Quintessence of Quadratics
MARCH 2017-JUNE2017
Introduction to Quadratics:
What is a quadratic equation?:
A quadratic equation is an equation involving the term x^2. It can be written as a binomial and is used in kinematic equations. We started off the project with a kinematic equation that described the height of a small rocket launched off the roof of a building as a function of time. The kinematic equation presented for this was h(t)=-16t^2+92t+160, which is, of course, a quadratic equation of the form y=ax^2+bx+c. After further analysis you can surmise that in the kinematic equation the coefficient A describes the force of gravity on the rocket, B describes the rocket's initial vertical velocity and C describes the rocket's initial height. Our original goals in this project were to find how high the rocket would go, how long it would take for it to get there, and how long until the rocket crashes into the ground. To solve these problems we would have to convert the equation into other forms.
What is Standard Form?
The standard form of a quadratic is y=ax^2+bx+c. It is useful for finding the y intercept of the parabola, which is always C and, as shown earlier, it is useful when creating kinematic equations because you can describe forces using the other coefficients.
What is Vertex Form?:
The vertex form of a quadratic equation is written as y=a(x-h)^2+k, where the vertex of the parabola is (h,k). It is very useful for finding the vertex of a parabola, because otherwise you would probably have to find the x value in between the two x intercepts and solve from there, but sometimes there are no x intercepts so you would have to deal with imaginary numbers which are needlessly complicated, so it's usually best to just convert to vertex form and save yourself a headache.
A quadratic equation is an equation involving the term x^2. It can be written as a binomial and is used in kinematic equations. We started off the project with a kinematic equation that described the height of a small rocket launched off the roof of a building as a function of time. The kinematic equation presented for this was h(t)=-16t^2+92t+160, which is, of course, a quadratic equation of the form y=ax^2+bx+c. After further analysis you can surmise that in the kinematic equation the coefficient A describes the force of gravity on the rocket, B describes the rocket's initial vertical velocity and C describes the rocket's initial height. Our original goals in this project were to find how high the rocket would go, how long it would take for it to get there, and how long until the rocket crashes into the ground. To solve these problems we would have to convert the equation into other forms.
What is Standard Form?
The standard form of a quadratic is y=ax^2+bx+c. It is useful for finding the y intercept of the parabola, which is always C and, as shown earlier, it is useful when creating kinematic equations because you can describe forces using the other coefficients.
What is Vertex Form?:
The vertex form of a quadratic equation is written as y=a(x-h)^2+k, where the vertex of the parabola is (h,k). It is very useful for finding the vertex of a parabola, because otherwise you would probably have to find the x value in between the two x intercepts and solve from there, but sometimes there are no x intercepts so you would have to deal with imaginary numbers which are needlessly complicated, so it's usually best to just convert to vertex form and save yourself a headache.
Converting between forms:
Factored to Standard:
Converting a factored quadratic to a Standard form quadratic can be done by distributing one of the factors into the other (this is sometimes called FOIL for binomials). For an equation y=(a+b)(c+d) you would multiply every term by each term in the other factor, resulting in y=ac+ad+bc+bd. When doing this with a quadratic in the form y=a(x-p)(x-q) you get y=ax^2-axq-axp+apq. |
Standard to Factored:
You can factor a standard quadratic in 2 main ways, by guessing and and experimenting with random potential factors or by using the quadratic formula. For an equation y=x^2+bx+c you might be able to guess the factors (x+p)(x+q) by recognizing that p*q=c and p+q=b. However, if you don't feel like doing that you can use the quadratic formula (-b+-sqrt(b^2-4ac))/2a, where a, b and c are the same constants from y=ax^2+bx+c and the equation is equivalent to the x intercepts of the parabola. You can prove this equation is correct by describing the factored form and standard form of a quadratic with the same constants and using equalities to write the intercepts in terms of those constants. |
Standard to Vertex:
1) Factor out coefficient. This is the inverse of distributing, therefore you divide each number by the coefficient, an area diagram can help you stay organized. 2) Fill out area diagram, this is what allows the problem to be converted into vertex form. 3) The fourth square in the area digram gave and answer of nine, but when added to the equation it is no longer equals the same equation we started with, so in this case we have to subtract 9. Since x^2-6x+9 is equal to x-3^2 we can put that in the parentheses. This leaves us with -9+8. 4) Distribute the coefficient, an area diagram can help this step remain organized. Final Answer: y=-2(x-3)^2+2 |
Vertex to Standard:
1) Make sure equation is written in vertex form (y=a(x-h)^2-k) 2) Expand (x-h)^2 3) Fill out area diagram using x-h for each side length. From there multiply side lengths together to fill out the diagram. The area diagram does not need to be drawn to scale. 4) Combine like terms, in this case it happens to be -3x+-3x 5) Distribute coefficient, an area diagram can be used to help keep things organized. 6) Combine like terms Final Answer: y=-2x^2+12x-16 |
Kinematic Problems:
A good example of a problem involving a kinematic equation is where we had to find the maximum height, the time until maximum height and the time until landing for a rocket launched off the roof of a building whose height follows the kinematic equation h(t)=-16t^2+92t+160. We know that the function is a parabola, so we can simplify the question to know that we need to find the coordinates of the vertex and the positive x-intercept. We can do this by converting the equation to vertex form to find the vertex, and factored form to find the x-intercept. By completing the square we know that, when written in vertex form, the equation is
h(t)=-16(t-2.875)^2+292.25, so we know that the maximum height of the rocket is 292.25 and the time until it reaches that height is 2.875. The equation can't be factored into nice looking numbers, but by using the quadratic formula we can find that the positive x intercept is roughly equal to 7.14883.
h(t)=-16(t-2.875)^2+292.25, so we know that the maximum height of the rocket is 292.25 and the time until it reaches that height is 2.875. The equation can't be factored into nice looking numbers, but by using the quadratic formula we can find that the positive x intercept is roughly equal to 7.14883.
Geometry Problems:
A lot of the problems we did involved some sort of geometry, ranging from finding the distance of a boat of a coast based on it's distance from two lighthouses to finding the area of a shave a variable side lengths. An example of this would be the problem shown to the right, where we were tasked with finding various bits of information about a cattle pen built against a preexisting fence. The person building the fence has a set amount of fencing they can use, and we are tasked with using equations to find the best way they can set up their corral to have the most area. You can do this by writing an equation describing the area of the corral based on a side length and then converting it to vertex form to find the maximum area.
Economics Problems:
The next real place we see correlation is in economics. Imagine you are an executive of a store and the store has 3,000 of the same mugYou are curious and want to sell these shirts to make the best profit. You originally say you want to sell them for 50 dollars each, but another store is selling them for 30. You want to find the best way to profit while selling the most shirts compared to the other store. This is a problem that could be solved with quadratic equations. You would use the different forms while conjecturing and testing. From here you can find the numbers.
Reflection:
I am proud of the work I did in this project. I collaborated with others, challenged myself, and learned to show my work in a clear and understandable way. I practiced self advocacy by speaking up when I was confused, and I am now confident in my ability to solve problems involving quadratics. This will help me with the SAT, as I have already solved SAT practice problems on Khan Academy that I was unable to solve before. I regret not keeping better track of my work throughout the semester. That would have helped me with this DP update immensely. As I move into 11th grade, I will remember the importance of keeping papers organized, even after they are graded. Overall, I am confident that I will be a better 11th grade math student because of this project.