Introduction
In this unit we learned about quadratic equations. First we did problems with kinematics. We learned how to find average velocity on a distance vs time graph and we learned how to find the average acceleration on a velocity vs time graph. The way to do it is to find the area of everything below the line on the graph. We also derived the distance formula by using trigonometry and two coordinates.
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Vertex Form
Vertex form is a form of a quadratic equation. It shows you where the vertex is inside of the equation itself. It is written as y=a(x-h)^2+k.
The a in the equation determines how narrow it is.
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The h in the equation determines how far to the left or right the vertex is from the y axis and is the x-coordinate in the vertex. When you subtract h it makes a positive number.
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The k in the equation determines how far up or down the vertex is from the x axis and is the y-coordinate in the vertex.
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Other Forms of Quadratic Equations
Converting Between Forms
Converting from vertex form to standard formTo convert into standard form from vertex form you need to distribute out the squared by using an area diagram. Then if there is an a you would distribute that into the new numbers. After you do that you then combine all like terms and you get standard form
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Converting from standard form to vertex formTo convert standard form to vertex form you must take the two numbers with x and make x^2 be alone by dividing, and that will be your a, and then reverse the area diagram, that will get you the (x-h)^2 part. After that you must take go back to standard form with the completed area diagram in parenthesis and take the newly added bit and either add or subtract it from n and out side of the parentheses and you get the k.
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Converting from standard form to Factored formto convert from standard form to factored form you would use the area diagram for the whole equation first getting the x^2 alone by dividing every thing by that number, but sometimes you can't. Then using the area diagram and guess and check until you guess both sides and then write it out.
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Solving Problems with Quadratic Equations
The proble i will solve is the victory celebration problem. We are given this quadratic equation(-16x^2+92x+160) and we had to answer three questions. They were, how high will the rocket go, how long will it take to get there and how long will it take to land. For the first 2 questions you can convert it into vertex form which is (-16(x-2.875)^2+292.25). From their we can see that it reached 292.25 feet upwards and it took 2.875 seconds to get there. Now comes the tricky part of finding how long it was in the air for. First we can make y zero and solve for x which gives us, 7.149 seconds.
Reflection
I had a lot of fun learning in this unit. the only thing I know about quadratics before this unit was that when x is squared then it makes a parabola. Now I know about how simple kinematics work, How to convert to different forms, What those different forms are, and how to use them to solve real world problems. I could potentially seeing myself using this skill in the future for school or even my work.
Habits of a Mathematician
Looking for PatternsI used this during the first challenge problem to help me solve the problem. I was looking for anything that I could go with to solve it.
Take Apart and Put Back TogetherWhen converting standard form into vertex form I would take apart the standard form to create all of the parts for the vertex form and then put them back together.
Describe and ArticulateWhen ever I convert with vertex form I draw the area diagram to helm me out.
Collaborate and ListenI used this when I worked in groups and when me and Carter debated on who got the right answer on a question.
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Start SmallEvery time I start a problem I convert the equation into vertex form so I may have an easier time.
Conjecture and TestI used this when I was converting to factored form because that is the only way to factor the standard form.
Seek Why and ProveI did this when i checked my work to see if I got it correct and to defend the answer I got.
GeneralizeI use this during conversions by creating steps in order to be able to use them for all equations.
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Be SystematicWhen ever I started a problem I would convert to vertex form and go on from there.
Stay Organizedi followed this when creating this DP update making sure everything looks nice and neat.
Be Confident, Patient, and PersistentI used this during the challenge problems that I did because i didn't know how to do them but I still tried to solve them.
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