SP #7: Unit Q Concept 2: Finding all trig functions when given one trig function and quadrant

This SP7 was made in collaboration with Adrie. Please visit the other awesome posts on her blog by going here. 



What is this problem about?
This problem will go over the steps on how to all trig functions when given one trig function and a quadrant by using identities. We will also try to fund the trig functions using SOH-CAH-TOA to show that both methods can give us the same answers.


Identity Work (Unit Q Concept 2)

SOH-CAH-TOA Work (Unit O Concept 5)

Make sure you pay attention to the rationalization, because we can never have a radical as a denominator. Along with that, make sure to remember your identities, so you know why we plug the values in those numbers. Also, keep in mind that 

Pay close attention to the relationships because in the end, SOh-CAH-TOA and trig function identities both give us the same answer to all trig functions.

SP #6: Unit K Concept 10: Writing Decimals as Rational Numers

In this section, we now get a number of repeating decimals and in order to find the sum of the infinite geometric numbers, we have to do a series of steps. First, we ignore the whole number in front, so we can use it later on. We work with the repeating numbers by using summation notation. This means that we first have to change our first time into a fraction and the ratio also needs to be included. Then, we do some division with these and make sure to cancel off properly. Always be on the look out because you can some across a problem that needs to be simplified and this problem requires of that. After that, we bring back the whole number and add it to the fraction. Be careful with your multiplication and addition. Other than that, this is the answer to your problem!

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

In this Student Problem, the same rules from Concept 6 apply, except that in this problem, we find repeated factors, which basically means that there are two or more factors, such as in this example (x+1)^3. There are still tricky sections in this problem though because there are parts that we have to include even though I did not go over it on the last blog post. For starters, it is important to remember that we need to count up to the exponent that the factor x+1 says, which is 3. Then we, have to make sure we factor out before distributing. In this problem, I got the resulting system and combined the first two problems and then I solved the last two through the elimination process. After that, I then combined those resulting individual equations and solved them by the elimination process. It is helpful to use the original equations, so make sure you do not leave them out!

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

                               

 In this student problem, I go over over partial fraction decomposition. We will break down the problem and then we will use our algebraic skills to finish solving the problem. Keep in mind that matrices are back, so make sure you remember how to check your problem with rref. When you start to break down the problem and then go on to foiling, remember to be very careful with basic math (distributing, multiplication, etc.) because it can throw you off. Also, leave the denominator factored out, so it is easier for you to break down the fraction. I used A, B, and C as my letters, but it really does not matter which one you decide to use. However, keep in mind you should avoid X because after all it's one of your variables.

SP# 3: Unit I Concept 1- Graphing the Exponential Function

          This Student Problem goes over functions and explains how to graph one. As you can see, the function is composed of an exponent and is in the form of a parent graph. Using the values, key points, asymptotes, x-intercepts, y-intercepts, domain, range, and range, the image provided showcases a correct graph for the function. You will find that my ordered pair (2,-17) is off the graph, so it is place at the bottom of the graph just as a guide to help draw the graph.
          In order to successfully solve this function, remember that there are some key factors that will determine the graph, such as the x-intercept. To find the x-intercept, start by substituting a 0 for the y-value and continue the process of solving the problem. After some addition (+5 to both sides) and diving (-3 to both sides), you will notice that you get -5/3. Move the exponent (x-1) to make it a coefficient and you will go over natural logs to get rid of the 4. However, the problem is that you cannot divide the logs, as LN -5/3 is negative and one cannot divide negative logs; it would be undefined, so there is no x-intercept. Remember that for the y-intercept, you have to solve the 4^-1 properly by finding its reciprocal of (1/4). For guidance, the "a" value will help you determine whether the asymptote is below or above, but in this case, it's above. For the key points, it's helpful to use ordered pairs that are not all so close together, so you can get a better view of your graph.

SP#2: Unit E Concept 7- Graphing Polynomials and Identifying All Key Parts

   This concept explains the end behavior, the multiplicities that derive from zeroes, y-intercept, and a graph to represent a factored polynomial. The multiplies and the zeroes help describe the shape of the graph and the end behavior describes the limit notation used to graph the polynomial. Meanwhile the y-intercept will simply be plotted on its corresponding ordered pair. This polynomial is a positive, fourth degree function, which means that there will be a total of 4 x-intercepts, or zeroes when the reader adds the multiplicities.
          The reader needs to pay attention to the small details, such as the correct factorization since this eventually leads into our zeroes. In this case, an x^2 can be factored from the polynomial, so the reader must remember that this means that there will be two zeroes since there is a variable. Along with that, label the ordered pairs with T, B, or C because this will determine if the graph will go through, bounce, or curve; in this case, it goes through, bounces, and goes through again. Lastly, remember that the end behavior describes how the graph acts at the extreme, so make sure that the direction are correct.

                                                                                                                       

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

Identifying x-intercepts, y-intercepts, vertex(max/min),axis of quadratics and graphing them

          This problem represents quadratic equations in standard form. This representation includes the detailed explanation of the steps included in order to identify the x-intercepts, y-intercepts, vertex
(max/min), axis of quadratics and how to graph them. We start by completing the square of the a standard form equation so we can then put in in the parent function form [f(x)= a(x-h)^2 + k]. Next, we will graph the equation by locating the vertex, the y-intercept, and up to two x-intercepts, as well as one dotted line, which will be the axis. This will result in an accurate representation of the function.
          Of course, there will be many aspects to this equation and you must make sure you notice the characteristics that must be included. For example, you must remember to complete the square correctly by first adding 2 and then factoring out the coefficient from "a". Then, you must find the parent function equation by remembering to subtract the 10 so it all equals to y. Also, one of the main points, is to notice that even though the "h" is negative, you have to get its opposite value and that the "k" remains as is. A common mistake that should not be left out is to get a positive and negative square root 5 and not just a positive value. Lastly, always check that there are at least five points in your graph before connecting the parabola's points (unless there's an imaginary x-intercept, which is not the case in this problem.