Download Prerequisites - Mr. Liu`s CCHS Classes

Prerequisites
What classes do you need to know:
Algebra 1
Geometry
Algebra 2
Precalculus is the combination of all previous mathematic classes
PREREQUISITE #1
REAL NUMBER
Review: Real Numbers
• Real number: Any number that can be written
as a decimal.
• Activity: Match the corresponding vocab to its
correct answer
– Integers
– Natural number(counting number)
– Whole number
{1,2,3…}
{0,1,2,3…}
{…-1,0,1…}
Answer:
– Integers
– Natural number(counting number)
– Whole number
{1,2,3…}
{0,1,2,3…}
{…-1,0,1…}
Review: Real Numbers
• What are the difference rational numbers and
irrational numbers?
• Activity: Identify which of these are rational
and irrational.
–
1
, 𝑒1 ,
11
3
5, -12, 1.75, 7.333… , 𝜋, 0, 8,
50
25
, log(6)
Answer
• Rational Number: number that either
terminates or infinitely repeating
–
1
,
11
3
-12, 1.75, 7.333… , 0, 8,
50
25
• Irrational Number: A number is infinitely
nonrepeating
– 𝜋, 𝑒 1 , 5, log(6)
New!!!
• {} this represent a set. It encloses the
elements or objects.
• Example: {0,1,2,3}
• Translation: This set includes the solution of
0,1,2,3
Review: Real Number
• Inequality ≥, ≤, <, >
• Activity: Match the symbol with the answer
•
•
•
•
a≥b
a≤b
a<b
a>b
a is less than b
a is greater than or equal to b
a is less than or equal to b
a is greater than b
Answer
•
•
•
•
a≥b
a≤b
a<b
a>b
a is less than b
a is greater than or equal to b
a is less than or equal to b
a is greater than b
Bounded Interval Example
Unbounded interval
• Graph the following:
– (3,∞)
– (-∞, 3)
– [3, ∞)
– (-∞, 3]
Properties of Algebra
Commutative property
Inverse property
Addition: a+b=b+a
Multiplication: ab=ba
Addition: a+(-a)=0
𝟏
Multiplication: a*𝒂=1 , a≠ 𝟎
Associative property
Distributive property
Addition: (a+b)+c = a+(b+c)
Multiplication: (ab)c=a(bc)
a(b+c)=ab+ac
(a+b)c = ac+bc
Identity property
Addition: a+0=a
Multiplication: a*1=a
a(b-c)=ab-ac
(a-b)c = ac-bc
Example:
• Expand: (a+5)8
• Simplify: 9p+ap
Answer
• Expand: (a+5)8
– 8a+40
• Simplify: 9p+ap
– p(9+a)
Exponential Notation
• 𝑎𝑛 = a*a*a*a*a…
• a = base, n is the exponent
• Example:
– 57 = 5 ∗ 5 ∗ 5 ∗ 5 ∗ 5 ∗ 5 ∗ 5
– (−3)6 = −3 ∗ −3 ∗ −3 ∗ −3 ∗ −3 ∗ −3
–
15
− (𝑐𝑎𝑟𝑒𝑓𝑢𝑙 𝑤𝑖𝑡ℎ 𝑡ℎ𝑖𝑠
2
– Why is this the case?
𝑜𝑛𝑒)=
−1
2
1
2
1
2
1
2
∗ ∗ ∗ ∗
1
2
Activity: Simplifying expressions
• −3𝑎𝑏 7 𝑐𝑑 −2 8𝑎−6 𝑏2 𝑐 −1 𝑑𝑑 3
•
𝑥𝑦 2 𝑧 −3
𝑥 −2 𝑦 3 𝑧 9
•
𝑥𝑦 2 𝑧 −3 4
( −2 3 9)
𝑥 𝑦 𝑧
Answer
• −3𝑎𝑏7 𝑐𝑑 −2 8𝑎−6 𝑏2 𝑐 −1 𝑑𝑑 3

•
𝑥𝑦 2 𝑧 −3
𝑥 −2 𝑦 3 𝑧 9

•
−24𝑏9 𝑑 2
(
)
5
𝑎
𝑥3
𝑦𝑧 6
𝑥𝑦 2 𝑧 −3 4
( −2 3 9 )
𝑥 𝑦 𝑧

𝑥 12
𝑦 4 𝑧 24
Scientific notation
• Scientific notation – related to chemistry where it
is written as the product of two factors in the
form a 10 n , where n is an integer and 1  a  10
• Activity: Convert the following into scientific
notation or expand them out
–
–
–
–
3.54 x 109
1.29 x 10−7
0.000000459
4,970,000
Answer
– 3.54 x 109
– 1.29 x 10−7
– 0.000000459
– 4,970,000
3,540,000,000
0.00000129
4.59 x 10−8
4.97 x 106
Homework Practice
• Pgs 11-12 # 2-44e, 48-54e, 58-64e
PREREQUISITE #2
CARTESIAN COORDINATE SYSTEM
• Given the recent unemployment rate reports
in California, describe the trend. What year
represent the biggest increase? Decrease?
What % can you predict about the future
unemployment rate? (Graph it and answer)
Year
Unemployment rate
2006
4.9%
2007
5.4%
2008
7.2%
2009
11.3%
2010
12.4%
2011
11.8%
2012
10.5%
Answer
• Unemployment rate increasing from year
2006-2010
• Unemployment decreasing from year 20102012
• 2008-2009 post the biggest increase
• 2011-2012 post the biggest decrease
• 10%-12% unemployment rate
• Scatter plot: plotting the (x,y) data pair on a
cartesian plane
• The previous question we just did is an
example of a scatter plot
What do you need to graph?
• Coordinates
– (X,Y), (input, output)
– Example (3,-2)
• Things to keep in mind:
– Always label!!!
Quick Talk: (30 second)
• What is absolute value?
Answer
• Absolute Value: Always positive, tells distance.
• Magnitude: size or distance
Activity
• l-4l =
• l16-7l=
• l9-27l =
Answer
• 4
• 9
• 18
Quick talk: How can you find the
distance between two places/points?
Answer:
• Measure
• Use distance formula
Distance Formula
• Distance Formula: It is derived from the
Pythagorean theorem.
D=
𝑥1 − 𝑥2
2
+ 𝑦1 − 𝑦2
2
Activity:
• Distance between (2,-5) and (-7,3)
• Distance between (-1,-7) and (-6, 8)
Midpoint Formula
• Midpoint Formula: A formula to find the
middle of the two points.
𝑥1 +𝑥2 𝑦1 +𝑦2
M=(
,
)
2
2
Activity:
• Find the midpoint between (6,8) and (-4,-10)
• Find the midpoint between
1 3
5 7
(- , )𝑎𝑛𝑑 ( , )
2 4
6 8
Review: Geometry Circle
• Standard form of a circle:
𝑥−ℎ
2
+ 𝑦−𝑘
2
= 𝑟2
– (h,k) is the center of the circle
– r= the radius of the circle
• Example: 𝑥 + 2
2
– (-2,9) is the center
– 5 is the radius
+ 𝑦 − 9 = 25
Quick talk: How do you find the
distance from the center to a point on
a circle?
Answer
• By finding the radius
• Use distance formula
Homework Practice
• Pgs 20-22 #5, 8, 9, 11, 13, 21, 23, 27, 31, 35,
39, 43, 49, 51, 55, 57
PREREQUISITE #3
SOLVING LINEAR EQUATIONS AND
INEQUALITIES
• How do you know if an equation/inequality is
linear?
Answer
• If the highest power or the highest exponent
is 1
Linear Equation
• 𝑦 = 𝑚𝑥 + 𝑏
• Note: the highest power (exponent) is one
Solving linear equation
• Solve for x
• 2 3𝑥 − 4 + 4 𝑥 + 1 = 8𝑥 + 3
Solve for y
•
6𝑦−1
8
=3+
y
4
Solving inequality
• Remember solving inequality is like solving an
equation.
• There are 3 ways of representing an answer
– Example:
• X>3
• 3, ∞
• graphically
Solve
• 2 𝑥 + 9 + 7 ≥ 5𝑥 + 16
Solve
•
𝑧
4
+
1
2
<
𝑧
5
−3
Solving a sandwich
• −7 <
2𝑥+15
3
<9
Note: Make 2 separate inequalities
Homework Practice
• Pgs 29-30 #3, 4-10, 17, 19, 25, 27, 39, 45, 51,
53, 55, 57
PREREQUISITE #4
LINES IN THE PLANE
Quick Talk: How do you find slope?
Answer
• You find slope by taking the difference of the y
divided by the difference of the x
• Formula for finding slope:
•
𝑦1 −𝑦2
𝑥1 −𝑥2
or
∆𝑦
∆𝑥
• ∆= 𝑑𝑒𝑙𝑡𝑎, 𝑖𝑡 𝑚𝑒𝑎𝑛𝑠 "𝑐ℎ𝑎𝑛𝑔𝑒𝑠” or
“differences”
Activity
• Find slope
(-2,5) and (9,0)
• (1,1) and (3,-4)
Important!!
• You can ALWAYS find an equation of a line
when you are given 2 points.
• What is the way of finding the equation of the
line?
Answer: Point-slope form
• Point-slope form
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
m=slope
A point on the line (𝑥1 , 𝑦1 )
Example:
4
𝑦+8= −
𝑥−2
3
Slope = -4/3
Point on the line = (2,-8)
Slope intercept form
• Slope-intercept form
𝑦 = 𝑚𝑥 + 𝑏
𝑚 = 𝑠𝑙𝑜𝑝𝑒
𝑏 = 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
Example:
Y=-3x+8
m=-3
b=8
y-intercept coordinate= (0,8)
Using calculator
• Graph the following:
1) 2x+3y=6
2) 𝑦 =
−4
( )𝑥
3
+9
Quick talk: The difference between
parallel and perpendicular lines
Answer
• Parallel Lines
– Slopes are the same, but different y-intercept
– Example:
– 𝑦 = 3𝑥 − 5 𝑦 = 3𝑥 + 98
• Perpendicular lines
– Slopes are opposite reciprocal.
– The intersect form 90 degree angle.
– Example:
– 𝑦 = 9𝑥 + 8 𝑦 = (−1/9) + 61
Solve it with your group raise your
hand when your group has an answer
• (3,8) and (4,2)
• 1) Give me an equation parallel to those
points
• 2) Give me an equation perpendicular to
those points
Answer
• 1) m=-6
• 2) m=1/6
Cracking word problems
1)
2)
3)
4)
Know what you are solving for
Gather facts about the problem
Identify variables
Solve
Ultimate problem
• In Mr. Liu’s dream, he purchased a 2014
Nissan GT-R Track Edition for $120,000. The
car depreciates on average of $8,000 a year.
1) Write an equation to represent this situation
2) In how many years will the car be worth
nothing.
Answer
1) y=price of car, x=years
y= −8000𝑥 + 120000
2) When the car is worth nothing y=0
X=15, so in 15 years, the car will be worth
nothing.
Ultimate problem do it in your group
(based on 2011 study)
• When you graduate from high school, the starting
median pay is $33,176. If you pursue a professional
degree (usually you have to be in school for 12 years
after high school), your starting median pay is $86,580.
• 1) Write an equation of a line relating median income
to years in school.
• 2) If you decide to pursue a bachelor’s degree (4 years
after high school), what is your potential starting
median income?
Answer
• 1) y=median income, x=years in school
Equation: y= 4450.33x+33176
2) Since x=4, y=50,977.32
My potential median income is $50,977.32 after
4 years of school.
Homework Practice
• Pgs 40-42 #1, 5, 9, 13, 19, 23, 27, 35, 37, 43,
45, 53, 57, 59
PREREQUISITE #5
SOLVING EQUATIONS GRAPHICALLY,
NUMERICALLY AND ALGEBRAICALLY
Quick Talk:How do you know if you
have a quadratic?
Answer
• When the highest power or highest exponent
is 2
• Example: 𝑦 = 𝑥 2 − 6𝑥 + 9
What are the different ways of solving
quadratic?
•
•
•
•
1) Factor (very important)
2)Quadratic formula
3) Completing the square
4) Graphing using calculator
Solve using factor
• 2𝑥 2 − 3𝑥 − 2 = 0
Solve using quadratic equation
• Quadratic equation: 𝑥 =
• Solve 2𝑥 2 − 3𝑥 − 2 = 0
−𝑏± 𝑏2 −4𝑎𝑐
2𝑎
Completing the square
• Why is completing the square useful? It is useful
because it is in vertex form. It makes it easy to graph.
• Completing the square:
• 𝑥 2 + 𝑏𝑥 +
•
𝑥−
𝑏 2
2
𝑏 2
2
=𝑐+
=𝑐+
𝑏 2
2
𝑏2
4
• Example: 4𝑥 2 + 20𝑥 + 17 = 0
Solve by graphing calculator
• 4𝑥 2 + 20𝑥 + 17 = 0
Those were cake…now for the fun stuff
• In your group, talk about which ways can you
use to solve for the following and actually
solve them.
• 𝑥 3 − 2𝑥 − 1 = 0
• 𝑥 4 + 3𝑥 3 − 2𝑥 + 5 = 0
Answer
• Graphing calculator
• P/Q way (remainder theorem)
How do you solve for absolute value?
• l2x-7l=10
Homework practice
• Pgs 50 #1-43odd
PREREQUISITE #6
COMPLEX NUMBERS
Quick Talk:
• When you use the quadratic formula, you
have a radical 𝑏 2 − 4𝑎𝑐.
• What if 𝑏 2 − 4𝑎𝑐 > 0?
• What if 𝑏 2 − 4𝑎𝑐 = 0?
• What if 𝑏 2 − 4𝑎𝑐 < 0?
Answer
• 2 real solutions
• 1 real solution
• 2 imaginary solution or 0 real solution
What is imaginary number?
•
Imaginary number is when you deal with 𝑖
•
Very important to know:
−1 = 𝑖
𝑖1 = 𝑖
𝑖 2 = −1
𝑖 3 = −𝑖
𝑖4 = 1
Remember, it is a cycle of 4. It just repeats itself.
Example 1: Find 𝑖 58
Example 2: Find 𝑖 97
Example 3: Simplify −81
Example 4: Simplify −24
Answer
58
•
= 14 𝑅 2
4
So that means it completed the cycle 14 times, with 2 left
over. So the answer is 𝑖 2 = −1
97
•
= 24 𝑅 1
4
So that means it completed the cycle 24 times, with 1 left
over. So the answer is 𝑖 1 = 𝑖
• 9𝑖
• 2𝑖 6
Complex number: where real meets
imaginary
• 𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝑁𝑢𝑚𝑏𝑒𝑟 𝑓𝑜𝑟𝑚: 𝑎 + 𝑏𝑖
• 𝑎 = 𝑟𝑒𝑎𝑙 𝑝𝑎𝑟𝑡
𝑏 = 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 part
Remember all algebraic operations are
the same!!! It doesn’t change
• Example:
7 − 4𝑖 + −3 + 17𝑖 =
• Example:
2 − 𝑖 − 9 + 21𝑖 =
Answer
• 4 + 13𝑖
• −7 − 22𝑖
Multiply complex numbers
• 7 − 4𝑖 ∗ −3 + 17𝑖 =
• 2 − 𝑖 ∗ 9 + 21𝑖 =
Answer
• −21 + 119𝑖 + 12𝑖 + 68 = 47 + 131𝑖
• 18 + 42𝑖 − 9𝑖 + 21 = 39 + 33𝑖
Dividing complex number
• Important to note: you can NEVER have the
denominator in imaginary form or radical.
• If the denominator is in imaginary form, you
have to multiply the numerator and
denominator by its conjugate.
Example:
if you have 𝑎 + 𝑏𝑖, the conjugate is 𝑎 − 𝑏𝑖
Finding conjugates
• 5−𝑖 =
• 3 + 2𝑖 =
• 10 − 𝑖 =
Answer
• 5+𝑖
• 3 − 2𝑖
• 10 + 𝑖
Simplify
• Simplify
1
5+2𝑖
Answer
• Since the denominator is 5 + 2𝑖, the
conjugate is 5 − 2𝑖. You have to multiply the
top and bottom by 5 − 2𝑖. So the answer is
5−2𝑖
29
Simplify
•
3
2−𝑖
•
5+𝑖
2−3𝑖
=
=
Answer
•
6+3𝑖
5
•
7+17𝑖
13
Homework Practice
• Pgs 57-58 #1-43odd
PREREQUISITE #7
SOLVING INEQUALITIES ALGEBRAICALLY
AND GRAPHICALLY
Solve (show all types of answers)
• 𝑥 − 8 > 18
Solve (show all types of answers)
• 2𝑥 − 9 ≤ 15
Solve (show all types of answers)
• 2𝑥 2 + 7𝑥 − 15 ≥ 0
Solve (show all types of answers)
• 2𝑥 2 + 3𝑥 < 20
Solve (show all types of answers)
• 𝑥 2 + 2𝑥 + 2 < 0
Solve (show all types of answers)
• 𝑥 3 + 2𝑥 2 − 1 ≥ 0
Homework Practice
• Pgs 64 #1-29 odd, 33, 37