D.E.V. By Shay, Tara, Sid

Shay's reflection

Farmer Ted problem

The two problems I chose to do were two problems that I had gotten wrong on tests. I know that doesn't sound like I would know how to do these problems. But since I got these problems wrong I took the time to FULLY understand them. I can now say that I get how to find domain with radical over radical functions and the applied function of finding two numbers to equal a given number and product. This project was ALOT of work and stress but I did like how we had to prove that we knew how to do a problem. I knew how to do the problems but it pushed me to understand why I knew how to do them.

Nathan Scott wants to build a basketball court behind his house. He wants to fence it in for privacy and has 2500 feet of fencing. One side is bordered by the house but the other three need to be fenced in. What dimensions will maximize the area?

Step 1: Draw out the problem in order to see it visually...it really helps!

*note that one side of the perimeter is not included in the equation because it is fenced in by the house

Step 2: Identify which variables and formulas will be used in the problem. In this case, you will use the perimeter and area formulas to find the dimensions that will maximize the area.

Find the inverse of the function, then use the composition method to prove that they are inverses.

2 numbers subtract to equal 62 and their product is a minimum

• Set your function equal to y.

In this problem you are looking for 2 numbers that will subtract to equal 62. Also these two numbers must multiply to equal a minimum. These are the equations that will be needed to find these two numbers

• Switch your x and y to solve for y.

• Pick which function you want to plug into the other.

Step 3: Fill in the variables you know into the proper equation.

• You see that the power's are the same and can cancel out. Take the 10 root and apply the 10th power to the function.

• Take the 10th root from both sides.

Next we want to solve for x in the x-y=62. We want to solve for x because we will plug the answer into the next equation

• You are left with x-6+6. The -6+6 cancel to zero leaving you with only x.

• To know if your functions are inverses you want them to equal x in the end.

• Subtract 6 from both sides.

Step 5: solve for one of the variables. You can pick either x or y but in this case we chose to solve for y

Now we know that x is equal to 62+y so we will now plug that into the xy=min for the x variable. We replace the x variable with a quantity so we have like terms.

Step 6: Now that we know what y is equal to we can plug it in for the y spot in the area equation

7a: Distribute x

• Change y to f inverse of x.

Step 7: Now solve for x using the area equation in order to find your first dimensions

Step 4: Identify the equation you will use to solve for one of the variables. In this case, you will use the perimeter, since you are maximizing the area.

7b: Now use the equation -b/2a to solve for the x value.

Step 8: Using the total perimeter and the x dimension, solve for the y equation. You do this by taking total perimeter of fencing and subtracting the x dimension from it.

Next we will distribute the new minimum equation

Step 9: Combine your 2 answer/ dimensions to get your final answer

Now that we have distributed, we can use a the equation -b/2a. This equation will give us the value of y. And we find y = 31.

Sid's reflection

The last part of this problem is to multiply x and y and get a minimum. So we plug our x and y values into the xy=min equation

Now to find x we plug -31(y) into the equation we found that x was equal to and we find that x is equal to 31

Find Domain!

Polynomials

Solve, find the x-intercepts, graph, then find the domain.

To find the domain of of this function, you must first find the individual domains of the numerator and denominator. And then see where the two overlap.

• First you divide x^4 by x, which is just like subtracting, giving you x^3. Put x^3 above the radical.
• In parenthesis line up and subtract x^4 b x^4.
• Then you multiply the -1, from the x-1, by your x^3, above the radical, to get a -x^3.
• Under -2x^3 in the parenthesis, put your -x^3.
• Subtract x^4-x^3 from (x^4-2x^3)
• Drop your answer of -x^3 and bring down -3x^2

I chose to do a question on dividing and graphing polynomials and inverses because I feel like those are the areas I did really well in. That came easy to me and were simple to understand. It was not hard to make up my own questions for these because I knew what needed to happen in each one. In the inverse problem not only do you have to try and solve for an inverse, but then you have to use the composition method to prove that they are truly inverses. In the polynomial problem, you get a bit of everything. You have to divide a polynomial, factor by grouping your answer, find your x intercepts, graph a polynomial, and find your domain and write it in interval notation. I did not think this assignment taught me much because I was doing problems that I already knew how to do and my groupmates did problems they knew how to do so I was not learning to do problems I struggle with. Maybe if we had to only do the problems that we did not know how to do so well, it would teach you and you would learn more from that.

Let's start with the numerator. Since the numerator is under a radical, we have to take the function out from underneath the radical and set it greater or equal to zero. We have to set the function greater or equal to zero because we DO NOT want a negative under the radical because it will give us a imaginary number.

• Divide -x^3 by x, which gives you -x^2. Put -x^2 above the radical.
• In parenthesis, line up and subtract -x^3 by -x^3.
• Multiply the -1 by -x^2 and get 1x^2.
• In the parenthesis, put 1x^2 after -x^3.
• Subtract -x^3+1x^2 from -x^3-3x^2.
• Drop your answer of -4x^2 and bring down 8x.

Next solve for x

*note that since we are dividing by a negative the inequality sign must be flipped.

• Divide -4x^2 by x to get -4x. Put -4x above the radical
• In parenthesis line up and subtract -4x^2 by -4x^2.
• Multiply the -1 by -4x to get 4x.
• In the parenthesis, put 4x after -4x^2.
• Subtract -4x^2+4x from -4x^2+8x.
• Drop your answer: 4x and bring down -4.

This is the domain of the numerator

• Divide 4x by x to get 4. Put 4 above the radical.
• In parenthesis, line up and subtract 4x by 4x.
• Multiply -1 by 4 to get -4.
• In the parenthesis, put -4 after 4x.
• Subtract 4x-4 from 4x-4.
• Drop your answer: 0.

• Re-write your answer.

• To factor your answer you must group. Place a pair of parenthesis around the first two terms and another pair around the second two terms; not including the (-) between them.

Now we have to find the denominator's domain. Finding the denominator's domain is a little different. We have to set the denominator greater to zero. We don't set it greater or equal to because we can't divide by zero.

• Then you find the most common factor in each set of parenthesis separately. In the first you remove a x^2, in the second you remove a 4.

• Since both parenthesis have the same terms inside you put them together. Then take what was removed and put it together.

• Factor x^2-4 completely.

• Then find what number makes each set of parenthesis equal to zero to find your x-intercepts.

• Plot your x-intercepts on a graph.
• Determine the degree by looking at the highest power of your answer: 3.
• Since it's odd the general shape starts negative and ends positive.
• The highest power of all our x-intercepts is 1 and they go right through at each point.

Solve for x

• To find your domain, look for where your graph is positive.
• Write your answers in interval notation.

This is the domain of the denominator

Tara's reflection

Now we have to see where the two domains overlap to find the overall domain. To help see where they will overlap draw a number line

*Open circle means not including 3.75 because this is where the denominator would equal zero and we cannot divide by zero.

*Closed circle means including 4.17 because the numerator can equal zero.

The two domains overlap between 3.75 and 4.17. So that means the final domain is (3.75, 4.17]

This is the final domain

*Parenthesis goes with the open circle, meaning not including 3.75

*Bracket goes with the closed circle, meaning including 4.17

I chose the problems that I had struggled with during the units

which were the completing the square and the farmer Ted problems.

I did this in order to help myself understand them better by having to

write the explanation. Once I had taught myself how to do them through

writing the explanations, I was able to show my improvements and I have

made these problems my strengths. I learned how and why the problems

work the way they do. However, the downside of this assignment was the

time it took and the stress that came with it. This was a difficult project that

took time away from my studying for other classes.

Complete the square and find the vertex. What is the domain and range of this function?

*important for this problem, 1/8 is the a value, -7/9 is the b value and 9/11 is the c value

Step 1: Subtract the c value, which remember is 9/11, from both sides of the equation

Step 2: factor out the a value, which is 1/8. As you can see in the picture the b value, which was -7/9 is now -56/9 because we factor out 1/8.

Step 3: In the original function there is three terms, or a, b, and c, right now we only have two terms. At this point we need to find the perfect c value. To do this you have the divide the b value by 2 and then square that value

The value that you will find is 784/81, this is the perfect c value. Add this value on to the end of your function

D.E.V. By Shay, Tara, Sid

Since we added to one side of the function we have to balance the other side. Now it looks like we only added 784/81 but we really added 784/81 times 1/8. So that is the value we have to add to the other side of the function. -9/11+98/81=349/891

Step 5: Now we have to put the function into vertex form. To do this we have to subtract -349/891 from both sides of the equation

Now that the function is in vertex form we can easily find the vertex.

*note that the x of the vertex is 28/9 not negative 28/9

Vertex: (28/9, -341/891)

Range: [-349/891)

Step 7: Now we have to find the range. To know whether this quadratic opens up or down we look at the a value. The a value in this case is positive, so the graph is going to open up. This means the quadratic opens up towards positive infinity. Then take the y value from the vertex (-349/891). This is the minimum of the vertex.

This is what the graph would look like

Domain:

Step 8: Find the domain. In this type of problem, this is simple because normal quadratics have a domain of all real numbers or negative infinity to infinity.