Quadratics

Vertex Form~Standard Form~Factored Form

Quadratics

We run in to Quadratics in our everyday life's like Sports, work, school e.t.c. we might not notice but parabolas are used all over the world. For example something you might see a lot in Toronto is the Rogers Center, the whole room is shaped as a parabola.

Table of contents

  1. Terms a definitions
  2. First differences/ Second differences
  3. Vertex Form
  4. Standard form
  • X-intercept- when the parabola meets with the x-axis
  • Y-intercepts- when the parabola meets with the y-axis
  • Zeros- the x value which makes the equation equal to zero
  • Axis of symmetry- the vertical line which cut the parabola down in the middle
  • Optimal value- The lowest or highest point of the parabola
  • Vertex- the maximum or minimum point on the graph

First differences/Second Differences

Really easy, just small mistakes will mess your work up. Always remember when finding differences always subtract bottom number from top. If it is a first difference then it is a linear relation (strait line), if it turns out as a second difference it is a quadratic relation (curved lines). Examine this picture very well as it shown how to do all of

1st diff.

if you find a relation in the first differences then it is a linear relation. always subtract the second number from first.
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first diff.

When you find a relation in the first diff. then it is linear. The way you know if you have a relation is if all the numbers are the same like in this picture all the twos. If they are different you would try doing second diff and see if you can find a relation there.

2nd diff.

If you find a relation in the 2nd differences then that means it is a quadratic relation.
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Neither?

if your find that there is no 1st diff or 2nd diff then that means it is neither no solution.

Vertex form

This is the most easiest type of form used in quadratics. View picture below as it will tell you what the equation is
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Axis of symmetry, Optimal Value and transformations

Axis of symmetry

- splits the parabola in half

optimal value

- the minimum and maximum value

Transformation

the h value k value and a value

All of this will be shown in a picture

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What can we get from the vertex form

vertex- the coordinates of the parabola
Quadratic Functions Transformations and Vertex Form

what if the a value is negative?

if the a value is negative then the whole parabola is flipped over, the right term used is reflected. The a value could be any number 2 or even 10000000000 but if it has a negative in front then it is flipped.
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How to use step pattern

To graph using vertex form you must use the step pattern which is very simple just small mistakes can mess you up. watch video below to learn the step pattern
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here is a video to help understand

Graphing Quadratics Using Step Patterns

What to keep in mind in step pattern

Always multiply a by 1,3 and 5. also when plotting make sure you go left one and go up by how much the number is. Then do that to the left side.

vertex form- Word problems

to do these word problems it requires all knowledge of vertex form here are examples of vertex form word problem
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factoring

we must know standard form for this

ax^2+bx+c=0

factored form

multiplying binomials and special products

to multiply binomials and special products we must use "foil"

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Algebra - FOIL Multiplying Binomials

Common factoring

opposite of of foil, when we were doing foil we were expanding and then simplifying but here we will bring the expanded simplified form to its original form which is factored form

y=a(x-r)(x-s)

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Common Factoring Tutorial
How do we know this is right? well if we solve our answer we got it will give us the number we started with, if we don't get that that mans we did it wrong
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simple trinomial facotring

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Factoring the Simple Trinomial

complex trinomial factoring

to do this we must learn the decomposition method
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Factoring Complex Trinomials using Decomposition

difference of suqares

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solving quadratics by factoring

finding zeros
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how to graph using factored form

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word problems

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standard form

standard form is

ax^2+bx+c=0

completing the square

we know how to get from standard to factored and factored to standard but what about standard to vertex form, thats what completing the square does
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Completing the Square - Solving Quadratic Equations

quadratic formula

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Using the Quadratic Formula

reflection on a unit and assesments

the most favorite assessment of mine was the vertex form tips assignment we got to use desmos app, worked in partners to find the transformations of a vertex form graphed. i scored almost perfect on the partner section and on the individual i scored perfect. vertex unit was my favorite because it was the easiest and it brought my mark up and i learned alot about what parabolas and how we come across them in life a lot.
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