Quadratic 101

By Harry G

Learning goals

  1. I will learn how to transfer vertex equation on to a graph using one of the transformation.
  2. I will understand how vertex equations work.
  3. I will learn the difference between 1st and 2nd difference
  4. I will learn how to isolate to solve for a variable

what is Vertex form?

Vertex Form is a equation that you can use to solve quadratic relations.


  • The equation to describe this is y=a(x-h)+k.
  • The value H gives you information on the X value of the Vertex of the relation and the horizontal translation from the origin.
  • The value A gives you information on which way the graph faces and its shape.
  • Finally the value K gives you information on the y value of the Vertex and the vertical translation from origin.

1st and 2nd differences

1st and 2nd Differences

Step Pattern

How to graph using step pattern

  • Identify and plot the Vertex (H,K)
  • Determine the step pattern (1a, 3a, 5a, 7a, etc)
  • Plot the step points (over 1 up/down 1a, over 1 up/down 3a, over 1 up/down 5a)
  • Reflect each point across the axis of symmetry
  • Draw a curve between points

Example

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Graphing Quadratics Using Step Patterns

Mapping Notation

Mapping Notation is another way of figuring out the points to sketch your graph. What you need for this is:

  • Your equatuation in Vertex Form

  • The formula (x+h, ay+k)

  • and a table showing you the points of y=x^2

As you already know, Vertex Form is y=a(x-h)^2+k. To do Mapping Notation, what you need to do is sub the values of your equation, into the formula (x+h, ay+k). This means that if you had an equation like (y=2(x-3)^2+5) your formula would look like this (x+3, 2y+5). this is because (a=2, h=3 and k=5). Now all you have to do is take the x values from the table that shows you the points of y=x^2 and sub them inot the mapping notation formula. (for example i would take -3 and sub it into the x part of the formula [-3+3] and the 9 into the y part of the formula [2*9+5]. This gives you your x and y coordinates (make sure that while you are subbing the points, you are making another table on the sides which clearly show what the x and y values are). you will continue to do this until you get the next 5-7 coordinates.

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Video on How to use Mapping Notation

Mapping Notation

Transformation

The transformations in the an equation that is in Vertex form are: a(x-h)^2+k

A - the represents the Verticle Stretch (narrow graph) or Compression (wide graph) of the graph.
(-)A - if a is negative then the Direction of Opening will be down and if a is positive then the Direction of Opening will be up.

H - The h in Vertex form represents the Horizontal Translation left or right.

K - the k in Vertex form represents the Vertical Translation up or down.

Transformation

Word Problem

How to isolate for a variable,

In the video below i solve for a H, you would use the same process if you wanted to solve for T
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Problem solving :)

Factoring

Learning goals

1. I will learn how to factor complex trinomials

2. I will learn how to convert from standard form to factored form

3. I will learn how to solve word problem using factored form

4. I will learn how to factor Perfect squares, difference of square and how to complete a square

Table of content

Expanding and Simplifying

  • Factoring Standard Form
  • Common Factoring
  • Factoring by Grouping
  • Simple Trinomials
  • Complex Trinomials
  • Difference of Squares
  • Perfect Squares
  • Completing the Square

Summary of the units

Factoring is the method to simplify an expression. There are multiple types of factoring including common factoring, factoring simple trinomials, factoring complex trinomials, prefect squares, and difference of squares.

Key things to know before we start:

  1. The value of a gives you the shape and direction of opening
  2. The value of r and s give you the x-intercepts
  3. Solve using the factors

Common factoring

Common Factoring is when in an equation, there is a number or variable that you can divide out evenly. Its basically finding the Greatest common factor and factoring it out of the equation, but that doesn't mean you get rid of it! That number is placed in front of the brackets.
Algebra- Factoring Greatest Common Factor
Converting Standard Form into Factored Form 5.5 gr 11 college 10 16 13

Factoring By Grouping

Factoring by Grouping - MathHelp.com- Algebra Help

Simple Trinomial

This is a very simple method, and you get the your answer in factor by grouping question form. It is like the opposite of factor by grouping. The point of it is to get 2 Binomials that multiply to get the given trinomial. It is a simple trinomial if there is 3 terms. One beingthe other being x with no power and it can have a number in in font, and the last being a number with no variable.
factoring trinomials

Complex Trinomials and How to factor them

Factoring Complex Trinomials

Completing the Square

3.14 Completing the square

Perfect Square

Perfect Square Trinomials

How to Factor Perfect Squares

An example of a perfect square is 16x^2+24x+9. This is a perfect square trinomial because when you square "a" and "c" in the equation and multiply them together as well as by 2 again, the answer should be equivalent to the "b" value.


An example:

100x^2+60+9

*100 square rooted is 10 and the square root of 9 is 3

*2(10)(3)=60

*(10x+3)^2

*To check if your final answer is correct, when you expand your answer(in this case) (10x+3)^2 into (10x+3)(10x+3) and then multiply both brackets (using F.O.I.L), you should end up with the original equation (in this case) 100x^2+60+9

Difference of Square

Algebra - Factoring Differences of Squares

Factoring the difference of squares

Example:

x^2-49

*the square root of 1(x) is 1 and the square root of 49 is 7

*(x+7)(x-7)

*Multiply both brackets(F.O.I.L) to find out if the answer is correct

*x^2-7x+7x-49 (collect like terms)

*x^2-49 therefore (x+7)(x-7) is the correct answer in this case

Word Problem using factored form

math work (its lit)

Standard Form

Learning Goals

1. I will understand how to solve using quadratic formula to find the x-intercepts

2. I will understand how to solve using completing the squares to find the vertex

3. I will learn how to solve word problems

Summary

The quadratic formula was created in order to find the the x-intercepts of a given parabola. In order to use the quadratic formula your equations must be in the form: y= ax^2 + bx + c. Completing the square is another way to find the vertex and is when you get the equation listed above and convert it into vertex form: y= a(x-h)^2 + k. This equation then allows you to find the vertex which is (-h,k) = (x,y)

Quadratic equation example

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Completing the square

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Word Problem

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In the word problem above you're using completing the squares to solve for the maximum height once you have the maximum height you can sub that in to find the time it takes and the initial height is already given in the original equation which is the + 1

Quadratic formula word problem

Algebra - Quadratic Equation - Word Problem Solving Problems Intuitive Math Help

Similarities between all 3 form of quadratics



Vertex form

  1. By setting a value to 0 (ex x=o) you are able to find the other (ex. x=0 and y can be determined by setting x to 0)
  2. Able to be turned into standard form simply by expanding
Standard form
  1. Standard form is expressed as y=ax^2+bx+c
  2. Is able to be turned into factored form (only in certain cases)
  3. When changed into a perfect square it then is in vertex form
Factored form
  1. Is able to be changed into standard form by expanding
  2. (A 2 step process) can be changed into standard form then vertex form


The word problems in all the form of quadratics were mostly asking for the same thing just in different forms of equations and different methods.


All three form of quadratics can be graphed, to graph them you need the same thing (X,Y and their intercepts) Once graphed all of them make a parabola.


if we use the quadratic formula we can solve to get x and then we can graph the equation while in vertex form.

Reflection

Throughout the course we looked at 3 different units each one relating to the other. I found it difficult to understand how to graph the parabola at first for all of them, or even solving for the vertex. The most difficult part was understanding how to do the word problems, in the way they were written sometimes the wording of the problem would confuse me and i would just mess up. I also found it difficult to isolate for a specific variable, something which i struggled with in every unit of quadratics. Besides all of the difficult things one of the easiest parts in quadratics was using mapping notation to graph the parabola, this part i found to be extremely easy because using the step pattern my teacher taught me, i just need to know the a value and times that by 1 and then 3 and then 5, once i have this i could just graph it move over which ever direction and then graph that point and keep doing this until i have a shape of parabola.

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