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CP Math three

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FC: Come cheek out what we learned this year in Mrs. Johnson's math class!!!

1: UNIT 1

2: SOLVE MULTIVARIABLE EQUATIONS ~ If you have more then one variable in an equations then you should first determine what variable you want to solve. ~ After you determine what you want to solve you will need to get it all by itself. ~ You can do that by adding subtracting multiplying or dividing. You should "undo" the equation. ~ You undo it by doing all the steps that you would do last to first. 5d = rt + 1 - 1 (5d - 1)/r = rt/r so (5d-1)/r=t | Examples

3: Recognize relationship between two variables as direct or inverse & Identify how one change in one variable affects the other. ~You may use A= b/c or a=b*c ~Replace b and c with any number and then if you change b find out if a will be opposite or the same. ~ If the reaction is the same it is a direct change and if it is opposite then it is inverse ~ As you change b or c describe what happens to a and explain if it increases or decreases. If A=b/c and you will fill in numbers for c. So a is unknown and b= 12 and c= 3 so a would be 4. If we make c be 6 then a will be 2. This reaction is inverse. But if we change b to a higher number then A will be higher. This is direct Wen A=b*c then if you increase b or c and if you decrease b or c then the effect of a is direct We know this because if b is 7 and c is 5 then a is 35 but if you increase b to 9 then it is 45. Also if you decrease c to 2 then a= 14. | EXAMPLE

4: Identify and solve for missing side and angle in special right triangles Always remember sohcahtoa which stands for sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. You will always use a^2+b^2=c^2 when dealing with soshcahtoa. Label your right triangle with a b and c, c will always be the hypotenuse which is the longest side. Then write what you know for each side and angles. Decide what you need to find out and use the equation to find it. | EXAMPLE

5: Find missing sides and angles in non right triangles using Law of Sines and Law of Cosine. c^2=a^2+b^2-2abcos(c) These are the equations you will be using. First determine what you know and fill in all the sides and angles that you already know. If all three sides are known or two sides and an angle in between then use the law of cosine. If two sides and one angle are known or two angles and one side then use law of sines. Plug in numbers and solve. Use the fish rule for law of sines. | EXAMPLES next page!!!

6: A = 20 + 13 2(20)(13)cos(66) A = 400 + 169 520cos(66) A = 569 211 A = 358

7: Compare multiple linear equations modeling real world scenarios to find the “best deal” Write all equations in th y equals format. Solve for y and x and graph the lines on the same table. Find what one has the better deal and at what points. Cell Phone Unlimited has a deal where you can pay a monthy cost for $55 flat for unlimited texts and calls, and Call Fun says that if you pay $30 a month and pay .05 cents for each text and call. If you have the cell phone service for 2 months and have called and text 534 times, which business would be cheaper? CPU: C=55x, CF: C=30x+.05z C=cost & x=month & z=text and call CPU: 55(2)=110=C, CF: 30(2)+.05(534)=86.7=C Fun city would be the better deal at $86.60

8: Solve systems of equations using: graphing and elimination To graph: graph each line on the same graph Find the point where both graphs intersect and write the solution as a point To eliminate: Add like terms together and solve for y or x. you will solve for y or x by adding the two equations together. Put the value of x or y into the equation and solve for the other number. Write the two numbers you got as a point. Examples:

9: Solve inequalities and plot the solution on a number line: Linear, Double, and fractional Linear- Plot the x value on the number line, 2.If the sign is or then the brackets are [], if the sign is < or > then the brackets are (). and then shade in the number line according to the correct place. Double: solve so you will have only one unknown and two inequalities. solve for x and y as you normally would. Put appropriate brackets and then plot on number line. Then shade Fractional: Multiply by the denominator to get rid of it. Then you will solve like a normal equation. Remember that if you change the negative sign to change the way the inequality goes. (6+4a)/5-4 (6+4a)/5*5-4*5 multiply by 5 6+4a-20 subtract 6 6+4a-6-20-6 4a/4-26/4 a-6.5

10: Linear Programming 1.Decide what object will be x and y 2.Write any constraints for x and y 3.Write the maximum constraints 4.Write total amount of each needed 5.Write the profit equation using the last part of the story 6.Graph all inequalities on the same graph 7.Test each corner or intersections where two lines cross in the FR and plug and chug until you find which set creates the most profit 8.In the area where they can receive the most profit, shade in everywhere else, and label that area the “Feasible Region” or “FR”

11: A company makes two products (X and Y) using two machines (A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B.At the start of the current week there are 30 units of X and 90 units of Y in stock. Available processing time on machine A is supposed to be 40 hours and on machine B is supposed to be 35 hours.The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95 units. Company policy is to maximise the combined sum of the units of X and the units of Y in stock at the end of the week.

12: Convert from Inequality Notation to Interval Notation 1.The lesser number or unknown goes on the left side of the point 2.If the sign is or then the bracket on the known number is [, but never put a closed bracket on infinity 3.If x is less than, then it goes on the left side with an open bracket ( and written as this symbol: which is infinity 4.If x is greater than, then it goes on the right side with an open bracket ) and written as 5.Whether the number is greater than or less than, put on the other side determined by the unknown number, with either a [] or () depending on the inequality sign.

13: UNIT 3 A

14: Functions: To determine whether or not a line is a function the rule is to do the vertical line test. In other words just take a ruler or a piece of paper and go from left to right on the graph, is there a point on the graph where there is more than one value for y on one value for x? And its that simple! thats it!

15: Domains and ranges: practical Domain: All x values possible for the equation. practical Range: All y values possible for the equation. Theoretical Range: All real numbers x values theoretical Domain: All real numbers y values

16: Piecewise Functions: The rule for the function depends of a set of interval so graph these functions you need to find the end points. since the y values for all x values between 2 and 10 are determined by the equation 2x-12 we will plug in 2 for x and get (2,-8) We will also plug in 10 for x and get (10,8) for 3: (-7,3) and (2,3) for 2x-12 (2,-8) and (10,8) If -7 < x < 2 If 2< x < 10 To see if something is a function from a piecewise function graph do the vertical line test

17: Simplifying and Standard Form - Standard form is putting the equation from largest exponent to smallest exponent. To simplify just add all the x and y values and combine until you have the simplest form. The degree of a polynomial is the largest exponent. With the equation above the degree is 6. 5x 7x 3x 7x 1x 9x The degree of a polynomial is the largest exponent. With the equation above the degree is 6.

18: Combining and Composition- To combine functions set them equal to each other and continue with the problem. Example: Profit for a recording company is also a function of sales. The label had the following conditions. Studio, video, touring, and promo expenses: 365,000 Pressing and packaging : 3.75 per CD Discounts for stores: 1.50 per CD Other discounts: 1.65 per CD Final combo: -356,000-3.75x-1.5x-1.65x= 15x Composition of functions (x)=-2x-1 g(x)=3x h(x)=x +1 1. f(g(x))=-2(3x)-1 The composition of f(x) will be written first, then where it has an x is where the composition of g(x) is. Then complete f(x).

19: Inverse- The inverse is simply the original function being solved for a different variable. To determine if the inverse of a function is a function do a horizontal line test. One-to-One is where both the function and the inverse are functions If there is a point where there are two x values for one y value the inverse in not a function F=9/5C+32 F-32=9/5C+32-32 5/9(F-32)=5/9(9/5C) Inverse of F=9/5C+32 is C=5/9(F-32)

20: Commutative, Associative, Distributive Commutative: When the numbers of the equation are switched. a+b+c = b+a+c Associative: When the order of the parenthesis is switched (a+b) +c + a + (b+c) (a*b)c= a(b*c) Distributive: When you multiply and distribute a number throughout the equation. A(b+c) +ab +ac a(b-c) = ab - ac


22: Unit 3b

23: Polynomials- Zeros: factored form: f(x)=x(x=7)(x-9) Zeros are -7 and 9, they are opposite of what is in the parenthesis from factored form. Also Ignore the x outside of the parenthesis because it does not matter at this point. When Given Zeros- If(2)=0 and f(8)=0 the formula is- F(x)=(x=zero)(x=zero) example f(x-)=(x-2)(x-8)

24: Foil: l (3x-y) (3x+y) In order to foil you need to multiply every number or variable by each other. So we will multiply 3x by 3x and 3x by positive y. Then we will multiply negative y by 3x and –y by positive y. Use the equation (a+b)(c+d) = ac+ad+bc+bd

25: Factoring Polynomials/ grouping: Plot- Product sum chart, When you have a polynomial that looks like ax2 + bx + c you need to know what ax2 times c is. When you get that answer it goes in the product part of your chart. Then you need to know the factors that add up to be bx. When you get it your equation should look like ax2 + one # you got *x + the other # you got*x + c. Then to group then do (ax2+#x ) * (#x+c) then take out like terms and numbers from each parenthesis and place them in their own parenthesis. You should get the same “left overs” in each one so you will multiply the left overs by the numbers taken out.

26: Identify the hypothesis and conclusion of a conjecture The hypothesis of a conjecture is everything after the if and before the then. The conclusion of a conjecture is everything after the then If it is raining outside then the side walk is wet. If a and b are any positive odd integers, then the sum of a and b will be even | conjecture

27: Quadratic Formula: Use the quadratic formula when you want to find zeros. The equation is x=(-b + or - square root of (b^2-4ac))/2a Use it by plugging in the numbers from the equation ax^2+ bx + c so x^2 + 3x -4 is

28: UNIT 4

29: Recognize the differences between inductive and deductive reasoning. deductive reasoning You can prove an conjecture is true inductive reasoning you CAN NOT prove a conjecture is true If it is raining outside then the side walk is wet. If a and b are any positive odd integers, then the sum of a and b will be even 3+5 | EXAMPLE

31: proof | Theorems | Transversal | Parallel | supplementary | linear pairs | congruent | Transitive | Vertical angle theorems and angle relationship theorems: If two angles are vertical angles then they are congruent. As you see in the example a and b are equal to each other because they are vertical angles. VERTICAL ANGLES: Are angles opposite of each other. B=19

32: Angle relationship theorems that result from two parallel lines being cut by a transversal. First find things in common with the sides, angles and triangles. then decide what the reason is for them being in common.

33: Know and use the triangle similarity and congruence theorems. Angle Angle Similarity- If three angles of one triangle are congruent to three angles of another triangle Side Side Side similarity- If 3 sides of one triangle are in proportion to 3 sides of the other triangle. Side Angle Side 2 sides are in proportion to other triangle. corresponding included angle of both is congruent Side side side congruence theorem - If all sides of both are equal. Side angle Side if 2 sides are congruent to the other triangle Angle side angle- both triangles angles and a side are the same.

34: To prove a conjecture is false.... FInd one thing that is a counter example for it. So one angle that doesn't match or a side. EXAMPLE!!! ASS

35: Converse of an if then statement. Switch the if and then parts but keep the if in front and then after. EXAMPLE!!! Before converse: If it is raining then the ground is wet. Converse: If the ground is wet then it is raining.

36: Central and inscribed angles of a circle Inscribed angle is an angle formed by two cords. Its vertex is on the circle. Measure of it is half of measure of arc. Central angle- and angle in a circle between two radii. the measure of a central angle is the measure of its arc. The angle is twice the inscribed angle.

37: FInding the arc length and area of the sector ARC LENGTH...

38: UNIT 5

39: STANDERD DEVIATION | To use with out calculator use formula To input into a calculator go to stat edit hit enter enter your data quit stat over to calc varstats enter twice!!! Standard deviation = amount of variation from the mean i.e how far you can go away from it. Example 4, 6, 8, 2, 4, 1, 3,3, 5, 7, 9 it will be 2.41

40: calculate z scores and percentiles: Use percentiles when you are trying to find how far away your standard deviation is from the mean and the percent to go with it. So if you need 2 standard deviations away use the table for the percent. For z scores to find use x- mean divided by the standard deviation. Then use that to look up on the chart.

41: Mutually exclusive events = events that do not depend on the first one in order to get the second. Ex. Rolling a dice because it won’t effect what you got on the first one. Addition rule: If you need the probabilities and it says or use addition by adding the fractions straight across. 3/4 + 1/2= 4/6 so 2/3

42: Unit 6

43: Real world situations for equations: Linear- a game going on forever= constant rate of zero, the heat of the marshmallow melting power- Pop fly, trowing the ball Inverse Power- ground ball= its speed Exponential- Getting better at baseball, making friends Trigonometric- running around the bases, ferris wheel

44: Graphs!!! Linear= positive, negative, zero and undefined slopes. Equation is mx+b Power = Y=ax^n n= even then graph smiles=) n=odd graph frowns a=positive then graph goes from left to right a=negative it is opposite. Inverse Power= Y=ax^n (n= -) or Y=a/x^n (n=+) Exponential- Y=ab^x a=starting value b=rate of change x=time Trigonometric= (sine, cos) Y=a*sin(bx)+c a=amplitude b= 2pi/period c= virtical shift sin graphs don't start at the max or min. to get a= max-min/2

45: Function Rule and vertical transformations for sine and Cos graphs!!! Y=a*sin(bx)+c Shifts upward when: c is positive Shifts downward when: C is negative B is the length of one complete cycle If you don't have A add it to the center of the graph and now you know the max. Then subtract A from the center to get the min.

46: EXAMPLES!!!!!!!!!!!!! 3cos(pi/4x)+2 a= 3 b=(pi/4x) c=2

47: Additional info

48: Math is interesting! The diameter of a circle is the length across the circle The ratio to a circles circumference to the diameter it is 3.14 Pi is an irrational number because it cannot be written as a simple fraction or as an exact decimal. Johann Heinrich Lambert came up with pi as irrational number. pi is transcendental and it means a class of irrational numbers.

49: Pie poem crispy,warm, sweet yummy smelling an apple pie is the home sweet home feeling. memories of making pies with my sisters and grandpa old days, but yet days never to forget. Thanksgiving day filled with pie being so stuffed but eating another delicious slice pumpkin pie in the fall and winter pudding pie to cool you off in the summer oh how fruity some pies can be they are so tart and tangy in my mouth yummy and delicious peacefully resting in my tummy pie altogether is satisfying in every way possible i want some pie!!!

50: The most memorable thing we did...... was when we had pi day because i have never done anything like it before and it was so much fun! I loved class this year altogether and will never forget the things i learned in CP math three!!! | CYA!!! | life lessons

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Ariana Tindall
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