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Chapter 6: Quadratic Functions

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Wiki Summary Assignments

6-1: Nate J. and Christian S.
6-2: Andrew E. and Paige H.
6-3: Courtney G. and Nick Y.
6-4: Ryan B. and Jacob L.
6-5: Amira A. and Jessica R.
6-6: Josh H. and Tricia L.
6-7: Amanda T. and Trisha T.
6-8: Jeremiah B. and Emma R.
6-9: Jacob R. and Tyler Y.
6-10: Brittani H. and Joe Q.

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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There are a couple forms of a quadratic. They are quadratic expressions, quadratic equations, and quadratic functions. Each of these have a certain form that they follow. This form is the standard form of a quadratic. A quadratic expression can be expanded too. It is called expanding the power. When expanding a quadratic one has to write the expression two times as if you were multiplying the two. Quadratic can also be written in two or more variables. It doesn't always have to be written in the variable x. By Nate J.

In this lesson we figured out Quadratic form can come in a asortment of different ways. In math today it can be found in an expression, equations, and functions. They all then follow the standard form of a quadratic. the expressions, equations, and functions are in degree 2 and are not in standard form but can be then rewritten into standard form. You can do this process by expanding it by multiplying by itself then followed by simplifying. This is then just tied in to finding the area of rectangles and squares using the process from above. By Chris S.

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

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Lesson 6-2 was about absolute value, square roots, and Quadratic Equations. The absolute value of n is the distance of n from 0 on the number line. For example the absolute value of 5 is 5 and the absolute value of -5 is 5. The absolute value function is f(x) = the absolute value of x. It's domain is the set of real numbers while it's range is the set of nonnegative real numbers. The graph of this function is in the shape of a V. When working with square roots, if x is positive, then the square root of (x²) = x, if x = 0, then the square root of (x²) = 0, and if x is negative, then the square root of (x²) = -x, which is a positive number . This brings us to the Absolute Value - Square Root Theorem. This states that for all real numbers x, the square root of (x²) = the absolute value of x.
- Paige H.

This lesson is about absolute value, square roots, and quadratic equations. The absolute value of a number is written like lnl, the letter n represents the absolute value. lnl would be the distance of n to 0 on a number line, such as l23l=23 and l-23l=23 because it can never have a negative value. The absolute value of a number is written as a function such as lnl={x when x is greater than or equal to 0} or lnl={-x when x<0}. The absolute value function is written f(x)=lxl and when graphed makes a v shape. Andrew E.

Section 6-3: The Graph-Translation Theorem

AA Section 6-3
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The graph translation theorem states that when you have a graph, for example y= (x+4) squared, the graph would be translated 4 units to the left. If the equation were y= (x-4) squared, then the graph would be translated 4 units to the right. When you replace "x" with "x-h" the "h" would stand for the number of units you would move left or right. When you have a vertical graph translation problem, you would change "y" out with "y-k." If you would need to do a horizontal and vertical graph translation at the same time, you would use the formula T(sub h, sub k) where h moves horizontally and k moves vertically.
- Courtney G.

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

AA Section 6-4
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There are two types of equations: standard [ y = ax^2 + bx + c ] and vertex [ y = a(x - h)^2 = k ] form. You can turn a vertex problem into a standard problem by factoring. If you factor all of the vertex problems by using a=a, b=2ah, c=ah^2 = k. (h,k) is the vertex of the parabola in a vertex or standard form equation. Newton also designed a formula where the graph is a parabola. His equation deals with height and acceleration. h = (-1/2)gt^2 + (v-sub 0)t + (h-sub 0). You can solve for certain variables in the problems by substituting values in for the variables.
-- Ryan and Jake L.

Section 6-5: Completing the Square

The expanded binomial square is known as a perfect square trinomial (we squared something, so it's a perfect square, and it has three terms, making it a trinomial).Theorem (Completing the Square): To complete the square x^2 + bc, add (1/2b)^2.

In this section we learned how to complete the square. To complete the square ax^2+bx+c, add (1\2b)^2. We did some examples to practice using this method. The process we used is;
1. Isolate the x terms
2. Make sure that a=1, then find b
3. Add (1\2b)² to both sides of the equation
4. Factor the perfect square trinomial and simplify
5. Check
And when a does not equal, we need to factor out a from both terms that have an x. Our perfect square trinomial is of the form x²+bx+c.
-Amira A.

Rewriting in vertex form

Section 6-6: Fitting a Quadratic Model to Data

Lesson 6-6
In this lesson we learned how to fit a quadratic to data. What this basically means is the question will give you a table or the information needed and then you need to put them into the systems to find what a, b, and c are equal to. These systems have to be fit to the quadratic model of y= ax²+bx+c. Make sure that you keep in mind a, b, and c are just the variables that you plug your information into. Your model may not always go through every point that is given but it should be reasonably close to make a good model. You and another classmate may do the same exact problem and get a different function because it is only one model. If you want to check your model, you can graph it in your graphing calculator and trace the points that you were given and see if they match up with your graph!!!!

-Tricia L. and Josh H.

Section 6-7: The Quadratic Formula

In lesson 6-7 we learned about quadratic formulas. We learned how to fit quadratic models to data as well as finding the y-intercept of the quadratic model. In the warm-up question we used a data table and got a quadratic model to fit. Going along with that we found the y-intercept and tried to find the x-intercept. Going through the process we found you can't find the x-intercept unless you use the quadratic formula. The quadratic formula theorem say if ax² + bx + c = 0, with a not equal to 0, then x = to the opposite of b, plus or minus the square root of b squared minus 4ac all over 2a. To learn the formula we used Pop Goes the Weasel but just plugged in the theorem.

Amanda T.

In the lesson 6-7, we explored the world of the Quadratic Formula. We learned how to match a Quadratic Formula to data using a system of equations and solving it. We were also taught how to find the y-intercept and, if possible, the x-intercept of the equation. To find the y-intercept you substitute 0 in for x and solve. For the x-intercept, you substitute 0 in for y and solve.

Section 6-8: Imaginary Numbers

Today's lesson found here.

Section 6-8 Summary:
This lesson's main purpose was to introduce us to imaginary numbers. An imaginary number is i = the square root of -1.
The theorem states that if k > 0, the square root of -k = the square root of -1 multiplied by the square root of k = i multiplied by the square root of k.
Also, all square roots of negative numbers are multiplies of i.
Here are some examples:
Solve x² = -25
the square root of x² = +/- the square root of -25
x = +/- the square root of -1 times the square root of 25x = +/- 5i

-Emma R., Jeremiah B.

Section 6-9: Complex Numbers

Complex Numbers

The main topic of this lesson was complex numbers and how to use them. A complex number is a value in the form of a+bi, where a and b are real numbers and i =√(-1). Any number can be written as a complex number ex. 0+5i 8+0i. When dividing complex numbers you must eliminate the I, by multiplying by the conjugate of one of the denominator or the numerator. Complex numbers have numerous applications in electric calculations. -Jacob R.

Section 6-10: Analyzing Solutions to Quadratic Equations

In this lesson we reviewed the ways to find solutions to quadratic equations. The three ways are graphing, solving a system, and using the quadratic formula. We then learned that the discriminant of the quadratic formula can determine how many solutions there will be for a quadratic equation. If b^2-4ac>0 There are two real solutions If b^2-4ac=0 There is one real solution If b^2-4ac<0 There are two complex solutions
Brittani H.