SP #7: Unit Q Concept 2 - Finding All Trigonometric Functions Using Identities

This SP #7 was made in collaboration with Sergio Sanchez. Please visit the other awesome posts on their blog by going here.

Our given is:

One of the first and foremost steps that must be undertaken, is identifying which quadrant we are working with. Although, using the values of the trig functions and their accompanied signs would do just as well.

Next, we'd just go right into solving this with trigonometric identities. It's pretty self-explanatory had one taken this course. But, for the sake of something: first, I list out the identity, and then, substitute in the value(s) that I already have. I then solve it until there is one unknown trigonometric function is on one side, and a value is on the other.

As for Sergio's work, I suppose that I shall put forth some sort of explanation, in lieu of the original creator, who just gave me this picture. Basically, he plugged in values that were given (from the tangent trigonometric function), into the Pythagorean Theorem, in order to get the excluded side's value. Having gotten all of the sides, he just plugged them into the rest of the unknown trigonometric functions.

It is important that one remembers what all of the trigonometric functions mean - what sides they represent, and how they represent each other. As in, you have to remember that sine is opposite / hypotenuse, and also, 1 / secant. This is important because if you could fail pretty easily if you use the wrong functions/identities. It's also very important for one to remember to rationalize (which I did), and reduce (which I didn't do). Apparently, it's the most important thing you could do for yourself, for forgetting to do so, would result in loss of points on your tests or whatever.

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

        Concerning this problem, one in which you are tasked with finding the fraction of a number with repeating decimals, there are not really many things one has to remember to do. Though, it is a bit important that one remember to add whatever you didn't account for in your geometric series. In my case, it was the 4, since it was not part of the repeating numbers. You must also remember to plug your answer back in, to see whether or not you did it correctly. I mean, you don't really have to, but it is quite useful for saving a bit of face when having your problem examined by your colleagues. Yep.

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

        First of all, we must remember that what distinguishes these types of problems from other ones is that they have repeated factors. Basically, all you do is count up, which means, as you separate the initial fraction, you repeat the factors, but with their exponent increasing incrementally by 1. It sounds confusing, but if you had saw an example, you'd get it right away. As always, with these types of lengthy problems, you must remember to calculate carefully, as one mistake could ruin you. Oh, and you can't use calculators for many of these problems, since most of the answers come out as decimals, which we are not permitted to utilize, unless they are definite (don't go on forever). Fin.

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

        There are several crucial things that one must remember in order to fully get through this struggle and emerge victorious. First of all, one must pay close attention to the calculations that they do in their head, because it's really easy to mess up. And if you mess up once during the problem, you'll have to go back and redo the entire thing from that point onward. Second of all, you have to remember to transfer the correct sign (+/-), and also utilize them correctly, for the next step. One mistake could mess up your entire problem. Lastly, if you're not utilizing a calculator to derive RREF (Reduced Row Echelon Form), you have to remember to utilize your elementary row operations without mistake, for one mistake could, again, cost you the entire answer, ergo, a whole lot of time. That is all.

SP #3: Unit I Concept 1 - Graphing Exponential Equations

        Alright, so some important things to remember when finding the components of exponential equations, and afterwards, graphing it, include: making sure that you have extracted the a, b, h, and k values correctly, making sure that you've got the right type of asymptote for the such equations, and remembering how to tell whether or not the equation will have an x-intercept or not. Basically, you just adhere to the parent graph closely (adapted for the exponential equations) to make sense of said values. The exponential yak died will tell you that exponential equations have the asymptotes of y = k. It will also tell you that the domain is never restricted, which is why y-intercepts will always be present in these equations. Lastly, you can easily tell whether or not an equation has an x-intercept by seeing whether or not the equation crosses the x-axis, which would only occur when the horizontal asymptote is not blocking the way to the axis.

SP #2: Unit E Concept 7 - Graphing A Polynomial And Identifying All Key Parts

        The purpose of this problem is to derive a polynomial function from a designated number of zeros and a designated power and coefficient. With this in mind, we graph a fourth degree polynomial with an even coefficient for its highest x-power, all without of the complications that a graphing calculator obviously presents. This is done by first extracting the factors from the polynomial, or formulating them from the zeros that were given, and then deciding the orientation of the ends of the graph. That is all.
        Some important facts to factor in are to pay attention to the: orientation, through / bounce / curve x-intercepts, and whether or not it looks right. An important thing that must be done (that is excluded from this problem) in order to graph a more accurate graph, is to try and figure out the extrema of the graph. Don't forget to utilize that, and of course, the y-intercept in more accurately displaying your graph. An extra thing that could be done would be to check whether or not your graph is correct on a graphing calculator. Don't forget to input the correct things though, for your calculator is only as smart as you.

SP #1: Unit E Concept 1 - Graphing A Quadratic and Identifying All Key Parts

        This problem is about deriving a parent function type equation from a regular trinomials, so that graphing the equation wouldn't be as hard as it would have been had you tried to graph the other way. Basically, it's about finding the values that we deem crucial in graphing polynomials, which include the vertex, x-intercept(s) (if there are any), the y-intercept (if there is one), and the axis of symmetry. Ultimately, it's about going through this infinitude of steps, and arriving at a simpler means.

        I think that the reader needs most, to pay special attention to whether or not my writing is intelligible to them or not. However, in all seriousness, though, I think that they need to know that all of the important values that we need, can be calculated simply on a calculator right from the start, without having to go through any of these steps. To each, their own, I guess. Also, it's pretty important that you realize imaginary numbers cannot be graphed regularly, such as regular x-values, on a regular graph.