Quadratics Relationships

By Alicia Persaud

Table of Contents

Introduction

  • What are quadratic relationships
  • Introduction to parabolas


Vertex form: y= a(x-h)^2+k

  • Axis of symmetry
  • Optimal values
  • Transformations
  • Step pattern
  • Graphing vertex form
  • Personal assessment/small reflection


Factored form: y= a (x-r)(x-s)

  • Zeroes or x-intercepts
  • Axis of symmetry
  • Optimal value
  • Word Problem
  • Graphing factored form


Standard form: y=ax^2+bx+c

  • Using the Quadratic Formula and Discriminants
  • Word Problem
  • Completing the square to turn to vertex form
  • Factoring to turn to factored form
  • Graphing standard form
  • Personal assessment/small reflection


Conclusion

  • Connections
  • Overall reflection

What is quadratics?

Quadratics originates from the word "quad" meaning square like (x^2) in quadratics the first term is squared making it quadratic. They are often used to graph flight paths of different objects travelling in the shape of an arc. For example, the flight path of an arrow, or a fireball. In order to determine if a relationship is quadratic or not, you can simply look at a table of value; if the first differences are unequal, but the second differences are equal, it is a quadratic relation.
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Introduction to Parabolas

Parabola's are all around us! They range from the arc of the McDonald logo to the arc of a roller coaster. A parabola is a curve where any point is at an equal distance from a fixed point and a fixed straight line. When you shoot an arrow, it arcs up and comes down again which creates the shape of a parabola. Some important features include:



  • Vertex: The maximum or minimum point of the parabola. It is the point where the parabola changes direction.
  • Axis of symmetry: Divides parabola in half.
  • Optimal value: Minimum or maximum value.
  • Y-intercept: When x=0, parabola crosses y-axis.
  • X-intercept: When y=0, parabola crosses x-axis.
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Vertex Form

Vertex form is normally expressed as y=a(x-h)^2+k. If the a value is negative, the parabola will open downwards; if the a value is positive, the parabola will open upwards. If the a value is less than 1, the parabola will be compressed, meaning the graph of the parabola will widen. However, if the a value is greater than 1, the parabola will be stretched, meaning the graph of the parabola will become more narrow.

Axis of Symmetry

The axis of symmetry is a vertical line that goes through the middle of a quadratic equation, and the parabola is then split into 2 symmetrical parts. It is also known as the x-value of the vertex. The axis of symmetry is determined by the h value of the equation. Note: in the equation, if h is negative, on your graph it is positive; if in the equation, is it positive, on your graph, it is negative.

Optimal Value

The optimal value is the maximum or minimum point of the parabola, also know as the y-value of the vertex. It is determined by the k value of the equation. If the parabola opens upwards, it has a minimal optimal value; if the parabola opens downwards, it has a maximum optimal value.

Transformations

Each variable in the equation y= a(x-h)^2+k is in control of a transformation.


  • The a value stretches/compresses the parabola, the direction of opening, and the step pattern.
  • The -h value translates the parabola horizontally. *RECALL*- if it is negative in the equation, it will be positive on the graph! Therefore you will move it to the right as opposed to the left, and vice versa.
  • The k value translates the graph vertically. It multiplies the vertical part of the step pattern.
Once the vertex is determined, you can use the step pattern to find other points. What is the step pattern, you ask?
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Step Pattern

A step pattern is a general rule you can use to determine how to plot your points. Check out this video on how to graph using the step pattern!
How to Graph Parabolas

Graphing Vertex Form

Quick Way of Graphing a Quadratic Function in Vertex Form

Personal Assessment and Small Reflection

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As you may be able to tell by my assessment, quadratics in vertex form was one of my stronger topics. I really enjoyed this topic because in my opinion, identifying the different parts of a parabola (vertex, direction of opening, etc.) was really fun and not much of a challenge, in addition to plotting points of a parabola with the help of an equation and the step pattern. I had fun doing this application question; not only did it put me to a challenge, but it had me identify the different parts of the parabola and their purpose, for example it asked when the football reached it's maximum height and what the height was, and it connected back to learning about the different parts of a parabola where I realized that it was simply asking for the axis of symmetry and optimal value. Overall, exploring the vertex form was my favourite form.

Factored Form

Factored form is normally expressed as y= a(x-r)(x-s). Much like vertex form, the a value determines the direction of opening, the stretch or compression, and the step pattern. Note that change of the a value will not affect zeroes or the axis of symmetry, however it does affect the optimal value.

Zeroes or X-intercepts

Factored form is expressed slightly differently than vertex form is. It introduces zeroes and x-intercepts, which are represented by the values r and s. In order to determine the zeroes, you must set y=0.
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Axis of Symmetry

Axis of symmetry should sound familiar by now; recall that it is the line that passes through the middle of the parabola, dividing it into 2 equal parts. In order to determine the axis of symmetry, you can add the 2 x-intercepts and divide it by 2 (r+s/2).

For example,

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Your two x-intercepts would be (-5) and (3).

As mentioned before, you must first add the two x-intercepts:

(-5)+(3)=-2

Now, you have to divide by two.

-2/2=-1

As shown in the diagram above, the axis of symmetry is -1.

Optimal Value

Unlike other forms of equations, determining the optimal value in factored form is a bit different. Once you establish the axis of symmetry, you must substitute it in to the original equation, in the place of the value x, then you must solve the equation to get a maximum or minimum point.

Word Problem

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How to Graph Factored Form

Step one: Find the zeroes. Recall- to find the zeroes, you must set y=0, therefore you will have to take the value of the x-intercepts and carry them to the other side of the equation to isolate x. This will give you the two zeroes.

Step two: Once you plot the two points on the graph, you must find the axis of symmetry. You learned earlier that to do so, you must add the two x-intercepts and divide that by 2, once you have done that, then you can create your axis of symmetry (h-value of the vertex).

Step three: You must now find the optimal value. It was stated earlier that you must substitute in your axis of symmetry to get this answer. After solving the equation, you will then get your optimal value (k-value of the vertex).

Here is a video to further your understanding:

https://www.youtube.com/watch?v=eFM2AqqnXYY
*I apologize for the terrible video taping.

Standard Form

Factored form is commonly expressed as ax^2+bx+c=0. The variables a, b, and c, are known values; a cannot be 0, x is an unknown value. There are 3 methods in regards of finding the solutions. They are:


  1. Factor the quadratic
  2. Complete the square
  3. Use the quadratic formula

Using Quadratic Formula and Discriminants

The quadratic formula is used to find x-intercepts. We can use it when we are unable to factor our equation. It is expressed as:
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How to Use the Quadratic Formula

Originally, you are given the equation in standard form, which is ax^2+bx+c. The first thing you must do is find the value of a, b, and c, in your equation. The equation is x² - 4x - 5 = 0

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Now, you must sub each of these values in to your quadratic formula.
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Recall how to simplify equations from the beginning of the unit. This is where it may come in handy! All you need to do now is simplify the equation.
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Finally, you can solve. Don't forget that you must both add and subtract whatever is in your square root.
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Check your answer! All you have to do is sub in your x-intercepts and make sure your equation equals zero.

Word Problem

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Discriminants

What is a discriminant and why is it useful?

A discriminant is the number inside of the square root (b^2-4ac) and it is useful because it helps up determine how many solutions a quadratic equation has.


  • If the discriminant is less than 0, there are NO solutions.
  • If the discriminant is greater than 0, there are 2 solutions.
  • If the discriminant is equal to 0, there is 1 solution.

Completing the Square to Turn to Vertex Form

Completing the square is a technique used to rearrange the quadratic equation. We must go from standard form (ax^2+bx+c) to vertex form (a(x-h)^2+k). Keep reading to learn how to do so.


For the equation y=-3x²+24x-27

Step one: Group like terms (x^2 and x terms) together by putting brackets around them.

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Step two: Common factor numbers only the constant terms inside the bracket so you are left with no value in front of x^2, and leave the factor outside the bracket.
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Step three: Complete the square inside the bracket; we must do (b/2)^2, then take your answer and add it in behind the bx. You must also subtract it afterwards because you do not want to change the equation.
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Step four: You must now simplify the equation you have. Take into consideration that you have a perfect square trinomial, so you must factor that.
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Step five: You can now distribute the term that is outside the brackets, into the brackets. You will also be left with two values, which you must add. You are left with an answer that is in vertex form.
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Factoring to Turn to Factored Form

There are many different types of factoring methods you can use, in many different situations. There is:


  • Common factoring
  • Simple trinomials
  • Complex Trinomials
  • Perfect Squares
  • Difference of Squares
Common Factoring

Common Factoring is used when there there is a number or variable that is common between the terms in a polynomial. When you factor, you are dividing it out of the set of terms, and putting it in front of the brackets. Watch the video below on how to common factor.

3.7 Common Factoring
Simple Trinomial

A simple trinomial often looks like so: x^2+bx=c.

When factoring this, we want to find two numbers that multiplies to give you the c term, and adds to give you the b term. Once you find those two terms, you can simply set out two pairs of brackets and distribute the x^2 into them so there is an x in each set. You can then place the two values you found into a set of brackets. Here is an example:

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Complex Trinomials

You must be wondering what's so different about a complex trinomial than a simple trinomial, or what makes it more complex. Well, in a complex trinomial, there is a coefficient in front of the x value that is not 1. Two methods you can use are the decomposition method and the trial and error method.

Decomposition:

Step one: Multiply the a and c value, use that answer as you would originally use your c term.

Step two: Just like in a simple trinomial, find two numbers that will both multiply to give you your c value, and add to give you the b value.

Step three: Now, replace your middle term with the values the terms you just got as your answer.

Step four: Common factor! You will know you did it right if you have the same answers left in brackets. Now you just have to put the values that are outside the brackets in a set of brackets, and in the other set you can leave the values that were the same in previous step.

Factoring Complex Trinomials using Decomposition
Trial and Error:

Trial and error is a method used in a wide variety of cases when it comes to the principles of mathematics. To use the trial and error method, you have to do exactly what it sounds like- try, and end up with an error; you must keep trying until you get it right. Here is a video that shows you how to factor using the trial and error method:

3.9 Complex Trinomial Factoring
Perfect Squares and Difference of Squares

1, 4, 9, 16, 25, 36, 49, 64, 81, 100... these are all examples of perfect squares.

This is useful because of something called perfect square trinomials.

Note: a^2+2ab+b^2= (a+b)^2

a^2-2ab+b^2= (a-b)^2.

Difference of squares is a squared number subtracted from another squared number. It can be expressed as (a+b)(a-b)=a^2-a^2.

Watch this video on how to factor perfect square trinomials and difference of squares:

Graphing Standard Form

graph parabola standard form

Personal Assessment and Small Reflection

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Disappointingly, exploring standard form was my worst experience, however after receiving back my mini-test and reviewing where I went wrong, I've learned from my mistakes. This was definitely one of my weakest areas when it comes to quadratic relationship, possibly because I personally am terrible at factoring. As you can see, I had no answer at all for question #10 and #11(b) because I had absolutely no idea what to do. However, know that I have gone over my test and discussed where I went wrong with my peers and tutor, I realize what I need to focus on, which is expanding and grouping, and substituting-in values. In conclusion, I did not enjoy this unit but I am learning, and hope to improve.

Connections and Overall Reflection

Connections

Converting Vertex Form to Factored Form
Changing a Quadratic from Standard Form to Vertex Form
LT27 Factored Form to Standard Form

Overall Reflection

Taking everything into account, I enjoyed the quadratic unit. Personally, math is one of my weaker subjects and normally one of my lower marks, however this term that definitely changed. In my opinion, quadratics was the most challenging unit I've explored so far, possibly because it is all new to me; initially, I was unfamiliar with terms such as 'parabola' and 'discriminant', however throughout the unit I have come to familiarize myself with them. As stated before, I had the most fun exploring quadratics in vertex form because despite the fact it was all new to me, it put me to a challenge where I was able to play around with the different parts of a parabola and it how it works. In addition, I really enjoyed exploring quadratics in factored form, because as opposed to the other forms, I got to play around with x-intercepts and finding axis of symmetry and the optimal value in a different way. Finally, I enjoyed exploring quadratics in standard form the least. Not only is factoring my weaker spot, but I also had trouble with the decomposition method when factoring complex trinomials. Having a lot of older figures in my life who have already experienced this, and I can now relate to it.