Quadratics

The Best Unit Ever!

By: Elizabeth Esedafe

Introduction To Quadratics

The origin of the term "quadratic" is Latin. It comes from the word quadratus, which is the past participle of quadrare which means "to make square."The quadratic equation is used to find the curve on a Cartesian grid. It is also used to find the curve that objects take when they fly through the air. For example a softball, tennis ball, football, etc. Other than that, the equation is used to design any object that has curves and any specific curved shape needed for a project.

Here's a link to a more detailed explanation of the word parabola!

Unit Table Of Contents

Throughout this website you will learn many aspects of quadratics. Here are some of the key topics of the quadratics unit:
Discovering The Vertex Form
  • Key Features Of Quadratic Relations (Vertex,AOS,Zeros,Optimal Value,)
  • 4.2 Quadratic Relations and finite differences
  • 4.3 Transformations of Quadratics
  • 4.4 Graphing Quadratics in Vertex Form
  • Finding x and y intercepts in Vertex Form
  • Word Problems for Quadratics in Vertex Form


Quadratics in Factored Form


  • 5.1& 5.2 Multiplying Binomials and Special Products
  • 5.3 Common Factoring
  • 5.4 Factoring Simple Trinomials
  • 5.4 Factoring Complex Trinomials
  • 5.6 Factoring Difference of Squares and Perfect Square Trinomials
  • 6.2 Solving Quadratics by Factoring (finding the zeros)
  • 4.5 & 6.3 Graphing Quadratics in Factored Form
  • Word problems for Quadratics in Factored Form

Quadratics in Standard Form

  • 6.1 Maximum and Minimum values (Completing the square)
  • 6.4 The Quadratic Formula
  • Word Problems in Standard Form (6.5)

Connections in the Unit


Reflection

Discovering The Vertex Form

KEY FEATURES OF QUADRATIC RELATIONS


These Key Features of Quadratics are the basics of the unit. To graph a quadratic relation is called a parabola. The parabola has some very important features:


-The zeros and y-intercepts

-The Axis of Symmetry

-The Vertex

-The Optimal Value

Vertex

The vertex is the point of the parabola where the axis of of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value. The vertex is also the point where the graph changes direction. Its also labeled as (h,k) on a graph.

Axis Of Symmetry

The axis of symmetry divides the parabola into two equal halves. It is also labeled as x=h since the axis of symmetry is the h-value/x-value of the vertex.
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The Zeros and Y-Intercepts

When y=0, the parabola crosses the x-axis. The zeros are the same as the x-intercepts or "roots" and are labelled as (x,0). When x=0, the parabola crosses the y-axis. The y-intercept is labelled as (0,y).

The Optimal Value

The optimal value is the value of the y co-ordiante of the vertex. It can be a maximum value or a minimum value depending on the position of the parabola. To label the optimal value you write: y=k (y-value of vertex)
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4.2 Quadratic Relations And Finite Differences

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Finite differences are shown through a table of values that can show whether an equation is a quadratic relation or not. In order to calculate first differences you must subtract the second y-value from the first y-value.Then continue doing this for each pair of y-values.


If the first differences are constant, then this means that the pattern is linear.Remember to have all your x-values in order!The first differences also tells you about the rate of change for the situation.If the second differences are constant,then this means the pattern is quadratic.

4.3 Transformations of a Parabola

Now that we've learned the key features of a parabola, we can decipher different transformations that can be made by a parabola.


The vertex form of a parabola is showed as: y= a (x-h)² +k

The basic parabola has the formula y=x² which means the value of a=1, the value of h=0, and the value of k=0.


Now, im going to show you how to find the values of a,h,and k and what transformations each value does to the parabola.

How does the value of (a) determine the orientation and shape of a parabola?
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How does the value of (k) determine the vertical position of the parabola?
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How does the value of (h) determine the horizontal position of the parabola?
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Let's Put It ALL TOGETHER!

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Now that you know how a, h, and k can transform a parabola,we can figure out how to write out equations given the transformations.
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In the example above, since we know the vertex is (3,4), we can figure out that there is a horizontal translation 3 units right, and vertical translation 4 units up. You can then sub -3 and 4 into y=(x-h)²+k to get the equation above.


*Note: Remember if the h value of the vertex is negative to switch it to positive when you sub the value into the equation and vice versa.*

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Example 2 shows how you can go from having the equation, to writing out different transformations. Example 3 on the other hand gives you the transformations but you have to figure out the equation.

4.4 Graphing Quadratics in Vertex Form

Step Pattern


Watch the video below to further learn about graphing using the step pattern.

Graphing using Step Pattern
Mapping Notation Formula

Mapping notation is a strategy used to accurately graph any quadratic relation.This method can be used instead of step notation. In general, to go from the graph of y=x² to y=a(x-h)²+k , you can use the mapping formula (x.y) ---->(x+h, ay+k)


Here's some examples about how to use mapping notation:


We will be using the equation: y= (x-4)²+2


The first thing we need to do is find the a, h, and k value. We learned how to find these values in the previous lesson of transforming parabola's(4.3).


So we know: [a=0 h=4 and k=2]


To find the (x,y) coordinate, we just sub in the a,h,and k values into the mapping formula

(x+h, ay+k).


You should then get (x+4, y+2)


TRY THESE QUESTIONS!


a) y=(x+7)²

b)y=5(x+2)²-3

c) y=1/2(x-3)²+1

Finding x and y intercepts in Vertex Form

We can find the x and y intercepts from the vertex form: y=a(x-h)²+k
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Word Problems for Quadratics in Vertex Form

FLIGHT OF A BALL WORD PROBLEM


*WATCH THE VIDEO BELOW TO LEARN HOW TO DO THIS TYPE OF WORD PROBLEM*


Flight Of A Ball Word Problem
BUSINESS APPLICATION (PROFIT QUESTION)

Profits(revenue) can be maximized and this relation can be modeled by a quadratic relation.


- The relationship between selling price and profit can be modeled by a quadratic

relation.

- The relationship between selling price and the number of units sold can be modeled by

a quadratic relation


Ex.1 Maple is selling sunglasses. Let P represent the profit and let s represent the selling price. The profit can be modelled by the equation:


P= -4(s-35)²+2500


a)What is the selling price that will maximize profits?


We know that the max profit $2500, so the selling price that will maximize profits is 35$.


b)What is the selling price when profits equal zero (also known as the break-even point)


*Solution Below*

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Quadratics In Factored Form

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Special Cases
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Perfect Squares :Squaring a Binomial


A perfect square is a number that can be expressed as the product of two equal integers.


The formula used for this concept is:

(a+b)²=a²+2ab+b² OR (a-b)²=a²-2ab+b²


Example 1: (2x+3)²= 4x²+12x+9


Example 2: (3x-5)²=9x²-30x+25

More special cases and differences of squares
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5.3 Common Factoring
Common Factoring
5.4 Factoring Simple Trinomials
Factoring A Simple Trinomial
Factoring by Grouping
5.4 Factoring Complex Trinomials
3.9 Complex Trinomial Factoring
5.6 Factoring Difference of Squares and Perfect Square Trinomials
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6.2 Solving Quadratics by Factoring (finding the zeros)
6.2 Solving Quadratic Equations by Factoring
4.5 & 6.3 Graphing Quadratics In Factored Form
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*Note: I found the vertex from the axis of symmetry value(h-value of vertex) and optimal value(k-value of vertex)
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Word Problems for Quadratics in Factored Form

Dimensions of a rectangle given the area
Dimension of a Rectangle given the area
Flight of an Object
FLIGHT OF OBJECT
Shaded region of a shape
SHADED REGION AND SHAPE QUESTION

Quadratics in Standard Form

6.1 Maximum and Minimum values (Completing the square)

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6.4 The Quadratic Formula
Quadratic Formula

Word Problems In Standard form (6.5)

REVENUE QUESTION UPLOAD

|Connections between Topics|

Quadratics in vertex form, standard form and factored form can all be easily be graphed. For example, when graphing in standard form, you still have to convert the equation into a factored equation. Also an equation can tell you all the different aspects of the parabola without even graphing it. For example if there is a negative a value you can figure out that the parabola will be reflected into the x-axis. Also many topics we learned can convert from one form to another. For example you can go from factored form, to standard form, to vertex form easily. When we learned simple trinomials and complex trinomials I discovered how similar they are. The only difference between them is the number in front of "a". When finding roots, finding the x intercepts in vertex form and standard form are very similar. In both units you could get 2 x-intercepts as well as you had to make sure to use the plus and minus sign in front of the square root symbol. Therefore this unit is extremely interconnected.

Reflection on the Quadratics Unit

Throughout this quadratics unit I’ve had various ups and downs while learning concepts and doing quizzes and tests. Going into quadratics, I expected it to be very challenging and to be honest, I was very uneasy about how my marks would be. In grade 9 we learned about lines and now we were moving on to parabolas. The first unit (vertex form) was actually quite interesting. Before, I was clueless that parabola’s were all around us, all the time. Once I got home after learning the first vertex form lesson, I started discovering parabola’s around my own home! When we began learning about key features of the parabola, some were quite obvious while others…not so much. I found myself always getting confused about the AOS and optimal value and why they were tied in with the vertex. Thankfully, I eventually understood it after a while. Learning about the transformations of a parabola reminded me of learning transformations in the 3rd grade. The words “translate” and “reflection” brought back so many memories! The concept of transformations was although quite simple and easy to grasp. Moving on to quadratics in factored form, I had a few issues. The FOIL method was easy, but once we got into common factoring I started to get confused. After the lesson on common factoring, I was unsure about why numbers were being factored out and many other issues. Knowing myself, I did start to get frustrated but didn’t stop persevering. It did take a good amount of practice to really understand common factoring but it was all worth it. Once we learned all the different factoring methods, it was very confusing to solve them when they were all jumbled together. It took me long hours of practice to decipher what method to use for each question. Graphing in factored form was longer to do then using the vertex form step pattern, but was still pretty simple. The only issue was that it had a lot more steps compared to step pattern, so if you messed up one step, you’re screwed. Moving onto quadratics in standard form, this was the best unit out of all of them. The only minor issue I had was with completing the square and distributing the negative in front of the bracket. I eventually did figured it out though. My favorite topic in the whole unit was the quadratic formula. I loved solving both knowledge and word problems using the quadratic formula and found the concept easy to pick up. My best accomplishment throughout this unit was getting a 96% on the last quadratics test. I was ecstatic to know my hard work had payed off.
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Useful Links For This Unit