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Analysing Quadratic Equations and Graphing Their Parabolas

A quadratic equation with its associated parabolaQuadratic equations, when graphed, produce parabolas.

Fortunately, parabolas have a very nice structure.  They are a smooth curve; they have just one tip (vertex); and they are symmetrical.  As we shall discover, there are a number of other quite lovely little facts that we can learn about them.  All these insights will help us to analyse quadratic equations and graph their parabolas accurately and swiftly!

Typically, mathematicians like to know where the vertex (tip) of the parabola lies, where its axis is, and the points where it crosses the x and y axes.  Later, you will learn of the geometric structure of a parabola, and this will allow you to locate its directrix and focus and, by drawing two squares (boxes), graph its latus rectum and sketch the parabola very quickly.  Not only that, but because squares grow in a predictable way we can utilise this knowledge to advantage, too.  For example, the gaps/differences between the squares 0, 1, 4, 9, 16, 25, etc. are all odd numbers!  More on this, later.

I should mention that, to save time with creating an image of a parabola and the quadratic equation that generated it (see image at left), I resorted to using an image from a rather useful site, You might like to visit them and see if what they offer is of help to you.  For example, they seem to have a large number of useful Math Lessons listed under Math Resources that you may find worth investigating.


Summary of the Three Series

The first series of videos will include explanations covering:

  • what features of parabolas are of interest to mathematicians (and why),
  • how to analyse quadratic equations to find these features (using a four-step procedure),
  • how to understand and convert between three forms of quadratic equation (general form, factored form and completed square form), and the benefits of each,
  • how to factorise quadratic equations,
  • how to complete squares, and
  • how and when to use the quadratic formula (and how to memorise it)

The second series will discuss parabolas as a locus of points obeying a geometric rule. I will explain how to understand and use equations of the form x² = 4ay and how to draw their graphs accurately.

The third series will discuss the parametric representation of the parabola and how to understand it. It will also show how to deduce some geometric properties of parabolas using this form.

The compete set of three series should prepare students fairly thoroughly for the NSW Higher School Certificate in Australia and help students achieve high competency with this topic.

I hope you enjoy the material.

Quadratic Equations and Parabolas ~ An Introduction

Many students struggle with analysing quadratic equations and graphing parabolas. I intend to produce a comprehensive series of videos that thoroughly explains the whys and hows of this aspect of mathematics. This is the first video.

In this video I give a brief overview of quadratic equations, tables of values and graphs of parabolas, explaining something of the connection between them. I then explain and demonstrate how our understanding of square numbers can help us draw basic graphs very quickly indeed.

It is a lengthy video but I am particularly keen to be thorough and not rush students. So, if you are new to this topic or struggle with it, I encourage you to watch each video right through. I am confdent that you will gain new insights and find the process to be much easier than you think. You should also find that your recall of this information is reasonably good.

Finding the Vertex and Structure of a Parabola (Equation in Completed Square Form)

In the last video we learned how to use the odd numbers 1, 3, 5 ... to draw a parabola. This time we learn how an equation written in completed square form is designed to show us where the vertex is located! Using both these ideas helps us to make accurate graphs of parabolas very quickly.

I have explained elsewhere how to convert any quadratic equation into this completed square form. At this early stage, however, I just want you to be able to recognise the competed square form of the equation ... and understand how to 'read' and analyse it in order to draw your graph.

Four Steps to Analysing a Quadratic Equation (Theory)

So far, in the previous two videos/posts, I have introduced you to four concepts/facts about quadratic equations:

  1. that parabolas are the graphs of quadratic equations,
  2. that parabolas have symmetry about an axis,
  3. that the odd numbers help us construct parabolas, and
  4. the vertex can be found easily from the completed square form

In this video I explain the four features of a parabola that are of particular interest to mathematicians (we will learn about some geometric properties later). I also explain why they are of interest, and then explain (and demonstrate) a systematic method for setting your work out on a page as you find them.

My next video will be much shorter. In it I will simply demonstrate this four-step analysis of a quadratic equation and then graph the corresponding parabola. Consider this video as the theory and the next video as a demonstration.

The reason the 'fun' starts here is that I will need to produce a series of videos explaining the different methods available to you to find each of these four features. Think of it like fishing or building a house. You will choose your rod and tackle, bait and fishing location based on the prevailing conditions ... or choose your tools based on the material that you are using to build the house. How you go about finding these four features will depend upon the kind of quadratic equation that you have encountered ... but, in this kind of analysis, you will always attempt to find these four features!

So, settle back, hear me out ... and I trust that it all makes sense to you.

Four Steps to Analysing a Quadratic Equation (Demonstration)

The four main features of a parabola that are of initial interest are:

  1. the y-intercepts,
  2. the x-intercepts (the roots or zeros of the quadratic equation),
  3. the axis, and
  4. the vertex.

In this video I demonstrate how a simple quadratic equation can be analysed in four steps and how to draw the corresponding graph (a parabola). My objective here is not to explain how to perform each step. It is simply to demonstrate how logically the process can be set out on a page.

I like to start each of the four sections with a heading that identifies what I am going to find. This is good for my own mental clarity as well as making it very clear for an examiner or whoever else may wish to read/follow my work. I also like to state how I am going to find each item and include that in the heading.

In the next few videos I will be discussing alternative methods for finding each feature ... but it is important that you remember that, overall, this four-step analysis underlies everything.

How to Find the y-Intercept of a Parabola (From Its Quadratic Equation)

I almost didn't produce this video. The concept is reasonably simple and the mathematics is not difficult, however, I encounter many students who have difficulty understanding graphs and thought I should not omit this detail!

Usually, when we have to analyse and graph a quadratic equation, the first feature that we identify is the y-intercept. This is because it is often the easiest point to find.

Because the quadratic equation is usually presented to you in one of three different forms, I show how to find the y-intercept from each form. In subsequent videos you will learn what each form is particularly designed to do. In this case, it is the General Form that makes finding the y-intercept very easy! It is simply the value of the constant.

You will notice that, towards the end of the video, I use the heading "y-intercept (x=0)." When analysing quadratic equations, I like to start each of the four sections with a heading that identifies what I am going to find. This is good for my own mental clarity as well as making it very clear for an examiner or whoever else may wish to read/follow my work. I also like to state how I am going to find each item and include that in the heading. So, for the first section of my analysis, I like to use this heading and encourage you to do likewise (or, at least, similarly).

The next few videos will deal with finding the x-intercepts (aka roots or zeros) by factorising, completing squares, or by using the quadratic formula.

I had already posted another video explaining this procedure when I discovered that I had recorded this one weeks earlier! Rather than waste the effort, I thought I would post this as well. There is no need for you to watch both versions. One will do!

How to Find the x-Intercepts (Roots or Zeros) of a Quadratic Equation (Introduction)

When analysing a quadratic equation and graphing its parabola, the second feature that we identify will be any points where the parabola crosses the x-axis. These points are called the x-intercepts. They are also known as the roots or zeros of the quadratic equation.

I will show you a clear way of presenting your work, and thinking through the process for finding them. You will discover, also, that the factored form of the quadratic equation makes finding these roots/zeros/intercepts very easy.

On another page I have demonstrated different ways that we know for converting the general form of a quadratic equation into a factored form. This process is called factorising and I encourage you to watch all the videos and choose the technique that you like most. You should then, of course, practise it until you can factorise quadratic equations with ease.

I cannot even begin to describe how much you’ve taught and continue to teach me through your videos. I’m going into year 12 this year and I feel like your way of teaching is so ‘crystal clear’ to me so thank you so much! Too bad you can’t be my teacher I’ve recently watched you playlist for derivatives and those videos are amazing! … Continue to make awesome videos because you’re a legend.
Chelsea N (student, 2015, in private discussion about CCM YouTube videos, with permission)

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