SP 7# :Unit Q Concept 2: Finding ALL Trig Functions when given one trig function and quadrant (using identities)

Please see my SP7, made in collaboration with Gisela Barroso, by visiting their blog here. Also be sure to check out the other awesome posts on their blog. :D

SP #6: Unit K Concept 10: Writing a Repeating Decimal as a Rational Number using Geometric Series

This problem is on how to write a repeating decimal as a rational number using geometric series. So basically, this is when the decimal repeats itself on the right side of the decimal point. My example shows 13 repeating itself over and over. Using geometric series makes the process so much easier to solve.
It is important to format the problem correctly and use proper notation. For problems like these, which repeat, we use the infinity symbol to represent that. There are several fractions, so it's important to be careful when dealing with them. Get rid of the denominator by multiplying the fractional denominator by its reciprocal. Multiply the numerator by the same reciprocal from the denominator. Because there is a whole number,3, convert it into a fraction with the same denominator as the new fraction and add them together.

SP #5: Unity J Concept 6: Partial Fraction Decomposition with Repeated Factors

This problem deals with how to solve partial fractions that have multiple repeated factors. This problem is somewhat similar to the previous concept in that Most of the steps are the same. However, there are some twists in solving it, due to the repetition of factors.
It is important to know how to count up the factors, which means when making a fraction for each factor on the denominator, we have the first of the repeated factors be itself, the second repetition will be squared, the third will be cubed, and so on, up the powers. Again, it's imperative to properly factor out each factor set and then distribute the given numerator of each fraction to the result. Combine like terms into sets of equations and cross out the x^2's and the x's. Use the matrix setting in the graphing calculator to help solve this type of problem.

SP #4: Unit J Concept 5: Partial Fraction Decomposition with Distinct Factors

The first picture shows how to compose a partial fraction. The second picture shows how to decompose the result of the composed problem in order to eventually end up with the original partial fraction we started out with to compose. Both problems require a lot of distribution and factoring.
It's important to make sure to carefully factor and distribute because there is a lo of factoring and distributing. When decomposing, it;s crucial to remember to take out the x^2 and x in the combined equations. Also, the decomposition part of it requires a matrix to show the steps. When using the matrix part, use a graphing calculator properly by pressing 2nd matrix, edit, plug in the matrix values where they go, 2nd, quit, 2nd matrix, math, scroll down o rref(, select it, 2nd matrix, enter, close off with the end parentheses, enter, and that's the answer.

SP #3: Unit I Concept 1: Graphing Exponential Functions

This picture shows how to solve and graph for exponential functions. This includes identifying the x-intercept (if there is any, not in this case), the y-intercept, the asymptote, domain, range, and plotting at least four points on the graph. The equation I chose is: f(x)=(.3)(.5^x+2)+7. "a"=.3; "b"=.5; "h"=-2; "k"=7. Because my pictures came out blurry (SORRY!) the 2nd picture reads "value of "a" is positive, so the graph will go ABOVE the asymptote".
It is important to know when there is no x-intercept: when the log is negative. When finding what "h" equals, it is never that same number expressed in the power (^x+2). "h" is actually -2, not +2 because, to find "h", we use the expression "x-h", therefore getting the opposite. "h" can be used as the 3rd key point on our table, and is therefore plotted on our graph. Also, the asymptote can't be touched at all.

SP #2: Unit E Concept 7: Graphing Polynomials

This problem is a polynomial that is factored and taken apart in order to be able to graph it. This problem includes x-intercept, y-intercept, zeroes (with multiplicities),and end behavior. We are able to pull out all of these things from the given polynomial which we then use to graph it.
It is important to pay special attention to the detail and how everything is necessary for plotting the graph. The term with the biggest exponential value (x^4) determines the end behavior of the graph. Zero multiplicities are especially important because they determine the middle behavior around the x-axis; this shows us how precisely how the polynomial is graphed so we don't have to guess.

SP #1: Unit E Concept 1: Identifying x-intercepts, y-intercepts, vertex (max/min), axis of quadratics and graphing them

This problem deals with how to identify x-intercepts, y-intercepts, the vertex (whether it’s the max or the min), and the axis. This problem also shows how to use all this information to graph a quadratic derived from standard form, which is then put into parent-function form. This problem has several parts to it that all go together to find the necessary information in order to plot the graph. A graphing calculator might be needed for a problem similar to this.

For a problem like this, it is important to remember all the steps that go into the end result. Completing the square is crucial for transforming the equation into parent-function form. It is necessary to know that if the result ends up being imaginary (x-intercepts), then there is nothing to graph, seeing as imaginary answers don’t exist.