zestymathblog

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 ...