Quadratics

By:Puneet Guleria

Intro

Quadratics are used graph the path of an object. This is very useful for finding where the object will land, finding the maximum or minimum height the object will reach and at what time it will reach the maximum or minimum height. Also you can use quadratic a to find the maximum and minimum revenue. This is very useful for those thinking about selling an item but don't know what price to sell it at. These were just a few ways quadratics are helpful

Table of context

Quadratics in vertex form

Key features of quadratic relations

Scatter plot

Line/curve of best fit

Finite differences

-linear relation

-quadratic relation

-neither

Transformations of quadratics

-writing equations given the transformation

Graphing using step pattern

Finding equations given the vertex

Quadratics in factored form

Multiplying Binomials and Special Products

Common Factoring

Factoring Trinomials

Factoring Difference of Squares and Perfect Square Trinomials

Solving Quadratics by Factoring (finding the zeros)

Graphing Quadratics in Factored Form

Quadratics in standard form

Maximum and Minimum values (Completing the square)

The Quadratic Formula

Key features of quadratic relations

Vertex(the poin where the parabola crosses the axis of symmetry)

Axis of symmetry(where the parabola is divided equally)

Optimal value(the y coordinate of the vertex, can be maximum or minimum)

Zeros/Roots(the x-intercepts)

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Scatter plot

A scatter plot has points that show a relationship between two sets of data x and y. These points are plotted onto a graph to organize the data. This is one of the ways to organize the set of data so that it is easy to understand. x= independent variable y= dependent variable. the independent variable is the variable that always stays the same and is not dependent on another. while the dependent variable changes depending on what happens to the independent variable.

Line and Curve of best fit

When you make a line or curve of best fit you must determine where to put your liner curve of best fit. This must be done in 2 steps. Step1: determine weather it is a line or curve of best fit by looking at your points and check if the points are more suited for a line or curve.

Step2: make the line/curve of best fit in a place where almost all points on both side of the line are equal. this is how to determine and make a line or curve of best fit.

line of best fit

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Curve of best fit

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Using finite differences to determine if the relation is linear, quadratic, or neither

To determine weather your difference is a quadratic or linear you must look at the x and y points that have been given. Then to find out if it a quadratic, linear relation or neither, you must look at your y points look at how much it increases or decreases moving from the last point and going up to the point above and continuously go up until you reach the top. remember to jot down the difference between the points every time you go up. when you have reached the top look at the new data you have collected from the differences if the differences are the same the first time it is called a first difference. if the first difference is the same it is a linear relation if not you must use the same method again with the first differences and see if your second differences are the same if they are it is a quadratic relation if not then it is a neither.

Linear relation

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This is what first difference will look like. Ignore what is in the brackets on the right hand side. if you look every time you go up from the bottom to the top all first differences are the same meaning it is a linear relation.

Quadratic relation

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If you look at this relation it shows a quadratic relation because you can see that the first differences were not the same so they tried again to find out if it was a quadratic equation. since second differences are the same that means it was a quadratic relation. if they were not to be the same the relation would be neither.

Transformations

Transformations

writing equations given the trasformation

If the transformations are given you can easily write what your equation will be.

The steps

1- identify weather your parabola is opening up or down.

2- identify the stretch or compression(a)

3- identify the transformation left or right (h)

4- identify the transformation up or down (k)

5- finally put your variables into the equation y=a(x+3)^2+3

Graphing using the step pattern

The step pattern shows how values change as the parabola stretches or compresses (a).
Graphing Quadratics Using Step Patterns

Finding an equation given the vertex

These are the steps to finding an equation using vertex form.

steps

1- sub in your vertex into your equation y=a(x-h)^2+k (plug x value as your h but make your x the opposite sign(negative-positive or positive-negative) then for k plug in your y value the same as it is)

2- sub in one of your x intercepts into the equation

3- solve until you find the value of (a) which will complete the equation

4- plug in a to you original equation to complete the equation

Finding x and why intercepts in vertex form

to find the x and y intercepts in vertex form you must always remember to make the other intercept 0 first. I suggest finding the x intercepts first by making y zero. the video below will explain more. then when you have your answer find the other intercept to find all your points. try and graph your points to see if you were correct.
Finding x-intercepts (Vertex Form)
y intercept from vertex form

multiplying binomials and special products

Binomials when multiplied form a trinomial, an equation with three terms. theses terms are used to expand and simplify the terms.

(x+5)^2---(x+5)(x+5)----x^2+5x+5x+25---x^2+10x+25 formula=a^2+@ab+c

(x+5)(x-5)---x^2+5x-5x-25----x^2-2 formula where equations are same but middle sign is different a^2-b^2

Introduction to special products of binomials | Algebra I | Khan Academy

common factoring

Factoring is used to simplify terms so that it is easier to use them.

for example

6x^2 − 2x
whats common is 2 and x so the new term would be
2x(3x-1)


Common Factoring Tutorial

factoring trinomials

steps to factoring simple trinomials

Step 1 find two numbers that multiply to give ac (in other words a times c), and add to give b.

Step 2 Rewrite the middle with the new numbers to form a equation with 4 terms

Step 3 Factor the first two and last two terms separately

Step 4 put the common factors together

here is an example

2x^2 + 7x + 3 sum of (a)(c)=6 (6)(1)=6 6+1=7

2x2 + 6x + x + 3

2x(x+3) + (x+3)

(2x+1)(x+3)

Factoring Complex Trinomials

factoring differences of squares

differences of squares is very easy to calculate these are the steps

1 look for a gcf

2 square root the two numbers

3 determine if the the new numbers can be factored further

4 put numbers into factored form (x+a)(x-a)

Example: 18x^2-98y^2

2(9x^2-49^2)

square root the middle terms

2(3x-7y)(3x+7y)

factoring perfect square trinomials

Example 4: Factoring quadratics as a perfect square of a difference: (a-b)^2 | Khan Academy

factoring overview

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solving quadratics by factoring

These are the steps to solving for x intercepts.

so the first step is to get one side to be zero- x^2-10x=-16---x^2-10x+16=0

next find two numbers that multiply to give ac (in other words a times c), and add to give b-

(x-8)=0 (x-2)=0

then solve for zeros-- x-8=0--- x=8 x-2=0--- x=2

Solving Quadratic Equations by Factoring - Basic Examples

Graphing Quadratics in Factored Form

To graph equations in factored for you must find all of the following, vertex, roots(x-intercepts), direction of opening, axis of symmetry.
Graphing Parabolas in Factored Form y=a(x-r)(x-s)

completing the squares

This is a form of turning a standard form equation into a vertex form equation.
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Step 1 Divide all terms by a (the coefficient of x2).

the quadratic formula

This formula will always work for any type of problem even if it is not factorable.
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The equation uses a,b, and c from the standard form equations that is ax^2+bx=c remember to always make one side of your standard form equation 0.
Solve Quadratic Equations using Quadratic Formula

chapter 1 test

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chapter 1 test review

My major mistakes in this test were the tips and the drawing of the graphs. I believe I could have done much better on the test i did not try to attempt a few of the graphing problems and the tips had me really confused although it was simple. i did really well on the application and communication components of the test

chapter 2 test

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chapter 2 test review

I found this test to be fairly simple i made too many silly mistakes that really affected what I did. Also I think I was over thinking because the word problems were fairly simple but i made simple errors by not putting the right signs. also I did not read the question properly which really affected my mark.

Chapter 3 test

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chapter 3 test review

I really understood the concept of completing the squares and the quadratic formula but what i have learned is that you have to be really careful with every step you take because if you miss or forget a step your whole answer will be wrong and any upcoming questions related to the same topic will most likely be wrong. Also what I have learned how to use the discriminate really well which helps to tell me how many x intercepts there will be. Finally what I learned was how you can use different methods, quadratic and completing the squares in the same problem. using quadratic formula to find x intercepts and using completing the squares to find the vertex

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