WPP#10: Unit L Concepts 9-14 - Basic Probability, And/Or Events, & Possibility of Union/Events

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WPP#9: Unit L Concepts 4-8 - Fundamental Counting Principle, Combinations, and Permutations

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SP#6: Unit K Concept 10 - Writing Repeating Decimals as Rational Numbers

Make sure to pay special attention to plug in the right numbers into the right formula. Since we'll be solving this without using a calculator, leave the numbers as fractions so they are easily simplified. Remember to add the fraction with the number in front of the decimal and to always check to see if your answer can be reduced.

Problem:
3.215215215215

1. Break apart each repeating unit. Ignore number in front of decimal for now. Arrange them into a geometric series.

2. Find the first term and the ratio. Keep them as fractions. (To find the common ratio, take any term and divide it by the preceding term.)

3. Use summation notation and plug in the numbers.

4. Plug into the infinite sum formula (a sub 1 divided by 1 minus r).

5. Simplify without a calculator. Multiply by the reciprocal to get rid of fraction in the denominator.

6. Check to see if resulting fraction can be reduced.

7. Bring back the number in front of decimal (3) and add it to the fraction. 

8. Can check answer by plugging in answer into calculator. Should end up with the number we started with.

Fibonacci Haiku: Time is Precious

Time

Fleeting

Always moving

Infinite yet limited

The past cannot be undone

The future is uncertain, the present a gift

http://t1.gstatic.com/images?q=tbn:ANd9GcS06v29eg3nyl5Wzg_egclW6ehG14oyml8E-H5FN-jxGKFLPgA8

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

Make sure to pay special attention to when you are simplifying and foiling. A single mistake could make your entire answer wrong. Also, check to see that you are solving the system of 4 equations correctly, that you grouped the right like terms and eliminated/substituted properly. There will be a lot of elimination involved with this problem, since there will be 4 equations and 4 variables.

Problem: (x^2 - 2)/(x-2)(x+1)^3

1. Since denominator is already factored fully, find the common denominator.

2. Separate each factor into a separate fraction with and assigned letter as each of its numerator. Since one of the factors repeat, you must count up the powers and include the factor as many times as the exponent.

3. Multiply each part(numerator and denominator) by what is missing from the common denominator.

4. Simplify. Set numerator equal to the numerator of the problem because the common denominator doesn't matter now.

4. Group all like terms together and set it equal to the corresponding like term on the other side.

5. Remove the coefficients, but leave the coefficients as they are in the equation. You end up with a system of 4 different equations.

6. Solve resulting system of equations by using elimination. We want to get rid of D first since there are only two equations with D, so that we end up with a new 3 variable equation.

7. Combine the first two equations to make another 3 variable equation.

8. Use elimination on the 2 new equations to make a new 2 variable equation.

9. Use elimination on the new 2 variable equation with either one of the 2 original 2 variable equation to calculate the value of one variable.

10. Substitute value back into the other equations to calculate the remaining values of the other variables.

11. Substitute values of variables back into original split up fractions with the letters.

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

Make sure to pay special attention when you are decomposing or composing the fraction. Check to see if you foiled and distributed correctly, and if you grouped all like terms. In addition, when solving, make sure your equations are correct, your matrix is set up correctly, and the values are correctly substituted back into the split up fractions.

Problem: -5/x + 2/(x+3) + 1/(x-2)

Part 1: Composing Fractions

1. Find the common denominator. Multiply each part by what it is missing in the numerator and the denominator.

2. Foil out the factors first and then distribute the numerator. Keep the denominator factored.

3. Simplify the expression by combining all like terms. 

Part 2: Decomposing 

Problem from Part 1: (-2x^2 - 6x + 30)/(x(x+3)(x-2))

1. Since denominator is already factored fully, separate each factor into a fraction assigned with a different letter. (Doesn't matter what letter as long as it's not x since it's already being used)

2. Find the common denominator. Multiply each fraction (numerator and denominator) by what it is missing from the common denominator.

3. Simplify by foiling out the factors first and then distributing the numerator. Keep the denominator factored. 

4. Set the numerator equal to the numerator of the problem. (Denominator doesn't matter since common now. Can ignore it.)

5. Group like terms with letters together and set it equal to the like terms on the other side. 

6. Take out the x's so that only the coefficients remain. (Letters stay too.)

Part 3: Solving System of Equation with Matrices

1. Take the coefficients and set them up in a matrix.

2. Plug into graphing calculator. (2nd Matrix, Edit, Select any letter, Input coefficients of original equations, Quit, 2nd Matrix, Press rref( feature, 2nd Matrix, Select edited matrix, Close Parenthesis, Enter)

3. Calculator will give Reduced Row-Echelon Form. Record values of letters.

4. Plug values of letters back into split up fractions from the beginning. (Should end up with original problem.)

SV#5: Unit J Concepts 3-4 - Solving 3 Variable Systems with Matrices

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Make sure to pay special attention to your matrices and that all the numbers you write are correct because even if you forget something like a negative sign, then your entire answer will be wrong. You should also be careful about your elementary row operations to make sure your able to get Row-Echelon form and check your graphing calculator to check if your answer is indeed correct.

WPP#6: Unit I Concepts 3-5 - Interest and Investment

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SV#4: Unit I Concept 2- Graphing Logarithmic Functions

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Make sure to pay special attention to the asymptotes and x/y intercepts. Since this is a logarithmic function, the asymptote will depend on h, not k like in exponential functions. So if you forget to take the opposite of h, then your graph will be wrong. For the x/y intercepts, you need to know how to exponentiate and solve logarithmic equations since they can get a little tricky or undefined.

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

Make sure to pay special attention to the asymptote and the x/y intercepts. The asymptote for exponential functions is from the k in the equation y= a x b^(x-h) + k, so you must remember to set it make it y=K not just k. When solving for the x or y intercepts, be sure to remember that is possible to end up with undefined answer; all that means is that there is no x/y intercept.

1. Find a, b, h, and k. Refer to y= a x b^(x-h) + k. *Remember that for h just put the opposite since it shows as x+1, it is actually -1 since 2 negatives make a positive.

2. Calculate the asymptote. Asymptote of exponential equations is y=k. So the asymptote would be y=  -2, not just -2 in its own. Then, draw the asymptote on the graph.

3. Solve for the x-intercept. Plug in 0 for y and solve. This will review your knowledge on solving exponential equations. Since you can't take the log or natural log of a negative number and zero, the answer will be undefined and no x-intercept. This is also logical because since the graph is below the asymptote of y=-2, then there in no way for it to cross it, let alone intersect the x axis.

4. Solve for y-intercept. Substitute x with 0 and solve. Plot the point on the graph.

5. Write domain in proper notation. Domain of an exponential function will always be (-infinity, infinity), which just means there are no x value restrictions.

6. Write the range in proper notation. This depends on the asymptote and graph. Since the graph is below the asymptote, it would be (-infinity, -2). 

7. Find 4 key points using a graphing calculator. Plot them on the graph.

8. Draw the graph by connecting the points and following the asymptote. 

SV#3: Unit H Concept 7 - Finding Logs Given Approximations

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This video will over how to find logs when given clues on how to find it. There are four clues that are given to you and two hidden clues that you can find using your knowledge of the proerties of logs. Using the clues and the power, quotient, anx product property, you will then be able to substitute the values/variables in and simplify to find what the log will equal.

Make sure to pay close attention to breaking down the numbers because some might already be a clue, but could still be broken down further. Also, ensure that you have the right signs when substituting in the clues: if it's in the numerator, then it will be positive because it multiplying; if it's in the denominator, then it will be negative because it is dividing.

SV#2: Unit G Concepts 1-7 - Finding all parts and graphing a rational function

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This video will go over rational functions and how to graph them. It will go also go over your knowledge on horizantal/slant/vertical asymptotes, holes, domain, and the x/y intercepts. Make sure you remember to put everything in its proper notation as well.

You need to make sure to pay close attention factoring and long division. This will ensure that your vertical asymptotes and holes are correct while also helping establish a boundary for your graph. Also, make sure that you have at least 3 points on each side of your asymptotes before making the graph.

SV#1: Unit F Concept 10 - Finding all real and imaginary zeroes of a polynomial

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This video will go over how you can find all the real and complex zeroes of a polynomial. It will draw upon your knowledge from Descarte's Rule of Signs, your p's/q's, synthetic division, and factoring.

You need to pay close attention to when you are factoring, so you can get all the right signs and numbers, when you are using synthetic division, and well basically everything. This is the type of problem that if you get 1 tiny detail wrong, the rest of your answers will be wrong as well.

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

Equation:

f(x)= x^4-5x^3+3x^2+9x

This problem is about is to see if you can use what you know about polynomials to make an accurate graph of it.  To solve it, you must need to know and understand how to factor a polynomial, the end behavior of certain polynomials, how to find the x-intercepts with multiplicities, the y-intercept, and how to graph the zeroes.

If you want to make your graph even more accurate, then you can find and use the extrema of the graph. In addition, be careful to make sure the end behavior is correct and the zeroes are correctly plotted. If a zero has a multiplicity of 1, then the graph goes through the point. If a zero has a multiplicity of 2, then the graph bounces off the point. Finally, if the zero has a multiplicity of 3, then the graph curves through the point. Make sure that all your zeroes match up to the highest degree (exponent).

1. Since we chose the zeroes(x-intercepts) first, multiply all the factors to get the equation.

2. Make note of the end behavior and how the graph will approach a zero (Thru, Bounce, Curve).

3. Find the y-intercept by plugging in 0 for x in the equation and plot it.

4. Draw the graph.

WPP#4: Unit E Concept 3 - Maximizing Area

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WPP#3: Unit E Concept 2 - Path of Football (or other object)

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

Problem: 

F(x) = 6x² + 13x - 8

This problem is to see if you can identify the x-intercepts, y-intercepts, vertex(max/min), axis of quadratics and how to graph quadratics using the parent function. Already, just by looking at this problem, you already know that the graph opens up because a is positive.

To continue, you must remember how to complete the square, so you can convert it from standard form to the parent function equation, which would then make it easier for you to graph the function. After that, you can easily solve for x-intercepts, y-intercepts, vertex(max/min), and the axis of symmetry. Be careful with the x-intercepts because sometimes you might have imaginary numbers, but you wouldn't plot it on a graph because there is no axis for them.

Work:

Parent function equation:

y = 6(x + 13/12)^2 - 361/24

Vertex: (-13/12, -361/24) min.

Y-intercept: (0, -8)

X-intercept(s): 

(½, 0) and (-8/3, 0)

or

(0.5, 0) and (-2,67, 0)

After you finish the work, all you have to do is to plot the points on the graph and exploit symmetry to graph the quadratic (as shone below).

WPP#2: Unit A Concept 7 - Profit, Revenue, Cost

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WPP#1: Unit A Concept 6 - Linear Models

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