Quadtratics

Vijayinder Singh Hara

What is quadratics?

Quadratics is a type of algebra that deals with quadratic equations. There are 3 main types of quadratic equations you will be learning about on my website, Vertex form, Factored form and finally, Standard form.

Learning goals

1. In the unit for vertex form, I wish to learn how to explain all transformations after understanding the equation given to me.

2. In this unit I would like to learn to use the step pattern so I can graph an equation.

Key Terms!

Vertex : The maximum or minimum point on the graph. It is the point where the graph changes direction.


Axis Of Symmetry: The axis of symmetry divides the parabola into two equal halves.


Optimal Value: The optimal value is the value of the y co-ordinate of the vertex. The maximum value is when the graph opens down. The minimum value is when the graph opens up.


Zeros: Where the parabola crosses the x-axis.


Y - Intercepts: Where the parabola crosses the y-axis.

Vertex Form

Equation :

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

Checklist!

  • Finite Differences
  • Axis of Symmetry: X = h
  • Optimal Value: Y = k
  • Transformations
  • X - intercepts or Zeros
  • Step Pattern / Mapping Notation

Finite Differences

In order to calculate 'First Differences' you need to subtract the second y value from the first y-values. If the first differences are constant then this means that the pattern is Linear. The first differences also tell you about the rate of change for the situation. If the second differences are constant then this means that pattern is quadratic.

Example 1 :

Linear Relation because the first differences are similar.

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Example 2 :

Quadratic Relation because the second differences are similar.
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Terms

Axis of Symmetry : x = h


  • It divides the parabola into 2 equal halves
  • Always the opposite of what appears inside the bracket


Examples:

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Optimal Value / Min Max Value : y = k

  • Value of the y - co-ordinate of the vertex
  • Maximum value is when the graph opens down
  • Minimum value is when the graph opens up


Examples:

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Vertex : ( h , k )

  • Maximum or minimum point on the graph
  • Point where the graph changes direction
  • H and K value


Examples:

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Transformations :

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' a '

Orientation :

If a > 0, the parabola opens up ( + )

If a < 0, the parabola opens down ( - )

Shape :

If a is less than one, the parabola is vertically compressed

If a is more than one, the parabola is vertically stretched

' k '

Orientation : If k Is negative, the graph goes down

If k is positive, graph goes up

Shape : If k is positive, graph starts higher off 0

If k is negative, graph starts lower off 0

' h

X intercepts/Zeros

Zero - Where the parabola crosses the x-axis.

To find x, you must set Y=0

Parabolas can either have 1,2 or 0 x- axis'.

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Step Pattern

Step pattern is a way to find out when your points will meet. The basic way to find your points is Up 1, over 1 and up 2, over 4.

You must take your A value from y=a(x-h)^2+k and multiply it by 1 for the first point, multiply it by 3 for your second point and multiply it by 5 for your third point.

1 x A=

3 x A=

5 x A=

Now you can plot your points on the graph and connect your dots.

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Word problem - Vertex form

At a baseball game, a fan throws a baseball from the stadium back onto the field. The height in meters of a ball (x) seconds after being thrown is modeled by the function: h= -6(x-4)^2+50

1. What is the maximum height of the ball?

- The maximum height of the ball is 50 meters. We know this because this is the k value.

2. When did the maximum height occur?

- The maximum height occurred at 4 seconds. We know this because this is the h value.

3. What is the height of the ball after 1 second?

- h= -6(1-4)^2+50

- h= -6(-3)^2+50

- h= -6(9)+50

- h= -54+50

- h= -4

Therefore, The ball is at 4 meters after 1 second.

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Video

Here is a video which helped me to learn how to graph using transformations.
Using Transformations to Graph Quadratic Functions

Factored Form

Learning Goals

What I will learn-

1. I will learn how to factor numerous numbers by grouping them and finding the GCF.

2. I will learn how to find and subtract other shapes in others to find total area or perimeter for the given shapes.

Factored Form

Examples of types of factoring:

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

Equation: y = ax + bx + c

- Zeroes or x - intercepts ( r and s )

- Axis of Symmetry : ( x = ( r + s ) / 2 )

- Optimal Value : (sub in )

- Simple Trinomials

- Complex Trinomials

- Difference of Squares

- Perfect Square Trinomaials

- Group Factoring

What is factoring?

In factoring, to find your vertex, you will need to find you AOS and Optimal Value.

To find your AOS you will do X= S + R /2.

To find the Optimal Value you sub the X value from the AOS into your bracketed, factored form and to find the Y intercept you set X= 0.

You will also have to know how to factor by grouping, multiply binomials, perfect square trinomials, and difference of squares. You need to know how to factor equations where A is 1 and if A is more than 1.

Factored Form Equation: y = a(x-r)(x-s), Axis of Symmetry, Optimal Value

Axis of symmetry- Finding A.O.S requires you to find the x-intercepts. (r,0) (s,0)

X=r+s/2

Optimal Value- To find the Optimal Value, you would substitute the AOS into the original equation and solve for y.

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Greatest Common Factor

When you find the factors of 2 or more numbers, and then find some factors are 'common', then they are common factors.

Step 1 - Find GCF (What is the greatest # in common)

Step 2 - Divide the numbers by your GCF

Step 3 - Write solution with brackets

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Simple Trinomials

Is in the form of ax + bx + c ( a value = 1 )

The a value will always have to be 1 to make it a simpe trinomial. The factored form of a simple trinomial will be in the form of (x + __ ) (x +​ __ )

When factoring a polynomial of the form ax + bx + c (when a = 1), we want to find:

1) 2 numbers that ADD to give b

2) 2 numbers that MULTIPLY to give c

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Complex Trinomials

Complex Trinomials have a coefficient more than 1 in front of the x2 term.

Solving Complex Trinomials can be done by the common decomposition method or with trial and error.

Decomposition method-

1.Multiply the a and c value together to receive your product.

2.Find common factors between the product and the b term.

3.Rewrite the middle terms with the 2 factors replacing b

4.Factor by grouping

5.Use your binomial common factoring skills to receive your answer

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Difference of Squares

(a+b)(a-b) = 2a^2 - 2b^2

To find a and b, simplify square root the first and last term. There is no middle term (bx), meaning it is 0x.

Example- 9x - 16

a = √9x = 3x

b = √16 = 4

(3x - 4) (3x + 4)

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Perfect Square Trinomial

The trinomial that results from squaring a binomial is called a PERFECT SQUARE TRINOMIAL. Perfect square trinomials can be factored using the patterns from expanding binomials. In a perfect square, make sure (1) The first and last terms are perfect squares & (2) the middle term is twice the product of the square roots of the first and last terms

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Group Factoring

Group factoring is done when you have 4 terms.

Your first job is to put the like terms together to find their GCF.

You next need to put the GCF outside the bracket and divide the GCF by the 2 value.

Your brackets on both sides should be the same and the left over numbers will be your other bracket.

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Application Problem

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Video

This video taught me how to graph with factored form.
Graphing Factored Form of Quadratic Functions

Standard Form

Learning goals

1.In this unit I will learn to find the X-intercepts with using the quadratic formula

2.In this unit I will use complete the squares to find the equation and then plot it on the graph

Standard Form

- Zeroes ( quadraric formula )

- Axis Of Symmetry : ( -b/2a )

- Optimal Value : ( sub in )

- Completing the Square


Equation: y = ax^2 + bx + c

Summary: A represents direction of opening

C represents the y intercept. Standard form x intercepts- found with quadratic formula and to find the equation you use complete the squares.

Quadratic Formula

Quadratic Formula is used to find the x-intercepts. To use the Quadratic Formula you must first plug in your a,b, and c.
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Quadratic Formula Example

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Complete the squares

We use this method to help us go from standard form to vertex form. This can be useful in many different ways such as, it can help us find the vertex and graph.

My video: How to complete the squares

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Discriminant

The number inside the square root (b^2 - 4ac) of the quadratic formula is called the Discriminant. Instead of using the whole formula, we can use this to find out how many solutions the equation has.

D<0 = 0 solutions D=0 = 1 solution D>0 = 2 solutions

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Word Problem

A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway?


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Therefore, the width of the pathway is 1.5 M.

Connections - How they all connect..?

Vertex Form- y = a ( x - h )^2 + k

- It can be changed to standard form by expanding

- It can be changed to factored form by setting y = 0


Factored Form- y = a^2( x - r ) ( x - s )

- It can be changed to standard form by expanding and simplifying


Standard Form- y = ax^2 + bx + c

- It can be changed to factored form by factoring, if possible

- It can be changed into vertex form by completing the square

Reflection: Quadratics and Test

Quadratics: Quadratics is a very challenging and engaging unit. I felt that the unit became easier and easier every sub-unit. Standard form was much easier because there is not much to it which makes it very simple. Factored and vertex form were a different ball game because it was a totally different thing we learned last year which was slope so changing from that to this was tough. The graphing in the start was also very confusing because you never knew which quadrant it would be in and you would always have to focus as to what numbers are negative and positive.


Test: My test that I have attached (standard form) wasn't done very well but my mistakes were very understandable and correctable. I did good on the knowledge part, scoring 14/16 but struggled on application and communication. I was very stumped on the communication part because I couldn't explain my reasoning for my answers and I completely overlooked one question.

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