Prerequisites

What classes do you need to know:Algebra 1GeometryAlgebra 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 {1,2,3…}– Natural number(counting number) {0,1,2,3…}– Whole number {…-1,0,1…}

Answer:

– Integers {1,2,3…}– Natural number(counting number) {0,1,2,3…}– Whole number {…-1,0,1…}

Review: Real Numbers

• What are the difference rational numbers and irrational numbers?

• Activity: Identify which of these are rational and irrational.– , -12, 1.75, 7.333… , ,

Answer

• Rational Number: number that either terminates or infinitely repeating– , -12, 1.75, 7.333…

• Irrational Number: A number is infinitely nonrepeating– , ,

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

• ab a is less than b

• ab a is greater than or equal to b• ab a is less than or equal to b• ab a is greater than b

Answer

• ab a is less than b

• ab a is greater than or equal to b

• ab a is less than or equal to b

• ab a is greater than b

Bounded Interval Example

Unbounded interval

• Graph the following:– (3,

– (-

– [3,

– (-

Properties of AlgebraCommutative property

Addition: a+b=b+aMultiplication: ab=ba

Inverse property

Addition: a+(-a)=0Multiplication: a*=1 , a

Associative property

Addition: (a+b)+c = a+(b+c)Multiplication: (ab)c=a(bc)

Distributive property

a(b+c)=ab+ac a(b-c)=ab-ac (a+b)c = ac+bc (a-b)c = ac-bc

Identity property

Addition: a+0=aMultiplication: a*1=a

Example:

• Expand: (a+5)8

• Simplify: 9p+ap

Answer

• Expand: (a+5)8– 8a+40

• Simplify: 9p+ap– p(9+a)

Exponential Notation

• = a*a*a*a*a…• a = base, n is the exponent

• Example:

– (– =

– Why is this the case?

Activity: Simplifying expressions

Answer

)

Scientific notation• Scientific notation – related to chemistry where it is written

as the product of two factors in the form , where n is an integer and

• Activity: Convert the following into scientific notation or expand them out

– 3.54 x – 1.29 x – 0.000000459– 4,970,000

na 10 101 a

Answer

– 3.54 x 3,540,000,000– 1.29 x 0.00000129– 0.000000459 4.59 x – 4,970,000 4.97 x

Homework Practice

• Pgs 11-12 # 2-44e, 48-54e, 58-64e

PREREQUISITE #2CARTESIAN 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 rate2006 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 2010-2012

• 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=

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.

M=( )

Activity:

• Find the midpoint between (6,8) and (-4,-10)

• Find the midpoint between (-

Review: Geometry Circle

• Standard form of a circle:

– (h,k) is the center of the circle– r= the radius of the circle

• Example: – (-2,9) is the center– 5 is the radius

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 #3SOLVING 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

• Note: the highest power (exponent) is one

Solving linear equation

• Solve for x

Solve for y

6 𝑦−18

=3+( y4 )

Solving inequality

• Remember solving inequality is like solving an equation.

• There are 3 ways of representing an answer– Example:• X>3

• graphically

Solve

Solve

( 𝑧4 )+( 12 )<( 𝑧5 )−3

Solving a sandwich

• Note: Make 2 separate inequalities

Homework Practice

• Pgs 29-30 #3, 4-10, 17, 19, 25, 27, 39, 45, 51, 53, 55, 57

PREREQUISITE #4LINES 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:• or

• ” 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

m=slopeA point on the line (

Example:

Slope = -4/3Point on the line = (2,-8)

Slope intercept form

• Slope-intercept form

Example:Y=-3x+8m=-3b=8 y-intercept coordinate= (0,8)

Using calculator

• Graph the following:

1) 2x+3y=6 2)

Quick talk: The difference between parallel and perpendicular lines

Answer

• Parallel Lines– Slopes are the same, but different y-intercept– Example:

• Perpendicular lines– Slopes are opposite reciprocal.– The intersect form 90 degree angle.– Example:–

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) Know what you are solving for2) Gather facts about the problem3) Identify variables4) 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=yearsy

2) When the car is worth nothing y=0X=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 schoolEquation: y= 4450.33x+33176

2) Since x=4, y=50,977.32My 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 #5SOLVING 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:

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

2 𝑥2−3 𝑥−2=0

Solve using quadratic equation

• Quadratic equation:

• Solve

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:

• Example:

Solve by graphing calculator

4 𝑥2+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.

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 #6COMPLEX NUMBERS

Quick Talk:

• When you use the quadratic formula, you have a radical .

• What if ?

• What if ?

• What if ?

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

• Very important to know:

Remember, it is a cycle of 4. It just repeats itself.

Example 1: Find Example 2: Find Example 3: Simplify Example 4: Simplify

Answer

So that means it completed the cycle 14 times, with 2 left over. So the answer is

So that means it completed the cycle 24 times, with 1 left over. So the answer is

Complex number: where real meets imaginary

• part

Remember all algebraic operations are the same!!! It doesn’t change

• Example:

• Example:

Answer

Multiply complex numbers

Answer

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 the conjugate is

Finding conjugates

Answer

Simplify

• Simplify

Answer

• Since the denominator is the conjugate is You have to multiply the top and bottom by . So the answer is

Simplify

Answer

Homework Practice

• Pgs 57-58 #1-43odd

PREREQUISITE #7SOLVING INEQUALITIES ALGEBRAICALLY AND GRAPHICALLY

Solve (show all types of answers)

|𝑥−8|>18

Solve (show all types of answers)

|2 𝑥−9|≤15

Solve (show all types of answers)

2 𝑥2+7 𝑥−15≥0

Solve (show all types of answers)

2 𝑥2+3 𝑥<20

Solve (show all types of answers)

𝑥2+2𝑥+2<0

Solve (show all types of answers)

𝑥3+2𝑥2−1≥0

Homework Practice

• Pgs 64 #1-29 odd, 33, 37