Introduction and Exploring the Vertex Form of the Quadratic Equation
The Quintessence Of Quadratics was a project course that teaches us the fundamentals of Quadratics. Everything from the algebra equations to the geometry style graphing. The learning objectives of this project was to learn about A couple things: First was learning about vertex form of a quadratic equation. This is where we discover how the a, h, and k parameters of quadratic equation affect the size and location of a parabola on a graph. Second is when we explored the other forms of quadratic equations like standard and factored. Soon after we that we learned how to convert all the quadratics forms into the other quadratics forms. Then the last part was where we took all the knowledge used it to solve real world problems.
We used both algebra and geometry to discover the physics displacement equation. For algebra we used the algebraic equation of y = a(x - h)^2 + k. The ‘y’ on the equation is representing the y-intercept of the parabola on the graph. There is only one y-intercept as opposed to the ‘x’ factor which means the x-intercept. There can be two x-intercept of the parabola. The ‘a’ parameter represents the the width and the direction of the parabola. The bigger ‘a’ value means that the parabola will be thinner, and if the ‘a’ value is smaller then it will be wider. If it is a negative number then the parabola will be curving downwards and if it's a positive number the curve will be facing upwards. ‘H’ and ‘k’ are very similar but are still a little different. They both determine the location of the vertex. ‘H’ is what determines the x location of the vertex. If the number is positive then it is going to be on the positive side of x-axis, and if it's negative then its going to be on the negative side of the x-axis. For ‘k’ it is determining the location on the y-axis. If the number is positive then it will be above the x-axis and if it's negative then its going to be on the negative side of x-axis.
We used both algebra and geometry to discover the physics displacement equation. For algebra we used the algebraic equation of y = a(x - h)^2 + k. The ‘y’ on the equation is representing the y-intercept of the parabola on the graph. There is only one y-intercept as opposed to the ‘x’ factor which means the x-intercept. There can be two x-intercept of the parabola. The ‘a’ parameter represents the the width and the direction of the parabola. The bigger ‘a’ value means that the parabola will be thinner, and if the ‘a’ value is smaller then it will be wider. If it is a negative number then the parabola will be curving downwards and if it's a positive number the curve will be facing upwards. ‘H’ and ‘k’ are very similar but are still a little different. They both determine the location of the vertex. ‘H’ is what determines the x location of the vertex. If the number is positive then it is going to be on the positive side of x-axis, and if it's negative then its going to be on the negative side of the x-axis. For ‘k’ it is determining the location on the y-axis. If the number is positive then it will be above the x-axis and if it's negative then its going to be on the negative side of x-axis.
For the following examples: The color next to the equation respents the same color parabola.
Examples of A
- The red parabola is the only one facing up because the 'a' value in the equation is not negative
- The green parabola is the thinest because the it has the biggest 'a' value
- The orange parabola is the widest because it has the lowest 'a' value
Examples of H
- The blue parabola is on the positive side of Y-axis because the 'h' value in the equation is positive
- The purple parabola is on the negative side of the y-axis because the 'h' value is negative
Examples of K
- The green parabola is the highest above the x-axis because it has the largest 'k' value
- The orange parabola is the lowest below the x-axis because is has the lowest 'k' value
Examples of everything combined
Here is the worksheets I did that thought me all about A, H, and K
Other Forms of the Quadratic Equation
The folwing equations above are in the form of what is called ‘vertex form’. There is also two other common equation forms that can respent a parabola. The other two forms are ‘standard form’ and ‘factored form’. Both of these have an unique advantage to them. Standard form can give you the y-intercept of the parabola, and factored form can give you the two x-intercpets. The equation for standard form is: y = ax^2 + bx + c. the ‘c’ is the y-intercept. The equation for factored form is: y = a(x - p) (x - p), the two ‘p’ are for the x-intercpets Note: for any parabola there is never more than one y-intercept, there can be only one intercept or none. But for x-intercept their can be can be one or two intercpets, or no x-intercpet at all.
Converting between the Forms
Converting a form into a different is useful in the sense that it allows us to get an intercept that the current equation can not provide.
Vertex form into standard form
1. take your vertex form and put it into the squares by making two (a -h), you can do this because there it is squared
2. You then combine your two H values and add together and then multiply then
3. with this you take your: x^2, your newly added H value and the multiplied H value and put then into the begging on the standard equation, everything that was left on the outside before still stays on the outside.
4. if there any distributing to be done then do that first, if not then take the old K value and add/subtract from the new k value
5. combine the two x values and your done
2. You then combine your two H values and add together and then multiply then
3. with this you take your: x^2, your newly added H value and the multiplied H value and put then into the begging on the standard equation, everything that was left on the outside before still stays on the outside.
4. if there any distributing to be done then do that first, if not then take the old K value and add/subtract from the new k value
5. combine the two x values and your done
here are some examples ^
standard to vertex
standard to vertex is quite the same
1. take your standard form and put parentheses around everything but y = and k
2. expand the equation into the box and get your new values and put them into a early vertex form
3. simplify just to get you new vertex form
Note: the steps for standard to vertex are the same, if you need more detail then go back up to the steps of vertex to standard
1. take your standard form and put parentheses around everything but y = and k
2. expand the equation into the box and get your new values and put them into a early vertex form
3. simplify just to get you new vertex form
Note: the steps for standard to vertex are the same, if you need more detail then go back up to the steps of vertex to standard
factored form to standard form
factored form to standard form is even easier
1. for each of the two different (x-h) give then their own side on the square and get you new values with that
2. like above take your new values and plug them into a early version of the standard form
3. simplify the equation to get your standard from
1. for each of the two different (x-h) give then their own side on the square and get you new values with that
2. like above take your new values and plug them into a early version of the standard form
3. simplify the equation to get your standard from
Solving Problems with Quadratic Equations
kinematics
most of the kinematics problems that are solved with Quadratics mostly ones resenting the travel of an projectile. the graph can show many things about the projectile. the x-axis is for time and the y-axis is for height. with the two axis set up like this makes is so that the key parts of the graph have a more important value. the two x-intercepts are to now resent the start from ground and when it hits the ground. the vertex now shows the projectile max height and at what time it took to get there.
Geometry
geometry can be used in the way that I can help find the max area for a rectangle. so for example, if your are given a amount of perimeter what width and height would give you the max area. you can also do this with a triangle, all you would have to do is work with the pythagorean theorem.
Economics
How economics can be used in Quadratics is very straight forward. lets say you have a product that you want to sell, but you don't know what to put the price at. if you make the price too low you wont earn any profit, even when you could actually sell a lot that way you still wouldn't get any profit. if you make the price too high then no one will buy it because its too pricey. so what you can do is use quadratics to find the perfect price point where people wont be turned away from the price and when it is actually bought you actually get good amount of profit
Leslie's Flowers
Leslies's Flowers was a problem that was very long and hard, it involved everything. including trig. , pythagorean theorem, and quadratics.
The problem is that Leslie needs to find the area of her flower bed design, but what must she first do is find the distance from the altitude base to the left base point.
1. what i first did was make a equation that would solve finding x
-(14 -x)^2 + 15^2 = -x^2 +13^2
2. what i then did was simplify it down to x
The problem is that Leslie needs to find the area of her flower bed design, but what must she first do is find the distance from the altitude base to the left base point.
1. what i first did was make a equation that would solve finding x
-(14 -x)^2 + 15^2 = -x^2 +13^2
2. what i then did was simplify it down to x
now that i have found x Leslie's can now find the area of her triangle garden.
Reflection
as I reflect back on this whole quadratics section i see that at first everything was very easy. I would get the worksheet and complete it with the correct answers very fast. I was so good at it that many classmates had come to me for help with the problems. but from what i remember somewhere around worksheet #17 I had started to have a lot of trouble figuring out the answers. so much so that i had to start asking the same people that i was helping awhile ago for help now. What I think what happened was that when we started to move into the other forms of quadratics is that i lost track on how to do every conversion, and then on top of that i did not know what to convert the equation into. so typing up this Dp was very hard for me because i asked so many different people for help that it made me more confused. everyone had their own way of solving the problems.
habits of a Mathematician
Look for Patterns:
patterns that i looked out for was how to identify what equation I was looking at, vertex, standard, or factored. each from had its patterns that help with finding out what it was
start small:
how we started off small is how the first few worksheet were easy and all they did was explain the different forms, but things got harder when we were to convert the forms into other forms
Be systematic:
we had to be systematic for this because by having the same system for all the problems greatly helps solving it. like using the four boxes for all the conversions helps out a lot. if we did not do this because having different ways for each different problem takes up little time we have.
Take apart and put back together:
for what i can say for this is when we would have to expand the equation first before putting it into the box for the new values. and then put back together the equation with the new values.
conjure and test:
testing was very important because we needed to make sure that the conversions were true, like how we used demos to prove that the changed equation is still the same.
stay organized:
staying organized was very helpful in making this DP update. over the course of the quadratics every time i finished a worksheet i would take a picture of it so i could never lose it and use it later.
Describe and articulate:
when i was helping my friends i would have to describe very well on how to solve the problem, same goes on when they had to help me.
Seek why and prove:
when i got help from a friend i would ask another friend to see what he got, just to make sure that the frend that was helping me wasn't wrong.
Be confident and persistent:
if i did not know what conversion to use I would use all of then and use the best one
collaborate and listen:
there were a few times where me and my friends didn't know how to solve a problem, so we had to work together to solve it.
Generalize:
when reading a problem i did two things, i first removed any useless items like names and settings. and then i would try to figure out how i was going to solve the problem.
patterns that i looked out for was how to identify what equation I was looking at, vertex, standard, or factored. each from had its patterns that help with finding out what it was
start small:
how we started off small is how the first few worksheet were easy and all they did was explain the different forms, but things got harder when we were to convert the forms into other forms
Be systematic:
we had to be systematic for this because by having the same system for all the problems greatly helps solving it. like using the four boxes for all the conversions helps out a lot. if we did not do this because having different ways for each different problem takes up little time we have.
Take apart and put back together:
for what i can say for this is when we would have to expand the equation first before putting it into the box for the new values. and then put back together the equation with the new values.
conjure and test:
testing was very important because we needed to make sure that the conversions were true, like how we used demos to prove that the changed equation is still the same.
stay organized:
staying organized was very helpful in making this DP update. over the course of the quadratics every time i finished a worksheet i would take a picture of it so i could never lose it and use it later.
Describe and articulate:
when i was helping my friends i would have to describe very well on how to solve the problem, same goes on when they had to help me.
Seek why and prove:
when i got help from a friend i would ask another friend to see what he got, just to make sure that the frend that was helping me wasn't wrong.
Be confident and persistent:
if i did not know what conversion to use I would use all of then and use the best one
collaborate and listen:
there were a few times where me and my friends didn't know how to solve a problem, so we had to work together to solve it.
Generalize:
when reading a problem i did two things, i first removed any useless items like names and settings. and then i would try to figure out how i was going to solve the problem.