Quadratics Realtionships

Grade 10 Academic Math

Summary

Throughout this website presentation, we will summarize the entire quadratics unit! We will learn all the topics throughout the entire quadratics unit using a variety of different videos, math questions (knowledge, application and communication based), variety of different media (text, photos and videos).

Introduction

This chapter deals with quadratic equations; quadratic equations that take form of y = ax^ 2 + bx + c or y =a(x - h)^2 + k or x^2+bx or y=(x+a) (x+b). During this time we will graph quadratic equations; as well as learn how to find out if an equation is quadratic. When graphed quadratic equations form a parabola shape. Furthermore in the unit we visit factoring; in which we will mainly focus on how to factor simple/complex binomials and tri-nomials. We will factor equations such as ax^2 + bx + c = 0 to find two binomial expressions, and use zeroes for graphing. Later on we will learn completing the square method to graph equations in y= ax 2 + bx + c form. Finally we will learn a formula for equations in form of ax 2 + bx + c = 0; which is called the quadratic equation.

Table of contents

Understanding a Parabola

-What is a parabola?

-Analyzing a parabola (second differences)

Types of equations

Vertex Form

- Understanding the step pattern

- Describing Transformation

- Graphing vertex form x

- Isolating for a variable

-Finding optimal value

-Finding axis of symmetry.

Factored form

- Factors and zeroes

- Finding axis of symettry

- Finding optimal value

- Factoring using algebra tiles

- Expanding factored form equations to standard form

- Factoring simple trinomials

- Factoring complex trinomials

- Other forms of factoring (differences of squares, perfect squares)

- Reviewing graphing with factored form

Standard form

- Using quadratic formula

- Completing the square

Word Problems

What is a Parabola?

A parabola represents the graphed form of a quadratic equation. A parabola takes shape of a symmetrical U shape. A parabola can open up or down in direction depending on a equation.
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A parabola can be compared to a soccer player kicking a ball, the arch the soccer ball represents the symmetrical shape a parabola forms. Parabola's have many different properties which are essential to learn for doing well in this unit!

Properties of a Parabola

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Labelled Parabola

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How to find out if an equation is quadratic

Using a table of values you can easily as to an equation is quadratic or not. A property all quadratic equations have in common is that they have the same second differences in Y values, and unequal first differences; compared to linear equations which have the same first differences.
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Extra Practice on calculating second differences

On a separate sheet of paper calculate the second differences of the values chart's below to determine if the equation is quadratic.
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Vertex Form

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Breaking down and understanding the equation & transformations

Term Representations

-The Y variable represents the Y-intercept

-The A variable represents whether the parabola opens up or down (positive a value results to parabola opening down, negative a value results to the parabola opening down), it also represents if the parabola will have a compression or stretch

- The X variable represents the x intercept

- The H variable represents where the X point is on the graph (right or left) (axis of symmetry) (If h is positive 6 the parabola will be horizontally translated 6 units to left)

- The ^2 creates the arch shape of the parabola

- The X variable is in charge of whether the parabola will be vertically shifted up or down (according to positive or negative sign) (positive K value will shift parabola up and negative K value will shift parabola down)


After you plot your vertex using the transformations explained you can use the step pattern to plot your vertex form equation!

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Understanding Transformations

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When a equation is in vertex form each letter variable is in charge of some type of transformation.


Y=a(x-h)+k

With all variables at 0 the parabola would be at the origin at 0,0


- The (a) value stretches or compresses the parabola. If the (a) value is greater than 1 the parabola will stretch by the factor of the value of the variable; This is what is called a vertical stretch. If the (a) value is less than 1 the parabola will compress by the factor of the value of the variable; this is known as a vertical compression.


- The (-h) represents whether the parabola moves left or right. *RECORD (When the h value is positive the parabola moves left and when the h value is negative the parabola moves right). This is called a horizontal translation.


- The (k) variable represents whether the parabola moves up or down. *RECORD (Whenthe k value is positive the parabola moves up, when the k value is negative the parabola moves down). This is called a vertical translation.


Did you know? The H value and K value make up the vertex?

The Step Pattern

The step pattern is used to create the arc-like shape of the parabola; the step pattern is as follows

Over 1 up 1

Over 2 up 4

Over 3 up 9


Note: *When using the step pattern we just square the number we are going over by to find the number we go up by

Note: *When there is a whole positive/negative: number, fraction, decimal in place of variable a value you just simply multiply the a value to the squared part of the step pattern

finding Axis of symmetry in vertex form

To find the axis of symmetry (y=a(x-h)+k) in a vertex form equation you need to look at the h value as explained briefly before. If the (h) value is positive you will simply need to change the sign, and what the value then becomes is the axis of symmetry.


For example:

y= 4(x+5)+6

----a----h----k

In the equation the value of h is +5, so when the sign is flipped the axis of symmetry is -5.

Finding optimal value of a standard form equation.

The optimal value in a standard form equation is entirely dependent on the equations k value. If the k value is positive it will go above the x axis. If the k value is negative it will go below the x axis. *NOTE: Do not be confused, we do not flip the signs in order to find the optimal value.


For example:

y= 4(x+5)+6

----a----h----k

Therefore the k value is positive 6, meaning the optimal value of the parabola will be +6 over the x-axis.

Isolating for x in vertex form (finding x intercepts/roots)

We isolate for the x variables to find the values of the x-intercepts in a vertex form equation. The next following steps will teach you to find the x-intercepts of a vertex form equation!
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Graphing vertex form

Graphing a parabola in vertex form

Factored form

During this section of the video we will learn how to factor a variety of equations into factored form equations; throughout the whole unit this is the most challenging and important part of the unit. It is very important that this section is understood thoroughly!

Finding Zeroes in factored form

The zeroes of a parabola are also known as the parabola's x- intercepts, and to find the x- intercepts you must the Y variable to the value of 0, In simpler words you just need to take the co-efficient's inside and change the sign and that would be your zero.

Finding axis of symettry in factored form

To find the axis of symmetry you need to use the zeroes we just learned to find and find the average of them both. To find the average you will add both a and b values and divide them by two.


For example: X int @(6,0), (-4,0)

(a+b)/2

(6+-4)/2

2/2

1=Axis of symmetry Therefore after averaging the x intercepts 1 is the axis of symmetry.

Finding the optimal value for equations in factored form

To find the optimal value in factored form we will need to sub the axis of symmetry as the x value in the original equation; this will allow you to solve for y (max/min).

Common Factoring

3.7 Common Factoring

Factoring simple trinomials

Factoring Simple Trinomials

Factoring Complex Trinomials

3.9 Complex Trinomial Factoring
I am sorry for using the video already shared on Edmodo, but I was determined after several attempts I was not able to find a video on complex factoring using the guess and check method learnt in class.

Special cases of factoring (difference of squares and perfect squares)

Factoring Perfect Square Trinomials and the Difference of Two Squares

Using algebra tiles to factor (particularly for visual learners)

How to Use Algebra Tiles For Factoring

Expanding Factored Form equations to standard form

Factored form of a quadratic equation makes life much easier when trying to find x intercepts/roots; but sometimes you may need to expand factored form equation for example to simplify the equation. Expanding factored form equations involves the distributive property of numbers. When complete you can expand a equation from (x+4) (x+5) into =x^2+9x+20. Below I have created a document lesson on how to expand factored form equations into standard form equations.
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video about Graphing with factored form (made by myself)

Graphing a Parabola in Factored Form

Standard form

Standard form equation's take the format of y=ax^2+bx+c. From these equations it is very difficult to graph and solve, so we can convert the equation into other forms of equation. For example factored form, but not all times is this easy, so we use the alternative methods such as the quadratic formula.

The Quadratic formula

The quadratic formula allows us to find for the x-intercepts and axis of symmetry using a standard form equation. Even if the equation is in different form, they can be expanded into standard form, and make use of quadratic formula to solve for. Below is a picture of the quadratic equation, which we will learn to apply.
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Quadratic Formula Song

quadratic formula

Finding x intercepts of standard form equations using the quadratic formula

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Quadratic equation : Using the Quadratic formula to find the x-intercepts
The video explains what I have taught using a documented lesson. This video is suitable for an audience that learns best through listening.

Finding the axis of symmetry of a standard form equation

To find the axis of symmetry of a standard form equation we can manually take both x intercepts (found from the quadratic formula) and find the average of them (add them together and divide by 2). But we can also use part of the quadratic formula to find for the axis of symmetry (-b/2a). If done correct both methods WILL give you the same axis of symmetry.


Lets do an example, let's find the optimal value for the equation we found x-intercepts for above (3x^2-4x+1).


Method 1 (adding both x intercepts and divide by 2)


Axis of symmetry: (1+1/3)/2

=0.666


Method 2- Using -b/2a (variables were identified in previous part)


Axis of symmetry: -b/2a

=-(-4)/2(3)

=4/6

=0.66


Therefore we applied both methods taught to find for the x intercepts of a standard form equation and we managed to get the same answer proving that our answer is correct.

Finding the optimal value of a standard form equation

With the axis of symmetry you just found out, you apply it as a variable value. You sub in the values and solve to find for a optimal point.


With the example used above for explaining how to calculate x intercepts and axis of symmetry, we will also use the same example to explain optimal value.


Equation being subbed into: (3x^2-4x+1=0 X=0.66 (Axis of symmetry)


3(0.66)^2-4(0.66)+1

=-.3332


Therefore you have your optimal value.

discriminants

So now that we know the quadratic formula, we can focus on finding discriminant's. Discriminant's are simply just to inform us on information we did not know of the standard form equation.
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As we can see the b^2-4ac term within the second half of the equation is being square rooted. The value of the within the square roots, before dividing by 2ac is the discriminant. The discriminant determines the number of solutions a equation has within it. If the value of the square term being squared rooted is:


  • negative there is no solution
  • is equal to 0 there is only 1 solution
  • if greater than 1 there is 2 solutions.
Discriminant of Quadratic Equations

Completing the Square

Completing the Square - Solving Quadratic Equations

Connections chart

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Word problems

Revenue Questions

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Optimization Fencing problem

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unit reflection

Overall, I found this unit very fun, exciting as I was exposed to so much new content. But as I was exposed to so much new content, I found it hard to know when to apply what. Because this unit was a very long unit it was important to do your homework up to date, some of the I found difficult to understand was factoring, and this was due to my lack of completed homework. But once you pay attention, Mr. Anusic, a great learning instructor makes all of the unit so simple and easy! Therefore my suggestions are to complete your homework at all times, and ask questions about what you don't understand, as Mr Anusic is always more than happy to answer them! The Quadratics formula was finally a unit which we can connect to in real life; making it so exciting. I found that we can apply concepts learned in the unit in real life for example: revenue questions. Even though I feel as though I did exceptionally well this unit, no body is perfect; therefore improvements can be made! One thing I encourage everyone to do is complete your homework, it gives you a better idea of what you are learning and an opportunity to ask any questions which you found difficult. Below is the factoring unit test, Although I did very well during application I struggled in knowledge, though I am not proud of this section, we learn from our mistakes! I set a goal to promise myself that I will go through all my factoring homework before exams and summative task to ensure that I will not make the same mistakes again!


Below is a picture of my factoring unit test:

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Though my application part of the test was exceptional, the knowledge part was not. I over thought most of the multiple choice questions which i found the correct answer the first time. Secondly, I also made silly mistakes like not factoring all of the terms. In the second, and third picture i did not factor all of the terms out.

conclusion

In conclusion you get to learn a variety of things during quadratic relations unit. I will always encourage you to do your homework and keep up to date, if you do not understand one concept, when it comes time for word problems like fencing and revenue questions (showed-explained above) you will do just one step wrong and in result get the wrong solution. I hope the concepts/terms I taught through a variety of media use (pictures, text, videos) helped you understand this unit! I would like to thank you for your time!