SV #1: Unit 7 Concept 10 - Finding all real and imaginary zeroes of a polynomial

      This problem is about solving for zeroes and factorizing the polynomial: -35x^4+89x^3-110x^2+63x-7. The zeroes for this type of polynomial will result to be real and imaginary/complex. The factorization will also result as complex for the most part. This problem will have multiple steps starting with finding the possible real/rational zeroes and finding the possible positive/negative real zeroes. From there, the possibilities make a more precise attempts to find the zeroes, and lead to factorization.

     The viewer needs to pay special attention in finding the possible zeroes: real/rational and positive/negative. Finding the real/ration zeroes involves p/q, which is the factored last term over the factored first term. Find the possible positive and negative zeroes involves Descartes' rule of signs. The positive and negative possibilities have a small different rule throughout the process. The last thing views should pay attention is that the quadratic formula is easier to use when the polynomial is reduced to the 2nd degree and there can be no negative in the radical.

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

     This problem is about graphing a polynomial with a degree of 4 and a positive leading coefficient. In order to do so, the polynomial must first be factored and the end behavior must be determined. From the factored equation, one must solve for x (by equaling the factored portion to 0 and solving) which will be the x-intercepts/zeroes with multiplicities for the graph. To solve for the y intercept, solve f(0) in the original equation. Have the points plot on the graph and sketch.
     Remember that the even/odd of a degree and the positive/negative of the leading coefficient will determine the end behavior of a graph (of a polynomial) Also note that the multiplicity of a number directs how the graph will "react" with the x intercepts/zeroes. An x value with the multiplicity of 1 will have the graph go THROUGH the x intercept; an x value with the multiplicity of 2 will BOUNCE off the x intercept; and finally, a multiplicity of 3 will CURVE through the x intercept. Remember that the graph must lead toward the end behavior, meaning don't make extra or minimize moves to reach the end behavior.

WPP #4: Unit E Concept 4 - Maximizing Area

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WPP #3: Unit E Concept 2 - Path of Ice-Cream Pint

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

*Notice- the scale on the graph was modified for the function to fit. There is color coding to distinguish: blue- parent graph; yellow- vertex; orange- y intercept; purple- axis of symmetry: and pink- x intercepts.

Finding the Parent Graph: Solving 3x^2-18x-22=5

1. Add 22 to both sides and factor out a 3. 

2 a. Use the vertex form equation [(b/2)^2] to create a prefect square, which will end up with 9. 

b. 9 will be added to both sides in the boxes and the prefect square can be completed on the right hand side: 3(x-3)^2. 

c. Add/multiply the numbers on the left to get 54.

3. The equation 3(x-3)^2=54 is what is left. Subtract 54 to get the parent graph: f(x)= 3(x-3)^2-54

Solving 3x^2-18x-22=5 for X Intercept/s 

*Do the previous 1-2 steps.

.4. The equation left off was 3(x-3)^2=54. Solve for x algebraically to find the x intercepts. There will be three possibilities: two answers, one answer, or none (imaginary numbers). Ex: (answer value, 0)

Make the original standard from equation equal to 0 and solve to find the y intercept. The vertex will be (h,k) from the parent graph and the axis of symmetry will be x= "h".

This problem is about sketching quadratics in standard form (f(x)= ax^2+ bx+ c) by converting it into parent form (f(x)= a(x-h)^2+k) for simplicity. In order to do so, one must fully complete the square. With the parent graph, you'll need to find the vertex, y-intercept, axis of symmetry, and x-intercepts in order to graphs. Several algebraic steps will be needed to be processed to find few of the requirements.

Pay special attention that (h,k) from the parent function form are the (x,y) values of the vertex. For "k", simply shift up or down with the corresponding number value. For "h" make x-h=0 and solve (note: the value from h should be the opposite from the original value) Also remember that the quadratic function will have two, one, or none (imaginary numbers) x intercepts. And thirdly, if "a" is (+) positive, then the graph will have a minimum, since the vertex will be on the bottom of the curve. If "a" is (-), then the graph will have a maximum, since the vertex will be on the top of the curve. With the parent form, the sketches will be more accurate and detailed.

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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