Introduction
The Quintessence of Quadratics project was made with the purpose of teaching us how to understand quadratic functions through using methods relating to volume and area, projectile motion, and pythagorean's theorem.
Volume: V = length x width x height
Area: A = length x width
Projectile Motion: A projectile is any object that is given an initial velocity, then follows a path that is entirely caused by gravity.
Pythagorean's Theorem: a²+b²=c²
By learning and using these different concepts, we better understood how to evaluate and solve quadratic functions.
In the beginning of this project, we learned how to calculate distance, velocity, and time using graphs. We then learned how to evaluate parabola's. There are two ways to write a parabola equation: standard form and vertex form. We learned about how each part of these equations affects a parabola's location, vertex, width/distance and its concave. We figured out what each variable represented. We then practiced how to convert from vertex and standard form using area diagrams.
Standard form: y=ax²+bx+c
Vertex form: a(x-h)²+k
Example Area Diagram:
Volume: V = length x width x height
Area: A = length x width
Projectile Motion: A projectile is any object that is given an initial velocity, then follows a path that is entirely caused by gravity.
Pythagorean's Theorem: a²+b²=c²
By learning and using these different concepts, we better understood how to evaluate and solve quadratic functions.
In the beginning of this project, we learned how to calculate distance, velocity, and time using graphs. We then learned how to evaluate parabola's. There are two ways to write a parabola equation: standard form and vertex form. We learned about how each part of these equations affects a parabola's location, vertex, width/distance and its concave. We figured out what each variable represented. We then practiced how to convert from vertex and standard form using area diagrams.
Standard form: y=ax²+bx+c
Vertex form: a(x-h)²+k
Example Area Diagram:
Exploring the Vertex Form of the Quadratic Equation
Through handout numbers 4 - 7, we learned how each part of the parabola equation affects the parabola.
Parabola Formula: a(x-h)²+k
Vertex of the Parabola = (h,k)
These handouts helped us understand what each part of the parabola equation does because, through practice problems, we found patterns that helped us determine what each variable of the formula meant.
Parabola Formula: a(x-h)²+k
Vertex of the Parabola = (h,k)
These handouts helped us understand what each part of the parabola equation does because, through practice problems, we found patterns that helped us determine what each variable of the formula meant.
Example: How to find the equation of a parabola.
The parabola has a vertex of (2,6).
The parabola concaves or faces down.
The equation of the parabola, using the formula a(x-h)²+k, would be: -(x-2)²+6.
The Relationship Between a Quadratic Equation and a Parabola
Quadratic Formula: y=(x-h)²
A quadratic equation can be written out and expanded to provide the same information that the standard form of a parabola's equation gives.
The parabola concaves or faces down.
The equation of the parabola, using the formula a(x-h)²+k, would be: -(x-2)²+6.
The Relationship Between a Quadratic Equation and a Parabola
Quadratic Formula: y=(x-h)²
A quadratic equation can be written out and expanded to provide the same information that the standard form of a parabola's equation gives.
Other Forms of the Quadratic Equation
We learned about all three forms of writing quadratic equations, not just vertex form. We also learned about standard form and factored form. Standard form was the first quadratic equation we learned about. It's written as ax² + bx + c. The a variable determines the width of the parabola, and c is the y-intercept of the parabola.
Factored form/intercept form is written as a(x-n)(x-m). The variables n and m are the x intercepts of the graph, and the a constant is the width of the parabola.
Factored form/intercept form is written as a(x-n)(x-m). The variables n and m are the x intercepts of the graph, and the a constant is the width of the parabola.
Converting Between Forms
Parentheses
Exponent
Multiply
Divide
Add
Subtract
Vertex Form to Standard Form
Vertex Form: a(x-h)²+k
Example: y= -2(x-4)²+3
1. Expand the equation by writing the exponents.
y=-2(x-4)(x-4)+3
Exponent
Multiply
Divide
Add
Subtract
Vertex Form to Standard Form
Vertex Form: a(x-h)²+k
Example: y= -2(x-4)²+3
1. Expand the equation by writing the exponents.
y=-2(x-4)(x-4)+3
2. Create an area diagram and wre-write the equation.
y=2(x²-4x-4x+16)+3
y=2(x²-8x+16)+3
y=2x²-16x+32+3
y=2x²-16x+35
Vertex Form: y=-2(x-4)²+3
Standard Form: y=2x²-16x+35
Standard Form to Vertex Form
Example: y=2x²+8x-3
1. Factor out the coefficient. Make it so the equation is back to distributive form.
y=2(x²+4x)-3
2. Fill out an area diagram to rewrite the equation.
y=2(x²+4x+4-4)-3
3. Convert to vertex form.
y=2((x+2)²-4)-3
y=2(x+2)²-8-3
y=2(x+2)²-11
y=2(x²-4x-4x+16)+3
y=2(x²-8x+16)+3
y=2x²-16x+32+3
y=2x²-16x+35
Vertex Form: y=-2(x-4)²+3
Standard Form: y=2x²-16x+35
Standard Form to Vertex Form
Example: y=2x²+8x-3
1. Factor out the coefficient. Make it so the equation is back to distributive form.
y=2(x²+4x)-3
2. Fill out an area diagram to rewrite the equation.
y=2(x²+4x+4-4)-3
3. Convert to vertex form.
y=2((x+2)²-4)-3
y=2(x+2)²-8-3
y=2(x+2)²-11
Factored Form to Standard Form
1. Plug the equation into an area diagram.
2. Re-write the equation.
3. Combine like terms.
4. Distribute the coefficient.
5. Convert to Standard Form.
2. Re-write the equation.
3. Combine like terms.
4. Distribute the coefficient.
5. Convert to Standard Form.
Standard Form to Factored Form
1. Create an area diagram and plug in the equations.
2. Rewrite the equation.
3. Create a second area diagram to organize the equation.
4. Convert to factored form.
1. Create an area diagram and plug in the equations.
2. Rewrite the equation.
3. Create a second area diagram to organize the equation.
4. Convert to factored form.
Solving Problems with Quadratic Equation
3 real world situations quadratics can be used in:
1. Kinematics
Finding the relation between velocity and time of an object in motion.
2. Geometry
Finding the volume of something to figure out how much space is needed for that certain thing. For example if you needed to fill up a fish tank, or bucket, of water, you'd need to know the volume to know how much water you need.
Finding the area of things can also help with things too. For example if you wanted to build a house, garden, fence, etc.
3. Economics
Finding the prices of things and the relation between how much of a product you sell and how much you earn. For example, if you own a business or company, this would be very useful for profit.
1. Kinematics
Finding the relation between velocity and time of an object in motion.
2. Geometry
Finding the volume of something to figure out how much space is needed for that certain thing. For example if you needed to fill up a fish tank, or bucket, of water, you'd need to know the volume to know how much water you need.
Finding the area of things can also help with things too. For example if you wanted to build a house, garden, fence, etc.
3. Economics
Finding the prices of things and the relation between how much of a product you sell and how much you earn. For example, if you own a business or company, this would be very useful for profit.
Reflection
I found this project very challenging. What I found challenging was the concept in general. It took awhile for me to understand, but I eventually got the hang of it. What really helped me understand the concept we were learning about was the area diagrams. Many people were able to do the equations and mental math in their minds, but I found using the area diagrams very organized and helpful. Although it took me awhile to learn, I put a lot of effort into understanding this topic. This impacted my goals about 11th grade math and SAT prep, because it influenced me to practice math more so I'm for sure prepared in the future. It took me awhile to learn most of these math concepts we were practicing, so it showed me that I need to start preparing ahead of time for college readiness.
Starting Small and Looking for Patterns- I used starting small by breaking down equations to convert one form of a parabola equation to another. For example, when we're converting vertex form to standard form.
Being systematic- I used being systematic by trying different general ways to solve the problems we were working on. Problems relating to volume, area, and parabolas.
Conjecturing and testing- I used conjecturing and testing when we were converting from standard to vertex and vertex to standard form to because I tried different methods of conversions. Whether it was using mental math to convert the equations, or using an area diagram.
Staying organized- I used staying organized by using an area diagram for every conversion problem. Instead of using metal math, like many people in our class did, I found using the area diagram more organized and the setup felt really helpful.
Describing and articulating- I used describing and articulating by using the area diagrams.
Seeking why and proving- I used seeking why and proving by working with my peers. Like I said earlier, this concept was challenging for me to understand, so when I asked why things happened the way they did to my peers, it helped me understand how to do the work we were doing.
Being confident, persistent and patient- I used being patient when I was having a really hard time figuring out how to do the conversions. Although it took time, I was able to understand how to do these conversions with different methods.
Collaborating and listening- As mentioned earlier, I used collaborating and listening by working with my peers to have a better understanding of what we were learning.
Generalizing- I used the habit of a mathematician generalizing by generalizing the different ways we could solve for volume and area.
Being systematic- I used being systematic by trying different general ways to solve the problems we were working on. Problems relating to volume, area, and parabolas.
Conjecturing and testing- I used conjecturing and testing when we were converting from standard to vertex and vertex to standard form to because I tried different methods of conversions. Whether it was using mental math to convert the equations, or using an area diagram.
Staying organized- I used staying organized by using an area diagram for every conversion problem. Instead of using metal math, like many people in our class did, I found using the area diagram more organized and the setup felt really helpful.
Describing and articulating- I used describing and articulating by using the area diagrams.
Seeking why and proving- I used seeking why and proving by working with my peers. Like I said earlier, this concept was challenging for me to understand, so when I asked why things happened the way they did to my peers, it helped me understand how to do the work we were doing.
Being confident, persistent and patient- I used being patient when I was having a really hard time figuring out how to do the conversions. Although it took time, I was able to understand how to do these conversions with different methods.
Collaborating and listening- As mentioned earlier, I used collaborating and listening by working with my peers to have a better understanding of what we were learning.
Generalizing- I used the habit of a mathematician generalizing by generalizing the different ways we could solve for volume and area.