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The Binomial Probability Theorem: There is no objective. Binomial theorem is  n C r (probability of success)^r  * (probability of failure)^n-r Binomial Probability Theorem example: In the old days, there was a probability of  \displaystyle{0.8} 0 . 8  of success in any attempt to make a telephone call.  \displaystyle{7} Calculate the probability of having  7  successes in 10 attempts. To calculate this all I need to do is plug this into the calculator as 10 C 7 (7/10)^7 * (3/10)^3 and get .266827932 then move the decimal place back two and round to get 27%. One misconception I had was I did not know when to use nPr or nCr. I figured it out when I asked a friend and they explained to me that nPr is used when the problem has to be in order and to use nCr when there is no specific order that you have to use. EQ: Give an example: There is a bag of candy that you want at the store. The bag contains 10 snickers, 12 kit kats, 10 m...

Empirical Rule:  There is no objective Empirical Rule Example: When using the empirical rule the problem has to call for normal distribution or else you can  not use the empirical rule. There are three numbers you have to remember for empirical rule those numbers are; 68%, 95%, and 99.7%. These numbers represent the intervals of information that is included in these margins. The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives  16 1 6  years; the standard deviation is  1.7 1 . 7  years. Use empirical rule to estimate the probability of a gorilla living longer than 14.3 years?  In order to find   the probability I would need to make a normal distribution table and place 16 years as the mean and add and subtract 1.7 as intervals. The numbers I would use would be 9.2, 10.9, 12.6, 14.3, 16, 17.7, 19.4, 21.1, 22.8. Now that I know the numbers I can figure out the percentage. The number 14.3 ...

Recursive and Explicit forms: There is no objective Recursive Example:  Find the fifth term of a recursive define sequence a(1)=1 a(n)=a(n-1)+2n+1 where n is greater than or equal to two. a(1)=1 a(2)=a(2-1)+2(2)+1=a(1)+4+1=1+5=6 a(3)=a(3-1)+2(3)+1=6/a(2)+6+1=13 a(4)=a(4-1)+2(4)+1=13/a(3)+8+1=22 a(5)=a(5-1)+2(5)+1=22/a(4)+10+1=33 33 is the fifth term Explicit Example: Find the first four terms of the sequence given by a(n)=2n(-1)^n         ______            _______            _______           _______                  a1                     a2                       a3                     a4 a1=2(1)(-1)^1= -2           ...

Arithmetic and Geometric Sequences: I will be able to understand arithmetic series and sequence. I will be able to understand geometric series and sequence. Example of arithmetic sequence: -9,-2,5,12,___,___,___,___                        a(n)=a(1) + (n-1)d I do not know what my end term[a(1)] is so that is what I am looking for. I do know what my first term [a(1)] is, it is -9, I have 8 terms [n] and I can find my common difference by subtracting -2 from -9. which is 7. To solve all I need to do is add 7 to 12 then to that number and so on until I have all the terms I need.12+7=19, 19+7=26, 26+7= 3, and 33+7= 40. Now that I have my four terms I can add them to my series. Solution: -9,-2,5,12, 19,26,33,40 Example of geometric sequence: Find two geometric means between -2 and 54 -2, __, __, 54 a(n)=a(1)*r^(n-1) I do know what my end term is, it is 54, my first term is -2, I do not know what the common ratio is I h...

Piecewise Word Problems: I will be able to be able to write piecewise functions Piecewise word problem example: The admission rates at an amusement park are as follows:                 Children 5 years old and under: Free                 Children between 5 years and 12 years ,inclusive: $10                 Children between 12 years and 18 years, inclusive: $25                 Adults: $35 To solve this this first thing I would do is to write down the ages. The ages will serve as the x because they change. So for the first I say 0 less or than equal to x less than or equal to 5, because you can not just say less than or equal to 5 because that would include negative numbers and no one's age is a negative number. So 0 would have to be the lowest number. For the second I  say 5 less than x less than or equal to 12. Th...

Writing Piecewise Functions: I will be able to write piecewise functions. An example of writing piecewise functions: In order to write a piecewise function the first thing I look at the higher function and figure out if the function goes on forever or has a ending point. The top line does have a ending point with a closed circle. This means that this is a real point and not a placeholder so the equation x if x is greater than or equal to 2 would work for the top line. The bottom line has an open circle so this mean it is not a real point it is just a placeholder. This line is pointing upward toward the second quadrant with an arrow. The equation -.5x+-.5 if x is less than 1 (not equal to because the circle is open) . One misconception I had was with when two points were at the same coordinate but from different functions and had different end points. For example lets say two graphs have converge on the same point but one graph has a closed circle and the other has a open...

The Law of Sine and Cosine: I will be able to use law of Cosine and law of Sine.                                                          Example of law of Cosine:                                                                                                                       Solve for C: a=2 b=3 C=60 degrees        I n order to solve I need to figure out whether the equation is sine or cosine. The triangle is cosine  has Side Angle Side . Because it is a cosine problem I will use the cosine ...

Graphing: Find the Critical Points: I will be able to graph and identify properties of sine and cosine graphs. The graph above is the image of a sine graph. I was only given the equation I had to graph it on my own in order to graph it I had to find out the period, amplitude, and the critical interval the outside number tells me that the amplitude of my graph is 3 and this is a vertical stretch not compression since it is a whole number. Also I know this is a sine graph so it will start at (0,0) not at it's amplitude like cosine. The inside number is 2, (this is my frequency). 2 pi divided by this number 2 will give me the period pi. To find the critical interval you take your period, which in this case is pi and divide it by 4. I take this number and add 1 until I get my period. These numbers will make up my x-axis numbers. Now that I have all of my information I can graph it. The graph above is what I got.  One misconception I had was I did not know I had to use the criti...

Sine and Cosine graph:                                               Sine Graph                                                   Cosine Graph The difference between sine and cosine graphs are sine graphs touch the x-axis at pi and 2 pi and one period is 2 pi, also the amplitude is pi/2. For Cosine the graph touches the x-axis at pi/2 and 5pi/3. Sine graphs start at (0,0) while Cosine graphs starts at its amplitude on the y-axis.

Transformations of sine and cosine graphs: I will be able to transform sine and cosine graphs An example of a transformation of a sine graph is y= 4cos2x. The amplitude is 4, the period is pi, with a horizontal compression by a factor of 2. The amplitude is the number in front of the trig so in this case 4. The period is found when you take the frequency which comes before the variable (x) and divide it by 2 pi. Also I know that 2 is a horizontal compression because it is inside the equation and is a whole number. One misconception I had was I thought the frequency was the same as the period until Mrs. Burton explained the difference and told us that you have to use the frequency to find the period. EQ: How do I transform the graphs of trigonometric functions? You have to find out what the amplitude, vertical and horizontal compression/stretches, period, and find out if the graph reflects or not. The amplitude is the number in front of the equation this number also tells you ...

Finding coterminal and reference angles: I will be able to find the coterminal and reference angles. One example of an Coterminal angle is: Find the coterminal angle 86 degrees. To find the angle all you need to do is add or subtract 360 degrees, 86+360= 446 degrees or 86-360 = -274 degrees One example of an Reference angle: Find the reference angle of 312 degrees.  To find the angle I would first need to find out what quadrant it falls into. 312 degrees falls into quadrant 4. 312 is close to 360 so I would just subtract 360 from 312 and get 48 degrees. Before I say this is my final answer I need to make sure it follows the 2 rules that the reference angle has, which is it has to be positive and less than 90 degrees and 48 degrees ticks both of those boxes so 48 degrees is the answer. One misconception I had was with negative degrees. I didn't know which way they would go on a circle. I figured out my misconception when Mrs. Burton clarified it to the class that negative d...

Evaluating and Converting: I will be able to evaluate trigonometric expressions and convert between radians and degrees. One example of evaluating is: Find Cot 45 degrees. Cot is the inverse of tangent so 1/tan(45)= 1 One example of converting radians to degrees: Convert 3pi/4 into degrees. First I would need to multiply 3pi/4 by 180/pi. The pi's would cancel and I would end up with 3*180/4 ---> 135 degrees is the answer. One example of converting degrees to radians: Convert 160 degrees in radians.  First I would multiply 160 by pi/180. 160 pi/180. This is no the simplified form so I would need to divide the top and the bottom by 20 and the final answer is 8pi/9 One misconception I had was with which equation to use to convert from radians to degrees or degrees to radians. I figured out my misconception on my own I figured the pi would be in the numerator when going from degree to radian because radians have pi and the pi would be in the denominator when going from r...

Trigonometric Word Problems: I will be able to find trigonometric ratios of acute angles. One example of a trigonometric word problem is: A 15 foot ladder rests against a tree on level ground and forms a 75 degree angle of elevation. What is the height of the tree? What is the distance of the tree to the base of the ladder. The first step in solving this problem would be to go through and find any important information. I know that the ladder leaning against the tree forms a right angle so I will be using a trig ratio. The ladder is 15 feet long and is the hypotenuse so I have one length. The ground under the tree forms a 75 degree angle of elevation. This angle is across from the 90 degree angle. To solve this problem I need to find height and the base of the tree from the ladder. To find height I need to find the Sine (opposite over hypotenuse) 75 degrees = x/15. If I multiply 15 to both sides I can cross out the 15 in the denominator: 15(sin75)=x, x is approximately 14.49...

Exponential Word Problems I will be able to use compound interest to solve exponential problems. One example of an Exponential word problem is: Find the accumulated amount of money after 5 years if $4,300 is invested at 6% per year compounded quarterly . Find the growth factor .  In order to solve this problem you have to find out the accumulated amount after (t) number of years, which in this problem is what we are supposed to find out, the principle (original amount) which is 4300 (P) , the nominal interest rate per year, which is 6% or .06 (r) , the number of periods or years (n) , which is 4 because the money is compounded quarterly which is 4 times a year, and the number of years (t) , which is 5. Then you plug these numbers into the compound interest formula. A=P(1+r/n)^n*t ------->  A=4300(1+.06/4)^4*5 --------->  A=5791.48 The accumulated amount after 5 years is approximately $5791.48. The growth factor is 1.015 One misconception I had...

Logarithmic Functions I will be able to convert between exponential and logarithmic equations. For example 8 the problem is given b^a=123. To change this into log form you have to look at the base which is b, so log base (b) is the first step. The next step would be to look at the number behind the = sign which in this problem is 123 this number would is known as the solution. The last step is to find the exponent which is a and put this behind the equal sign. So your final answer is log(b) 123=a One misconception I had was if the base and solution is the same you can cross them out and your answer would be the exponent. I understood my misconception when Mrs.Burton explained it to the class. EQ: Exponential and logarithmic functions are related because exponential functions are the inverse of logarithmic functions. To convert exponential to logarithmic Log(b)x=n : b^n=x. You have to take the base b and the solution n would turn into the exponential. then the x value would tur...

Exponential and Linear Functions I will be able to know the difference between linear and exponential functions. In one example we are given a table with x and y values In this table you can see that the graph is decreasing by a negative 3: 5-2=3, 2-(-1)= -1, -1-(-4)=-7. Because -3 is being added consecutively this is an linear function.  One misconception I had was when I was putting the exponential value in the correct form. Instead of putting the function in the f(x)=Ca^x format I did 2.5x +50. The answer was 50(5/2)^x. I over came this misconception when Mrs.Burton explained the correct answers to the class. EQ: To determine if the function is a linear or exponential function you have to look at the y values if they are going up or down by adding then it is a linear function. If it is going up or down by multiplying it is exponential, If it is not going up or down in an even order by multiplying or adding then it is neither linear or exponential.

Domain and Range I will be able to identify the domain and range of relations and functions. This function has a open ended circle and a arrow on the other end. The end with the open circle means that that point is not a real point but more like an invisible barrier that shows where the function ends. This means that it is not equal to 5. On the other of the function is an arrow and because it is going up means that is considered positive infinity.You find domain be look how far left to right the function goes. The domain is parenthesis -3, positive infinity close parenthesis.They have parenthesis instead of brackets because the points are not actually real points. If the were they would brackets. You find the range by looking at the bottom to the top of the graph.The range is bracket -4, positive infinity parenthesis. One misconception I had this week was when we were doing an activity on transformation of functions I thought the function negative outcome was either negative o...

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