Algebra 2
This year, Algebra has been very easy for me to understand. I actually think I should be in a higher math class because of how simple I find it to be. Sometimes, I have a really hard time getting work done for Algebra because I don't see a point in studying it. One thing that I definitley gained from Algebra 1 was a work ethic. Sometimes in life there are things that we just have to do, regardless if it is applicant to us or not. I have learned that work comes in all forms and is not always supposed to have a purpose. One piece of work that I believe best shows my growth as a learner is the final project for this semester. Although it was an easy task for me to accomplish, this project was extremely difficult to work on. I become so easily distracted in class that it was almost impossible to finish it. At home, I also had a really hard time finding inspiration and motivation to do it. You can see the problems I created below:
Final Problems:
5x2+37x+10=-4
Factor & Solve for X
Hard
y-7=4(x-4)
Graph
Medium
y=4x-12
y=-2x+12
Where do they intersect?
Medium
2x-3y+4z=-4
4x+y-2z=6
-x+4y-9z=7
Solve using elimination
Ridiculous
2√45x7√400
Medium
y>3
x<-4
y<x+9
Graph on a coordinate plane; Shade the region in which all three inequalities intersect.
Medium
(p4s3)-3
Simplify.
Medium
3-√3
√(-2x5)
Simplify
Hard
6x2-7x+4=7
Solve By Completing the Square
Crazy Hard
Dave has a company named Dave’s Dogs. He sells hot dogs for 2.09 dollars each including tax. He sells burgers for 3.60 each including tax. Dogs require 10 minutes to assemble, and burgers take 20 minutes. It takes 1 onion to assemble 5 burgers or 6 hotdogs. He has 10 whole onions. Dave has 5 hours of time to assemble his goods. Assuming that every good he makes is bought, how many should he make of each to maximize profit?
Hard
Factor & Solve for X
Hard
y-7=4(x-4)
Graph
Medium
y=4x-12
y=-2x+12
Where do they intersect?
Medium
2x-3y+4z=-4
4x+y-2z=6
-x+4y-9z=7
Solve using elimination
Ridiculous
2√45x7√400
Medium
y>3
x<-4
y<x+9
Graph on a coordinate plane; Shade the region in which all three inequalities intersect.
Medium
(p4s3)-3
Simplify.
Medium
3-√3
√(-2x5)
Simplify
Hard
6x2-7x+4=7
Solve By Completing the Square
Crazy Hard
Dave has a company named Dave’s Dogs. He sells hot dogs for 2.09 dollars each including tax. He sells burgers for 3.60 each including tax. Dogs require 10 minutes to assemble, and burgers take 20 minutes. It takes 1 onion to assemble 5 burgers or 6 hotdogs. He has 10 whole onions. Dave has 5 hours of time to assemble his goods. Assuming that every good he makes is bought, how many should he make of each to maximize profit?
Hard
Polynomial Art Project
Overview
In this project, we were told to look out for pictures with curves. Once everyone had a picture that they found suitable to draw polynomials over, we began making points on the pictures. After the points were all plotted out, we drew curves over them. We then analyzed the curves and found the equation that went with it. From there, we translated that equation and line onto a transparent piece of graphing paper. From that we were able to see if it all matched up to check our work. The last step was to re draw the lines and create another piece of art out of it. I drew a radical dinosaur.
The Math
1st Polynomial:
Zeros
(6.5,0) (4.75,0)
Factored Form With Scalar of Polynomial #1
Y=-2(x+6.5)(x+4.75)
Maximum: (-5.625, 1.531)
Multiplication Process:
-2(X*X+6.5x+4.75x+30.875)
Standard Form Equation of Polynomial #1
Y= -2x2-22.5x-61.75
2nd Polynomial:
Zeros
(-2.5,0) (1,0) (7.5,0)
Final Factored Form With Scalar:
Y=-(1/35)(x+2.5)(x-1)(x-7.5)
Maximum: (4.929732, 2.144107)
Minimum: (-0.9297322, -0.7298212)
Multiplication Process-
Y= -(1/35)(x+2.5)(X-1)(X-7.5)
Y=-(1/35) (X*X)+(2.5x)-(x)-(2.5)(x-7.5)
Y=-(1/35) (X2+1.5X-2.5) (X-7.5)
Y=-(1/35)(X*X2)-(1.5X*X)-(2.5x)-(7.5*X2)-(1.5X*7.5)+18.75
Y=-(1/35)(X3-6X2-13.75x+18.75)
Standard Form for Polynomial #2
Y=-(1/35)X3+0.171428514X2+0.3928571429X-0.5357142857)
Explanation
Polynomials can cross the X-axis multiple times. If imaginary numbers aren’t accounted for, the least a polynomial can cross the X-axis is once. If imaginary numbers are used, then it doesn’t have to. A vertex is the highest or lowest point of a parabola, depending on weather or not it is positive or negative. The x-axis in any polynomial can be used to construct an equation that describes the line. For example, a zero for a polynomial that crosses the x-axis at 4 would be (x-4). This means that when x=4, y will be zero because they will cancel out and any other number that is involved in the polynomial that is multiplied will be canceled as well because anything multiplied by 0 equals 0.
Minimum and maximum are referred to as local minimum and maximum because a polynomial can have multiple minimums and maximums. So, the “local” minimum is the lowest part of one of the curves, and the “local” maximum is the highest point of one of the curves in a polynomial. Because my second polynomial had an X3 variable, it had a maximum and a minimum. My first was a simple parabola, so since it was facing downward, it only had a maximum.
Reflection
This project was alot of fun and I definitley got more practice using polynomials. One thing that I enjoyed was the fact that I was able to construct whatever I wanted after we had our polynomial curves. It left a bunch of room for creativity which I appreciated greatly. I found it really hard to put equations into a computer because I didn't necessarily know where the different symbols were.
In this project, we were told to look out for pictures with curves. Once everyone had a picture that they found suitable to draw polynomials over, we began making points on the pictures. After the points were all plotted out, we drew curves over them. We then analyzed the curves and found the equation that went with it. From there, we translated that equation and line onto a transparent piece of graphing paper. From that we were able to see if it all matched up to check our work. The last step was to re draw the lines and create another piece of art out of it. I drew a radical dinosaur.
The Math
1st Polynomial:
Zeros
(6.5,0) (4.75,0)
Factored Form With Scalar of Polynomial #1
Y=-2(x+6.5)(x+4.75)
Maximum: (-5.625, 1.531)
Multiplication Process:
-2(X*X+6.5x+4.75x+30.875)
Standard Form Equation of Polynomial #1
Y= -2x2-22.5x-61.75
2nd Polynomial:
Zeros
(-2.5,0) (1,0) (7.5,0)
Final Factored Form With Scalar:
Y=-(1/35)(x+2.5)(x-1)(x-7.5)
Maximum: (4.929732, 2.144107)
Minimum: (-0.9297322, -0.7298212)
Multiplication Process-
Y= -(1/35)(x+2.5)(X-1)(X-7.5)
Y=-(1/35) (X*X)+(2.5x)-(x)-(2.5)(x-7.5)
Y=-(1/35) (X2+1.5X-2.5) (X-7.5)
Y=-(1/35)(X*X2)-(1.5X*X)-(2.5x)-(7.5*X2)-(1.5X*7.5)+18.75
Y=-(1/35)(X3-6X2-13.75x+18.75)
Standard Form for Polynomial #2
Y=-(1/35)X3+0.171428514X2+0.3928571429X-0.5357142857)
Explanation
Polynomials can cross the X-axis multiple times. If imaginary numbers aren’t accounted for, the least a polynomial can cross the X-axis is once. If imaginary numbers are used, then it doesn’t have to. A vertex is the highest or lowest point of a parabola, depending on weather or not it is positive or negative. The x-axis in any polynomial can be used to construct an equation that describes the line. For example, a zero for a polynomial that crosses the x-axis at 4 would be (x-4). This means that when x=4, y will be zero because they will cancel out and any other number that is involved in the polynomial that is multiplied will be canceled as well because anything multiplied by 0 equals 0.
Minimum and maximum are referred to as local minimum and maximum because a polynomial can have multiple minimums and maximums. So, the “local” minimum is the lowest part of one of the curves, and the “local” maximum is the highest point of one of the curves in a polynomial. Because my second polynomial had an X3 variable, it had a maximum and a minimum. My first was a simple parabola, so since it was facing downward, it only had a maximum.
Reflection
This project was alot of fun and I definitley got more practice using polynomials. One thing that I enjoyed was the fact that I was able to construct whatever I wanted after we had our polynomial curves. It left a bunch of room for creativity which I appreciated greatly. I found it really hard to put equations into a computer because I didn't necessarily know where the different symbols were.