Quadratic Exploration

By: Kesigan

What is quadratics and how does it apply to everyday life?

The word Quadratic comes from "quad" meaning square, because the variable gets squared. The graph of a quadratic which is a parabola. Quadratics can be used to find the flight of objects. When throwing a ball the path the ball creates is a parabola. Parabolas can help find: the highest point reached by the ball, how long it takes to reach the highest point, what is the initial height of the ball, etc. Other real life scenarios can be used to calculate the parabola of roller-coasters or suspension bridges. Furthermore, quadratics are important in the business world. Quadratics inform companies about how many customers to expect, how to increase their cost, etc. In addition to business, quadratics increase our understanding dimensions of shapes. It allows us to find the area and perimeter to new shapes. By going through this website you will be a quadratic master!

Table Of Contents

Section 1: Introduction into quadratics


  • Introduction to vertex form
  • Graphing by step pattern
  • Word problems using vertex

Section 2: Factored form equations



  • Expanding and simplifying
  • Monomial Factoring (GCF)
  • Binomial Factoring (GCF)
  • Factoring by grouping (4 terms)
  • Simple trinomial factoring
  • Complex trinomial factoring
  • Special Product- Difference of Squares
  • Special Product- Perfect Square Trinomial
  • One Application problem


Section 3: Standard form and quadratic formula


  • Quadratic formula
  • Standard form equation
  • Completing the square
  • Discriminant
  • Application word problems

Learning goals of section 1

This website has 3 sections and for each section we have a committed learning goals for you to know once you have completed the section.


The learning goals for section 1 include:


  • I can graph
  • I know the features of a parabola
  • I can solve word problems using vertex form
  • I can create mapping notation equation

Features of the Parabola

  • Axis of symmetry: Is a imaginary line that divides the parabola into halves equally. The x co-ordinate of the vertex.
  • Vertex: Part of the parabola where it changes direction and is the minimum/maximum point of the parabola. The axis of symmetry pierces through the vertex.
  • X-intercept(s)/Root(S)/zero(es): The point where the parabola passes through the x-intercept. The could be up to two x-intercepts.
  • Y-intercept: The point where the parabola passes through the y-intercept.
Big image
Equation Video 1

Two videos on Step pattern graphing

I made two step pattern graphing videos for you to further understand Quadratics!

Chapter 2: Word Problems

The height of the rocket, y in meters, x seconds, and this is the equation:

y=-3(x-9)²+21.5


1. What is the maximum height of the rocket?

2. How long does it take for the rocket to reach maximum height?

3. What is the initial height of the rocket?

4. How long is the rocket in the air?

5. What was the height of the rocket after 3 seconds?

The solutions:

1. The maximum height of the rocket is 21.5 meters. To find maximum height we need to know the Y-intercept which is the K value hence the maximum hieght of the rocket is 21.5 meters.


2.The time taken to reach maximum hieght is 9 seconds. To find time to reach maximum hieght we just need to find the x co-ordinate of the vertex. Therefore, the time taken to reach maximum hieght is 9 seconds.


3. Initial hieght means hieght when time is equal to 0.

y=-3(0-9)²+21.5

y=-3(-9)²+21.5

y=-3(81)+21.5

y=-243+21.5

y=-221.5


Therefore, the initial hieght is -221.5 meters.


4. We need to find how long the ball was in the air hence we need to set the hieght to 0.

0=-3(x-9)²+21.5

-21.5=-3(x-9)²

-21.5/-3=-3(x-9)²/-3

7.166=(x-9)²

-7.166=(x-9)²

+-2.676=x-9

x=11.68


5. To find the hieght after 3 seconds we need to replace x with 3 seconds to find hieght.

y=-3(3-9)²+21.5

y=-3(-6)²+21.5

y=-3(36)+21.5

y=-86.5


Therefore, the hieght of the rocket after 3 seconds is -86.5 meters.

Mapping notation video

Mapping Notation.mp4

Chapter 2

In chapter 2, we are advancing from vertex form and moving to factored form. Throughout chapter 2 you will master the necessary grade 10 factor form math skills.

Learning Goals of section 2

  • I can expanding and simplifying
  • I can monomial Factoring (GCF)
  • I can binomial Factoring (GCF)
  • I can factoring by grouping (4 terms)
  • I can simple trinomial factoring
  • I can complex trinomial factoring
  • I can solve special Product- Difference of Squares
  • I can solve special Product- Perfect Square Trinomial
  • I can solve one Application problem
  • I can graph factored form equations
  • I understand factored form equation
  • I can find the vertex and axis of symmetry in factored form

Factored form equation

This is the factor form equation! This equation can you help you easily find the x-intercepts of a parabola. If you have an parabola in standard form and you need to find the x-intercepts, you can convert standard to factored by factoring! In factored form, you can find the vertex and the axis of symmetry making graphing simple in this form.
Big image

Different types of factoring tutorial

Expanding and simplifying

Expanding is to remove brackets from an algebraic expression.

Simplifying is to put an mathematical terms to the simplest form.


Here I will show you expanding and simplifying:


(x+3)(x+5)

=x²+5x+3x+15

=x²+8x+15


First I used the expansion method which is multiplying both of the terms in the first bracket to the other two terms. The "x²" came from (x)(x), the "5x" came from (x)(5), the "3x" came from (3)(x), and the "15" came from (3)(5). The 3x and 5x were common so, I added them together to create 8x which is called simplifying.


Another way a question might we asked is like this:

(x+9)²

=(x+9)(x+9)

=x²+9x+9x+81

=x²+18x+81


The "²" represents there are two copies of the question therefore, on my first step I wrote (x+9) twice. From here i just expanded the question and simplified.


The question could be asked like this:

3(x+7)(x-4)

=3(x²-4x+7x-28)

=3x²-12x+21x-84

=3x²+9x-84


I used the same expanding method and multiplied 3 around the whole expanded equation.


Monomial factoring:

In monomial factoring we factor out the GCF(greatest common factor) as shown below.

2x+2y

=2(x+y)


3s²+5s²

=s²(3+5)

Binomial factoring:

In binomial factoring, we put the common factor in brackets. The two remaining terms are also put in brackets.

4x(x+5)-2(x+5)

=(4x-2)(x+5)


2x(x+2)-7(x+2)

=(2x-7)(x+2)

Factoring by grouping:

To factor by grouping we just remove a common factor out of the equation.


10x²+5x+4x+2

5x(2x+1)+2(2x+1)

=(5x+2)(2x+1)


In the brackets we have the same expression so we write that once in our final answer. Then the two remaining terms outside the bracket are written in brackets with the other final answer and put them to together as shown above.


Simple trinomial factoring:

Simple trinomial factoring have an a value of 1.


x²+7+10

=(x+2)(x+5)


We need to find two numbers that multiply into 10 and those two same numbers add to 7. These two numbers go into the answer.


Complex trinomial factoring:

Complex trinomials have an "a" value higher then 1.


3x²+5x+2

3x²+3x+2x+2

3x(x+1)+2(x+1)

=(3x+2)(x+1)

The method i used to solve this complex trinomial is to decompose it. To do the compose method you multiply the a value and c value. After we need to find two numbers (same numbers) that when multiplied into 6 and add up to 5. The 5 came from the b value which once again the two numbers must add up to. The two numbers were 3 and 2.

3x2=6

3+2=5

We rewrite 5x as 3x and 2x as shown above. Then you just factor by grouping and now you have solved a complex trinomial!


Special products- Difference of squares:

Differences of squares can be defined as two squares being subtracted. By looking at these types of question we can factor them. Difference of squares formula is this: a²-b²


x²-16

=(x+4)(x-4)


We first factored the x and used a negative 4 and positive 4 to cancel a b value and equal negative 16. With the c value being negative one number in the bracket will be positive and the other will be negative.


A=1 B=0 C=-16

(x+4)(x-4)

=x²-4x+4x-16

=x²-16

=x²-4²


Special products- Perfect square trinomials:

To factor a perfect square trinomial we need to use the decompose method with breaks down the b value so we can later on factor by grouping.


To find out if your equation is a perfect square trinomial we can put it into this formula:

a²+2ab+b² or a²-2ab+b² depending on your b value. If your b value is negative then you use the negative 2 formula otherwise, you use the positive 2 formula.


Example: x²+6x+9

The b value(+6x) is positive hence we use the positive 2 formula. (x)²+2(x)(3)+(3)²

If the b value of the formula equals the original equation then that means it is a perfect square trinomial. The b value of the original expression is 6x and the formula tells us 2xXx3 which equals 6x.


One application problem:


The area of the rectangle is x²+15+56, find the length and width.

To find the length and width you just factor the expression.


x²+15+56


8x7=56

8+7=15


=(x+8)(x+7)


Therefore, the length and width of the rectangle are (x+8) and (x+7).


If the value of x=3, what is the area of the rectangle?

(3+8)(3+7)

(11)(10)

=110cm²


Therefore, the area of the of the rectangle is 110cm².

How to graph a factored form equation

Graphing Factored Form of Quadratic Functions

Factoring complete

You have learned how to factor like a pro! The next part of the website will teach you standard form and the quadratic formula.

Chapter 3

In chapter 3, we learn about how to convert standard form to vertex form. Throughout this chapter, you will learn how to complete the square and use the quadratic formula. This is the final chapter in the quadratics unit.

Learning Goals


  • I am able to use the quadratic formula
  • I understand the standard form equation
  • I can completing the square
  • I can find the discriminant
  • I can solve application word problems

Standard Form

This is the standard form equation ax²+bx+c. When you expand a factored form equation, you get the standard form equation. This equation has 3 terms: ax², bx, and c.

Quadratic Formula

The quadratic formula is used to identify the x-intercepts and by using this method to find your x-intercepts, you can get exact values of the x-intercepts. The quadratic formula has the variables a, b, and c. Therefore, you just sub in the matching value into the equation and solve it.


Example: y=3x²-8x+2

x=8+/-/(8²-4(3)(2))/2(3)

x=8+/-/(64-24)/6

x=8+/-6.3

x=2.4

x=0.3


Therefore, this equation has two x-intercepts at (2.4,0) and (0.3,0)

Solve Quadratic Equations using Quadratic Formula

Completing the square

Completing the square converts standard form into vertex form. Converting into vertex form makes it very easy to graph the parabola.


Steps required to complete the square:

First, put the first two terms in brackets and factor out the a coefficient in the entire bracket.

Next, you divide the new b value by 2 and then square it.

Write a positive value of the number in the bracket and add a negative value of the number outside the bracket. We do this because we can not create new numbers out of no where so the value added is subtracted.

Now, multiply the negative b number with the coefficient in front of the bracket.

After, factor everything in the bracket into a perfect square trinomial and add the numbers outside the bracket.

Finally, you have completed the square and have converted standard form to vertex form.

Completing the Square Example 2 Solve Quad. Equations

Discriminant

The discriminant value tells you how many solutions are in the given equation. The discriminant formula is the terms in the square root of the quadratic formula. If the discriminant value is under 0 then there are no solutions.

Example:2²-4(2)(3)=-20

If the value of the discriminant is 0 then there is one solution to the equation.

Example:4²-4(1)(4)=0

If the value of the discriminant is above 0 then there are two solutions to the equation.

Example:4²-4(1)(2)=8

Graphing

To graph a standard form equation, you need to put the equation into the quadratic formula and then solve to find the x-intercepts and graph the points. When the equation was in standard form plot the c value because it is the y-intercept.
Graphing Parabolas In Standard Form (Quadratic Functions)

Word problems using the quadratic formula

The width of a rectangle is 4m less than its length. The area of the rectangle is 240m². Find the dimensions of the rectangle.


Let x represent the length.

(x)(x-4)=240


Here I have the let statement explaining what the x value represents. I made this equation because lxw=area, hence the formula above is the same.


x²-4x=240

x²-4x-240=


a=1

b=-4

c=-240


x=4+-//-4²-4(1)(-240)/2(1)

x=4+-//16+960/2

x=4+31.24/2 x=4-31.24/2

x=17.62 x=-13.62


I used the quadratic formula to find the x-intercepts. As you can see we have two x-intercepts and we use the positive x-intercept the find the dimensions.


(17.62)(17.62-4)=280

(17.62)(13.62)=280

Therefore, the length is 17.62m and the width is 13.62m.

Word problems by completing the square

Jacob has a burger shop business that charges $12 per burger and sells usually 36 burgers. He discovers for every $0.50 decrease in price, he sells 2 more burgers per day. At what price can Jacob maximize his revenue?


Let x represent

Let y represent


y=(12-0.5x)(36+2x)

y=432+24x-18x-1x²

y=-x²+6x+432

y=(-x²+6x)+432

y=-(x²-6x)+432

y=-(x²-6x+9)-9+432

y=-(x²-6x+9)+9+432

y=-(x-3)²+441


Let r represent the maximum revenue.

r=(12-0.5(3))

r=12-1.5

r=10.5


Therefore, if Jacob sells at $1.50 less then he can maximize his revenue.

Reflection

In the end, the quadratics unit was eye opening because I didn't realize the real-life application of quadratics. I learned to understand quadratic, you have to learn many concepts before you can completely be a master at quadratics. In the beginning it was confusing but as I progressed i got more comfortable with quadratics. Overall, It was a very cool unit and i enjoyed learning the subject.

Connections

How factored form connects to graphing:


A factored form equation include the x-intercepts of the equation and tells you the opening of the parabola which helps in graphing. Once you expand the factored form equation, you get the standard form equation which tells you what the y-intercept is.


How vertex form connects to graphing:

Vertex form identifies the vertex of the equation. This point can also tell you if it is the highest or lowest point in your graph.


How standard form connects to graphing:


Standard form tells you what the y-intercept is and standard form can easily be converted to factored form or vertex form. You can factor to find the x-intercepts and furthermore, you can complete the square to find the vertex of the parabola.

Quadratic unit's assessments

Big image
Big image
Big image
Big image

End of the website

This is the end of Quadratic Exploration website and now that you have completed going through this website, you are a quadratic professional. If you still have doubts come again and if anyone needs quadratics help refer them to us!

Thank you for learning from our website!

Works cited

MrsVideoMath. "Mapping Notation.mp4." YouTube. YouTube, 2013. Web. 09 May 2016.


MathMeeting. "Solve Quadratic Equations Using Quadratic Formula."YouTube. YouTube, 2011. Web. 09 May 2016.


PatrickJMT. "Completing the Square Example 2 Solve Quad. Equations."YouTube. YouTube, 2008. Web. 09 May 2016.