Quadratics For Beginners

Learn all about the grade 10 math curriculum!

Introduction

Quadratic relationships are different than linear relationships as they are curved line instead of straight ones. Also unlike linear relationships, quadratic relationships can have 2 x-intercepts. In this website, you, a math enthusiast, will learn about different quadratic equations, the forms they may take and how to convert and solve them.

Different Quadratic Equation Forms

Vertex Form:

y= a(x-h)*2+k

This allows you to easily identify the vertex (h,k)


Factored form:

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

This allows you easily identify and solve for the x-intercepts


Standard form:

y= ax*2+bx+c

This allows you to solve using the quadratic formula

Let's Get Started On Chapter 4

In this chapter you will find:

-First Differences

-Vertex form & transformations

-Graphing vertex form {y=a(x-h)*2+k}

-Graphing factored Form {y=a(x-r)(x-s)}

-Word Problems

First Differences

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In a linear relationship, only the first differences are equal, in this example the first differences are all 3.

In a quadratic relationship, the first differences are not equal but the second differences are. In this relationship the second differences are all equal to 2.

When the first differences and second differences do not all have the same number, then the relationship is neither linear or quadratic.


Vertex Form & Transformations

Vertex form is {y=a(x-h)*2+k} and the vertex in this equation is (h,k).

If the variable "a" is positive, then the parabola (the curved line) opens upwards, and so if the same variable is negative, the parabola opens downwards.

The variable "a" also controls the transformations of the parabola.

If "a" is less than one, the parabola is thinner.

If "a" is more than one, the parabola is wider.

If "h" is positive, then the parabola will translate to the left.

If "h" is negative, then the parabola will translate to the right.

If "k" is positive, then the parabola will move up.

If "k" is negative then the parabola will move down.

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Graphing Vertex Form

In order to graph vertex form, you must first identify the vertex (h,k).

Secondly you must identify the step pattern (Up_ Over_).

Then with this knowledge you will be able to plot the vertex, and two other points in this graph thus making a parabola.

Below is a wonderful video explaining graphing from vertex form by the amazing Mr.Anusic and Mr.James.

Graphing from Vertex Form

Graphing Factored Form

Factored form looks like this y= a(x-r)(x-s)

This form gives you the x-intercepts of a parabola.

To graph this, you have to add your "r" and "s" values and then divide by 2.

This gives you the value of x, allowing you to substitute it and solve for your y coordinate.

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Watch this video that I made on what to do after finding your vertex!
https://www.youtube.com/watch?v=vVqay25lvNE&feature=youtu.be

Word Problems

Example:

A bridge has an arch which is represented by the relationship y=-2(x-2)(x-4). x represents the distance horizontally from the start of the bridge in meters and y represents the height of the arch in meters.

a) what length of the bridge does the arch cover?

b) what is the height of the bridge at 3m in distance?

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Let's Get Started On Chapter 5

In this chapter you will find:

-Expanding polynomials

-Expanding special products

-Common factoring binomials

-Factoring by grouping

-Factoring simple trinomials

-Factoring complex trinomials

-Factoring a perfect square

-Difference of square

Expanding Polynomials

This is expanding factored form expressions into standard form.
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Expanding Special Products

There are three types of special products.

1. (a+b)*2

2. (a-b)*2

3. (a+b)(a-b)


1. (a+b)*2

To expand this, you can use Sam Doesn't Pull Strings.

S - Square the first term

DP - Double their product

S - Square the second term

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2. (a-b)*2

To expand this, you can use the Sam Doesn't Pull Strings method as well!

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3. (a+b)(a-b)

To solve this, you just multiply the first two terms, and the last two, and DO NOT multiply the inner and outer terms as they cancel each other out.

If you have difficulty with this, just spread the rainbows and simplify.

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Common Factoring Binomials

In order to common factor a binomial, you must take out the GCF (greatest common factor) from both terms.
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Factoring By Grouping

To factor by grouping, you need four terms.

You then factor each group, and factor again as the numbers and variables in the brackets should be the same.

This gives you your final answer.

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Factoring Simple Trinomials

A simple trinomial is an expression in standard form and the first coefficient starts with the number 1 {x*2+bx+c}

In order to factor a simple trinomial, you must find two factors of "c" that add up to give you "b".

You substitute "bx" for those two factors and then factor by grouping, as taught above.

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Factoring Complex Trinomials

A Complex trinomial is an expression in standard form and the first coefficient starts with the number greater than one 1 {ax*2+bx+c} (The sign does not matter, as long as the number is anything other than one, it is a complex trinomial).

In order to factor a complex trinomial, you must first common factor if it is possible.

Secondly, you find factors of "a" that multiply with factors of "c" that add up to give you "b".

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Factoring Perfect Squares

To factor a perfect square you must first be sure it is one.

The first and last terms must be able to be square rooted, and the middle term must be the product of those roots, times two.

Do do this, you can use the Sam Doesn't Pull Strings method.

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Difference Of Squares

In order to factor a square, and find the difference you must first know if it is a square.

Both terms should have a square root.

You may use the Sam Doesn't Pull Strings method here, but you only need to use the Sam and Strings part.

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Let's Get Started On Chapter 6

In this chapter you will find:

-Finding zeros in vertex form

-Completing the square (Standard form to vertex form)

-The quadratic formula

-Optimization word problems

Finding Zeros In Vertex Form

Vertex form: {y= a(x-h)*2+k}

In order to find zeros in vertex form, you must substitute "x" for 0 and then solve for "y."

After doing this, you can substitute the value of "y" into the equation and solve for the value(s) of "x."

NOTE: when finding the value(s) of "X" you might end up with no value, one value, or two values, as a parabola can intercept the x axis these many times.

When square rooting a number, you can get 2 zeros as the square root of 16 is 4 and -4.

Make sure you branch off your equation into two at this point and continue solving!

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Completing The Square (Standard Form to Vertex Form)

In order to convert standard form to vertex form, we must complete the square.

Firstly you have to add and subtract half of "b*2" to your equation.

You then factor the equation, without the last two terms and simplify.

This then gives you the same equation in vertex form!

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The Quadratic Formula

The quadratic formula can be used to solve all quadratic equations.

However this formula can only be used to solve equations in standard form.

All you have to do is substitute values into the formula and solve!

NOTE: Don't forget that square rooting a number will give you two answer, one will be negative and the other will be positive. This gives you the value of the x-intercept(s) if there are any.

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There are two ways to solve quadratic equations.

you can give an exact answer, which is leaving the question in formula form, or an approximate answer, which is solving for the value of x, sometimes resulting in a decimal answer.

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Optimization Word Problem

In chapter 6, we are introduced to a new kind of word problem: Optimization Word Problems.

Attached below is a video demonstrating how to solve these problems!

Optimization Problem #4 - Max Area Enclosed by Rectangular Fence

Reflection

I personally feel like I have done really well throughout the quadratics unit and I feel that my second mini test reflects this. I found the first and last mini units of quadratics much easier than the second unit, yet I worked hard and did really well on the test. I feel like I had problems with the second unit because there were a lot of new terminology and ideas introduced, and sometimes I would struggle with keeping the rules to each one straight, and often got the mixed. I also had problems with using the quadratic formula because I would often make really small mistakes that would cause the whole equation to be wrong, but as i continue to use the formula, I make less and less mistakes!
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Time To Connect The Dots

Quadratics 1, 2, and 3 are all connected, they are after all one big unit divided into three smaller units. As we moved onto each unit, we took what we learned from the previous one and expanded our knowledge. in quadratics 1, we were introduced to parabolas and the different forms of relationships. In quadratics 2, we learned how to interchange these equations and expressions by factoring and expanding. In quadratics 3, we learned how change into and out of vertex form, and we learned how to find x-intercepts. These three units were made into small chunks so that students could learn and process quadratics in manageable bits. Quadratics 1, 2, and 3 are all connected.

Conclusion

Thank you for visiting my quadratics website, and I hope you know more about quadratics than you did before, now go forth and share your newfound knowledge!