May 15, 2014
CONIC REVIEW p.638
#21­35 odd
179
Pre­Calc 7 and 8 May 15, 2014
180
Pre­Calc 7 and 8 May 15, 2014
8.6 TOPICS
3-D Cartesian Coordinate System
• midpoint
• distance
• vectors
181
Pre­Calc 7 and 8 May 15, 2014
3-D Cartesian Coordinate System
182
Pre­Calc 7 and 8 May 15, 2014
183
Pre­Calc 7 and 8 May 15, 2014
Distance Formula
Midpoint
Example:
184
Pre­Calc 7 and 8 May 15, 2014
vectors
185
Pre­Calc 7 and 8 May 15, 2014
Example:
186
Pre­Calc 7 and 8 May 15, 2014
Assignment 8.6 p.635
#5,6,9,10, 23­32
187
Pre­Calc 7 and 8 May 15, 2014
Sphere...(like the circle)
188
Pre­Calc 7 and 8 May 15, 2014
Writing the Equation of a line in 3­D space
ro is like the b. It is the starting point
v is like the m. It is the component form between the two given points.
r is like the y, but since you are in 3­D it will be called r but it means <x, y, z>
You need to fill in the ro and the v. The r and t are part of the
equation.
189
Pre­Calc 7 and 8 May 15, 2014
Example:
solution
190
Pre­Calc 7 and 8 May 15, 2014
Example
191
Pre­Calc 7 and 8 May 15, 2014
192
Pre­Calc 7 and 8 May 15, 2014
WARM­UP
Solve each system (do at least one with substitution)
-4x + 4y = 20
2x - 6y = -18
5x + y = 17
-9x + 6y = -15
-10x - 6y = 0
-20x - 12y = 20
193
Pre­Calc 7 and 8 May 15, 2014
7.2 TOPICS
Introduction
to Matrices
No we are not watching the movie!
194
Pre­Calc 7 and 8 May 15, 2014
WARM­UP ~ aka Practice Quiz
Solve each system (do at least one by substitution)
x + 3y = -4
4x + 4y = 16
7x + y = -5
-8x - 4y = 20
3x - 3y = 0
12x - 12y = -12
A plane traveled 480 to Minneapolis and back. The trip there took 6 hours with
the wind. The trip back took 10 hours into the wind. Find the speed of the plane
in still air and the speed of the wind.
Lincoln is selling tickets to a play. On Thursday, the school sold 2 adult tickets
and 8 child tickets for $104. On Friday, the school sold 12 adult tickets and 7
child tickets for $173. What is the price of each ticket?
A = B =
C = 1. What is the order of matrix A? Matrix B? Matrix C?
2. What is b12? What is c31? What is a21?
3. Find the following: A + B
2A - 4B
AC
CA
AB
195
Pre­Calc 7 and 8 May 15, 2014
SOLUTIONS
1) (0, ­5) 2) (8, ­4) 3) No solutions
4) plane is 64 mph, wind is 16 mph
5) Adult tickets $8, child tickets $11
Matrices
1) a: 2x3 b: 2x3 c: 3x2
2) b12 = ­3, c31 = 0, a21 = 5
3) [
[
[
1 ­5 0
11 ­1 1
[[ [ [
2 8 6
­14 ­14 ­4
0 ­10
14 ­28
­21 7 4
7 ­7 2
­15 9 0
[ [
can't do
[
196
Pre­Calc 7 and 8 May 15, 2014
Example:
197
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.2 p.540
#23­33, 35, 36,
43, 44, 64­67
198
Pre­Calc 7 and 8 May 15, 2014
199
Pre­Calc 7 and 8 May 15, 2014
WARM­UP ~ aka Practice Quiz
1)
2)
Write a matrix equation for the system of equations
Solve each system
a)
-3x - y - 5z = 4
-4x + 3y - 5z = -4
5x - 2y + 5z = 7
c)
The arcade uses three different colored tokens for its game machines. For $20 you can
purchase any of the following mixtures of tokens: 14 gold, 20 silver and 24 bronze; OR
20 gold, 15 silver and 19 bronze; OR 30 gold, 5 silver and 13 bronze. What is the
monetary value of each token?.
d)
Last Tuesday, Regal Cinemas sold a total of 8500 movie tickets. Proceeds totaled
$64,600. Tickets can be bought in one of three ways: a matinee ticket costs $5, a
student ticket costs $6 and a general admission costs $8.50. How many of each type
were sold if twice as many student tickets were sold as matinee tickets?
b)
-4x + 2y - 4z = 12
-5x - 4y - 3z = -9
5x + 2y - 4z = 12
200
Pre­Calc 7 and 8 May 15, 2014
WARM­UP ~ aka Practice Quiz
Doctors have become increasingly concerned about the sodium intake in the US diet. Recommendations
by the American Medical Association indicate that most individuals should not exceed 2400 mg of
sodium per day.
Liz ate 1 slice of pizza, 1 serving of ice cream and 1 glass of soda for a total of 1030 mg of sodium.
David ate 3 slices of pizza and 2 glasses of soda for a total of 2420 mg of sodium. Melinda ate 2 slices
of pizza, 1 serving of ice cream and 2 glasses of soda for a total of 1910 mg of sodium. How much
sodium is in one serving of each item?
Set up a system of equations to solve this problem
a)
Set up a matrix equation to solve this problem
b)
Solve this problem
c)
Find the partial fraction decompositions of the following problems
b)
a)
201
Pre­Calc 7 and 8 May 15, 2014
What do the graphs of these two functions look like?
What do you think the graph of this function look likes?
202
Pre­Calc 7 and 8 May 15, 2014
Matching
C
1)
2)
A
3)
D
4)
5)
B
E
203
Pre­Calc 7 and 8 May 15, 2014
Assignment 7.4 p.563
#5, 6, 13, 15, 21, 22 ­­ fractions
#37­42 ­­ graphs
204