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Chapter 6: Quadratic Functions

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Wiki Summary Assignments

6-1: Tarah L. and Dale N.
6-2: Gage A. and Marissa B.
6-3: Kyle L. and Sarah P.
6-4: Crystalene E. and Kim K.
6-5: Nick H. and Marisa M.
6-6: Dominic F. and Autumn T.
6-7: Sarah K. and Brittany K.
6-8: Elissa M. and Shannon M.
6-9: Dakota E. and Austin L.
6-10: Mitch K. and Thomas S.

Quadratics Research Project

Link to audio reading of assignment (this is an example of what you will do)
Or download it: Quadratic Research.mp3

Assignment: Quads Research Assignment.pdf
Rubric: Quadratics Research Rubric.pdf

Section 6-1: Quadratic Expressions, Rectangles, and Squares

AA Section 6-1
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In this lesson we learned about the Quadratic Expression which is ax²+bx+c, the Quadratic Equation which is ax²+bx+c=0 and the Quadratic Function which is f:x→ax²+bx+c. All of these are for the variable x. We also learned how to make (x+7)² into a expression. You can just use the foil method. F- first O- outer I- inner L- last. We also learned a quicker way to do this by using the Binomial Square Theorem. (x+y)²=x²+2xy+y² and (x-y)²=x²-2xy+y². DALE N.

Section 6-2: Absolute Value, Square Roots, and Quadratic Equations

AA Section 6-2
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absolute value- Proxy-Connection: keep-alive Cache-Control: max-age=0 e distance of n from 0 on the number line.
absolute value function- its domain is the set of real numbers and the range is the set of nonnegative real numbers.
irrational numbers- a real number that can't be written as a simple fraction.
absolute value graphs make a "v."
the simplest quadratic equations are of the form x squared = k.
absolute value-square root theorem - for all real numbers x, the square root of x squared = the absolute value of x.

= ] marissa b. and gage a.!!!!!!!!!!!!

Section 6-3: The Graph-Translation Theorem

AA Section 6-3
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Graph-Translation Theorem: In any equation using x and y, the following are the same:
1. Replace x with x-h and y with y-k
2. Apply Tbk to the original base equation(Tbk is the translation of h horizontal units and k vertical units)
Vertical Form of a Parabola: y=a(x-h)² + k or y-k=a(x-h)², gives a horizontal shift of h units and vertical shift of (h,k) (The vertex will be (h,k)
Axis of Symmetry: A line that goes through the vertex that shows symmetry of a parabola; equation is x=h
Maximum/Minimum: Occurs at the vertex; minimum when it opens up and maximum when it opens down
When graphing a parabola by hand: 1. Find the vertex 2. Find the symmetrical values on either side of the vertex 3. Fill the graph
-Sarah P.

Section 6-4: Graphing y = ax² + bx + c

AA Section 6-4
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Standard Form for the Equation of a Parabola: y-k=a(x-h)² can be written in standard form, y=ax²+bx+c.
Example: y=2(x+3)²-8, and y=2x²+12x+10
Step 1:Expand the binomial, and simplify the right side of the equation.
Step 2: Put the Equations together. y= 2(x²+6x+9) - 8 Step 3: Expand it: y=2x² +12x +10
Make sure both equations are equivalent.
Then Check:
Graph both equations to see if the graph is the same.

Theorem: The graph of the equation y= ax² + bx + c is a parabola congruent to the graph of y= ax².
2x² +12x +18 - 8 Step 4: Find the equation: y

Crystalene E.

y-k=a(x-h)² equations fir a parabola also standard form for that is y=ax²+bx+c The graph of y=ax²+bx+c is congruent to y=ax²
Kim K.

Section 6-5: Completing the Square

Rewriting in vertex form
Completing the Square
There are five main steps to completing the square.
1. First you isolate the x terms. say you have the equation y= x² +18x + 90 you would first subtract 90 from each side to get the x's by themselves.
2. Make sure a=1, then find b. which in order to find b you must use the formula (1/2b)² so you would plug in 18 for b and do the math. In this case you would get that to equal 81.
3. Then add 81 to both sides * A must be equal to 1*
4. Then you factor the perfect square trinomial and simplify. y-90 +81= x² +18x + 81 then it becomes x² +18x +81 which when taking the square root of it will give us the equation y=(x +9)²
So there are overall the theorem you need to know is
x²+bx+ (1/2b)²= (x+1/2b)²= (x+b/2)²
Marisa M.

Completing the SquareThe main way of completing the square is to have the equation you are working with in standard form of a parabola. When you have the equation in standard form you then take (1/2b)². Once you get what this part of the equation is you then subtract what you get to the other side of the equation. From this point you put the other side of the equation into something that looks like this. (x-5)² then you take the other side of the equation and subtract or add it from the other side. Which looks like this y-7. So the final outcome should look like a vertex equation in the from y=(x-5)²+7. From this point you can easily find the vertex of the parabola.
Nick H.

Section 6-6: Fitting a Quadratic Model to Data

Fitting a Quadratic Model to Data.
You learn how to find equations of parabolas that go through three different points. You also use techniques for solving systems of equations.Then you take a 3-by-3 system can be solved using linear combinations, substitutions or matrices. You then have to try and eliminate a variable and then you have to solve the 2-by-2 system like you always would. Then when you find the variable you must substitute it back into the equation with two variables to find the second. Once you have found both variables you need to them substitute them back into the original equation and solve for the third and final variable. Autumn T.

Dominic F.
6.6 Fitting A Quadratic Model to Data
In lesson 6-6 you learn how to find the equation of a parabola with three given points. You learn different methods to solving systems of equations. You learn how to take a 3x3 system and solve it using three different ways, linear combinations, substitutions, and matrices. Then once you work the 3x3 down to a 2x2 you then can solve for one variable. Once found the variable can be plugged in to one of the previous equations to solve for another variable. Then you take both the variables to find the last one and then make your final equation.

Section 6-7: The Quadratic Formula

The Quadratic Formula.
X equals negative b plus or minus the square root of b squared minus 4 times a times c, all over 2a.
And, you should deff sing this to pop goes the weasel, because its fun:]
You will take the standard form of a quadratic equation and then apply it to the quadratic formula. If the equation equals zero then you must plug in a b and c into the standard form of the quadratic equation. This will then split into two equations, you have to take the positive and negative of this equations and you will get two answers.
BRITTANY K.:]]]]]]]]
ps, i want the purple space man thing, bad.

Quadratic Formula First things first! The Quadratic formula theorem is x= to -b +or- the square root of b to the squared - 4ac all over 2a
*and easy way to remember is to sing it to pop goes the weasel! x equals negative b plus or minus the
square root of b squared minus 4ac all over 2a
This formula is only used when y IS equal to zero. What happens when it isn't??? What do we do when the square root of b squared- 4ac is not a perfect square???Here's an EXAMPLE (yah examples!!!!!!!!)
On page 382 we are given the Pop Fligh's problem where y is equal to 50. The only thing you have to do is to
subtract 50 from each side. It's that simple!
Sarah K.

Section 6-8: Imaginary Numbers

Today's lesson found here

Theorem: If k>0, the square root of negative k= i times the square root of k. In this lesson we learned that i equals the square root of -1.
We know that the square root of a times b is equal to the square root of a times the square root of b, but does this work when they are negative? The square root of -16 times the square root of -25 is not equal to the square root of -16 times -25 which then simplifies into the square root of 400 and then 20 which we know is not equal.
Shannon M.

Big Question what is the square root of negative k?Here is the theorem: if k is negative then the square root of negative k is is equal to the square root of -1 which is represented with the variable i. the biggest thing to remember is that you cannot combine radicals that have negatives inside
Elissa M

Section 6-9: Complex Numbers

Complex Numbers

Well first of all this whole lesson is about Complex Numbers OoooOOOOooOO!!!!!!!! Can you guess what a complex number is? NOPE, WRONG! A complex number is as followed, a+bi. With a and b being real numbers and i is the square root of -1. Not that complicated right? Now moving on to Equal Complex Numbers, which are: Two complex numbers where the real parts are the same and the imaginary parts are the same. Let's look at an example now: 2 + 4i = y + xi (or something along those lines) what does y and x equal? y=2 and x=4. Now the final part of the lesson, YAY!! Complex Conjugate (remember that word from English class? WELL DON'T WE'RE TALKING ABOUT MATH STAY FOCUSED!) okay so a complex conjugate is a complex number but the sign thing-y in between a and bi... Let me verify that, If the complex number is a + bi then its conjugate iiiiiiiiiiiis
DUN DUN DUN! ... its a - bi. YOU LEARNED THE LESSON SORTA!!!! yay... woot, ahem...

Dakota & Austin L. (Dakota didn't put his last name because he is the only dakota)

Section 6-10: Analyzing Solutions to Quadratic Equations

We already know how to find the real solution to quadratics. In three ways including
1) Calculator
2) Systems(y=0 and quadratic is in standard form)
3) Quadratic Formula
But we learned in this lesson that the discriminant(from the quadratic formula the stuff under the radical) helps us tell how many solutions there there will be for the given quadratic.The discriminant for the quadratic formula is b²-4ac.When the quadratic is in standard for it should be in the form of ax²+bx+c = 0, when using the quadratic in standard form there are three different possible outcomes.
1) b²-4ac > 0 (When the discriminant is greater than zero) 2 real solutions
2) b²-4ac = 0 (When the discriminant is equal to zero) 1 real solution
3) b²-4ac < 0 (When the discriminant is less then zero) 2 complex solutions
This also helps when you are solving for the solutions later on because you already solved the discriminant, which is the bulk of the quadratic formula which we all love to sing(not really...) x = -b+/- √b²-4ac

This is a lesson where you find out how certain outcomes of quadratic equations effect the graph and the number of solutions to the quadratic equation. We learned the discriminant, which is the stuff inside the radical, or b² - 4ac. If that is positive, there will be two real solutions (no "i" to deal with) and two x intercepts. If it's negative, there's two complex solutions and no x intercepts. And if it's zero, the vertex of the parabola is on the x axis, so there's just one x intercept.

-Mitch K.