SP#7: Unit Q Concept 2: Pythagorean/Ratio/Reciprocal Identities with SOH CAH TOA

This SP was made in collaboration with Leslie N. and can be seen by clicking this word right here. While you're there, be sure to check out her awesome math posts too!

SP#6: Unit K Concept 10: Converting Repeating Decimals into Rational Fractions

 

Step 1: Start with the repeating decimal 5.212121... and find the a sub 1 and a sub 2 terms by taking the repeating terms out as chunks, using zeroes as placeholders where necessary.
Step 2: Divide the a sub 2 value by the a sub 1 value to get the ratio.
Step 3: Plug the terms into the summation notation for infinite geometric sums.
Step 4: Plug the terms into our sum formula and simplify.

For this problem, you needed to pay attention to the formulas used. Make sure you use the right ones! If you don't, you will get the wrong answers.

SP#5: Unit J Concept 6: Partial Fraction Decomposition (Repeated Factors ver.)

*FYI: The step numbers do not necessarily correlate with the numbers in the image.*

Step 1: Seperate your denominator factors into fractions- in this case, one of the factors is cubed, so when writing the fractions out remember to count up the factors. [1]
Step 2: Find a common denominator. In this case, you simply have to multiply the numerator by the factors that aren't already in the denominator. Then once you've multiplied and FOIL'd all the numbers and letters, set it equal to your original equation.
Step 3: Seperate the like terms into 4 different equations. At this point, you can take out the 'x cubed', 'x squared', and 'x' values and leave just the coefficients. [2]
Step 4: Two of your equations should have 2 terms and the other two should have 4 terms. Take the latter two equations and use elimination to get rid of a term. In this case, I multiplied the purple equation by two, then added it to the other 4 term equation to get rid of the D term. [3a]
Step 5: Now you have a 3 term equation, but no other one lying around ready to use. Add the 2 term equations together to make another 3 term equation. [3b] Use elimination again to get rid of another value- hopefully you should see where I'm going with this. In this case, I multiplied the 3 term equation by 4 to get rid of the C term.
Step 6: Now you have a 2 term equation. Take any one of your first 2 term equations and use elimination again to get rid of a term. In this case, I multiplied the pink equation by 4 to get rid of the B term, leaving me with the A term.
Step 7: Solve for that final term, then back substitute until you have solved for all the terms. That's it, you're done~!

SP#4: Unit J Concept 5: Partial Fraction Decomposition (Distinct Factors ver.)

Step 1: Find a common denominator. In this case, you have to multiply the numerator by the factors that aren't already in the denominator. Then once you've multiplied and FOIL'd all the numbers, add the common terms together.



Step 2: Set the numerators to letter values (in this case, A, B, and C) and do the same factoring and/or FOILing method as before. However, instead of adding the common terms together, set the fractions equal to your answer from Step 1. Then separate the common terms into three equations.



Step 3: Take the coefficients and put then into matrix form. Input the values into your calculator and put it into reduced row echelon form. Finally, take the answer column and put each one over a factor in fraction form. Look familiar? It should, that's the equation we started with.

SP#3: Unit I Concept 1: Solving an Exponential Equation

To solve this problem, you need to understand how to find the a, b, h, and k values. This is because they are key to finding the asymptote, domain, and range. You also need to pay attention to the coloured parts of the image, because chances they are important.

 

Step 1: Start with the equation (shown in the box titled 'equation') and determine if the graph lies below or above the asymptote. In this case, since the value of a is negative, the graph lies below the aymptote.
Step 2: Determine if the asymptote is vertical or horizontal by checking if it a log or exponential. In this case, the equation is an exponential one, and it's asymptote is equal to the k value. Therefore, the asymptote is 'y = -3'
Step 3: Solve for the x-intercept. Set y equal to 0 and add 3 to both sides. Then divide both sides by -2. At this point, we would normally take the log of both sides, but since you can't take the log or a negative (-3/2), there is no x-intercept.
Step 4: Solve for the y-intercept. Set x equal to 0 and simplify the exponent to 3. 1/2 to the power of 3 equals 1/8, and when you multiply that by -2 you get -2/8, or -1/4. Subtract 3 from -1/4 (I changed 3 to 12/4 to make it simpler) and you end up with -13/4. This means (0, -13/4) is your y-intercept- you can convert this to decimal form on your calculator to make it easier to graph.
Step 5: Determine the domain and range of the graph. Since this is an exponential equation, the domain is unrestricted. The range, however, depends on the asymptote. In this case, the range is -∞, -3.
Step 6: Graph the graph, haha that's an amusing phrase. Plug the equation into your graphing calculator and press the buttons labelled '2ND' and 'TABLE'. This will give you a table of all the points on the graph. Select a few, put them in your table, and plot them. Draw the aymptote, connect the dots, and voilà, you have solved this exponential equation.

SP#2: Unit E Concept 7: Zeroes and Multiplicity

This problem is about using randomly selected zeroes to create a polynomial. The zeroes should also be used to find the x-intercepts and the y-intercept. Finally, it should be possible to glean the end behavior and graph the equation as well.
For this problem, you should make sure you know how the multiplicity of a zero affects the graph of the equation. A multiplicity of one, for example, means that the graph will go through that point. Knowing the meanings of the multiplicity will make the graph more accurate.

SP#1: Unit E Concept 1: Quadratic

This problem is about putting a function, f(x)=x^2-6x+5, into the parent function equation and using the parent form to find the vertex, axis of reflection, y intercept, and x intercepts. To solve the function, we must first use the 'completing the square' method (as shown on the left side of the image) to factor the equation out. This will make it easier to put it in the parent function equation (as shown on the right side of the image) and subsequently find the needed values.
One thing you, dear viewer, might want to keep an eye on is the x intercept values. Make sure you do the math carefully and don't end up with a strange square root or an imaginary number (neither of which are in the answers.) Don't forget that there are two x intercepts as well, unlike the singular y intercept we all know and love.