SP#7: Unit Q Concept 2 - Finding all Trig Function Values Using Identities

Please see my SP7, made in collaboration with Vanessa C. by visiting her blog HERE. Also, be sure to check out the other fabulous posts on her blog.

SP#6: Unit K Concept 10 - Write Repeating Decimal as Rational Number Using Geometric Series

Remember to pay close attention when I'm substituting the numbers into the equation. Don't forget to add five at the end when you're done solving because there was a five in the beginning of the problem.

In the above picture, I found that "a" sub one is 0.25. However, you must convert it to a fraction which would be 25/100. "r' is 1/100 because I took the preceding term and divided it by the first.

This picture is self explanatory, after I found 25/99, I still had to add five. I multiplied five by 99 to find the common denominator and then added to find my final answer.

SP#5: Unit J Concept 6 - Partial Fraction Decomposition w/ Repeated Factors

Remember to pay close attention when adding all the like terms and remember to show all your work because it is very easy to make mistakes, especially when you have four variables.

Step 1: You first find a common denominator and multiply it to the four varaibles. Once you've done that you add up all the like terms and separate them into the common terms. Then, you can start canceling out the like terms and the results will be the system.

Step 2: Next, you take the first and second equations and add them together. Then, you can use the elimination method in the third and fourth equation, in the picture above, I eliminated D. Once you have the two equations, you use elimination on them as well and I eliminated C.

Step 3: This picture shows what I did to solve for the four variables. The picture is pretty much self-explanatory. I just solved for A first and then used the substitution method to find the other equations.

This will be your final answer in the end.

SP#4: Unit J Concept 5 - Partial Fraction Decomposition

Be sure to pay close attention in distributing the numerator and be very careful when plugging in the numbers into the calculator. Also, remember to use the rref feature and check if the answers are correct.

Part 1: First, you would have to find the common denominator. In this case it's (x+4)(x-2)(x+1). Then, you distribute them to the numerator but remember to foil it out first. Once you have completed this step, you add up all the like terms.

Part 2: Now you have to replace the numerator with letter values. Then, you multiply by the common denominator again. This time, instead of distributing a number, you'll be distributing a letter. Next, you set the distributed product equal to the original and add up any like terms (like terms highlighted in above picture). After, you've found the like terms, you can start canceling out and the end product will be the system (as shown above).

Part 3 : The last step is to plug in the system into the calculator. Then you use the rref feature to find your answer. Once you have found what A,B, and C equals, plug them back into the original 3 fractions.

SP#3: Unit I Concept 1 - Graphing Exponential Functions & Identifying All Needed Parts

In this problem, you need to make sure to check if there's an x-intercept or not. You can tell by checking if there's a negative log, if there is, then there is no x-intercept. Also, make sure to write your domain and range correctly. On an exponential graph, domain is always negative infinity to infinity.

The first step you would have to do is label your variables (a=-3, b=2, h=1, k=-2). Next, you find your y-intercept, this is basically just y=k (y=-2). Then, you find your x-intercept, and from the picture below, you can see that there can never be a negative natural log. As a result, there is no x-intercepts. The next step is to find your y-intercept. From the step-by-step work that I've shown below, the y-intercept is (0, -7/2) or it can also be written as (0, -3.5). The domain would always be negative infinity to positive infinity since it's an exponential graph. Lastly, the range would be negative infinity to negative two. It's negative infinity because the graph is below the asymptote and the negative two came from the asymptote.

SP#2: Unit E Concept 7 - Graphing a polynomial and identifying all key parts

This picture will show you how to find the end behavior as well as the x intercepts and multiplicities. You will also learn how to tell if the graph will go "Thru", "Bounce", or "Curve". The picture shows the basic steps of how to sketch a graph based on multiplicities and x intercepts.

The first step to solve this equation is to factor out the equation. Once you have done that, the answer should be: (x-2)(x-2)(x+2)(x+1). As you can tell from the equation, it is even positive. This means that both the end points go upwards. To find the x intercepts, you put each factor equal to zero. The answers should be: (2,0) M2, (-2,0) M1, (-1,0) M1. Next, you find the y intercept is 8 because you substitute all the X's for 0's; the answer should be: (0,8). To graph it, you look at the multiplicities if it's a multiplicity of one, then it's through. If it's a multiplicity of 2, then the graph bounces and if it's a multiplicity of 3, it's curving.

SP#1: Unit E Concept 1 - Graphing a quadratic and identifying all key parts

This problem is showing how to find x-intercepts, y-intercepts, and the vertex (Maximum/minimum).

~Completing the Square~

In this problem, I first added 8 to the other side of the equation. Then,I took out a four and put it outside of the parenthesis. I then divided by 4. After,I square rooted everything and subtracted 2 to the other side. Because the equation is positive, it will be a minimum. The axis of symmetry will be at -2. To find y-int, you substitute 0 in for x, and find that it's -8.