Quadratic Relations

By: Ishani Sharma

Table of Contents

Quadratics In Vertex Form


4.1 Intro Into Quadratics
  • Introduction to Parabolas
  • Key Features of Quadratic Relations
  • Types of Equations: Vertex form
  • Types of Equations: Standard form
  • Types of Equations: Factored form
  • Ways to Represent a Quadratic Relation


4.2 Quadratic Functions

  • Investigating Vertex Form
  • Graphing in Vertex Form
  • Finding an Equation in Vertex Form Using a Graph


4.3 Transformations of Quadratics in Vertex Form

  • Transformations of Parabolas
  • Writing Equations from Given Transformations
  • Graphing Transformations of Quadratic Relations
  • Mapping Notation


4.4 Graphing Quadratics in Vertex Form

  • Graph using Step Pattern
  • Graph using Mapping Notation
  • Word problems for quadratics in vertex form
  • Finding x and y intercepts in vertex form


Mini Test #1


Quadratics in Factored Form


5.1 Multiplying Binomials


5.2 Special Products


5.3 Common Factoring


5.4 Factoring Simple Trinomials


5.5 Factoring Complex Trinomials


6.2 Solving Quadratics by Factoring (finding the zeros)


6.3 Graphing Quadratics in Factored Form

  • Graph using x intercepts
  • Finding the Vertex
  • Word Problems for Quadratics in Factored Form


Mini Test #2


Quadratics in Standard Form


6.1 Maximum and Minimum Values (Completing the Square)

  • How to go from Standard Form to Vertex Form


6.4 The Quadratic Formula

  • Finding the x intercepts when factoring is not possible


6.5 Word Problems in Standard Form


Mini Test #3


Reflection


Useful Links

  • Khan Academy
  • Math Homework Help

4.1 Intro into Quadratics

Introduction to Parabolas

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About Parabolas....

  • Parabolas can open up/down
  • The zero is when the parabola crosses the x-axis
  • Zeros can also be called x-intercepts or roots
  • The axis of symmetry divides the parabola in half equally
  • The vertex of a parabola is where the parabola and axis of symmetry meet
  • The vertex is where the parabola is at its maximum or minimum value
  • The optimal value is the value of the y co-ordinate of the vertex
  • The y-intercept is where the graph crosses the y-axis

Key Features of Quadratic Relations

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Ways to Represent a Quadratic Relation

1) Table of Values

A table of values is used to find the first and second differences, When the first differences are the same it is a linear relation and when the second differences are the same it is a quadratics reaction. In order to find the first differences you must subtract the terms in the y-value from each other, and to find the second differences you must subtract the first difference terms from each other.

2) Graphs

When graphing or looking at graphs the line unlike a linear relation must be curved.

Types of Equations

4.2 Quadratic Functions

Investigating Vertex Form

The Vertex Form y= a(x-h)^2 + k gives us the the value of the axis of symmetry and optimal value. The value of "h" gives us the axis of symmetry. The value of "k" gives us the optimal value, and together "h" and "k" gives us the vertex of a parabola. The vertex form also gives away, the vertical stretch or compression by a factor of "a".

Graphing in Vertex Form

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Finding an Equation in Vertex Form Using a Graph

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4.3 Quadratic Functions

Transformations of Quadratics in Vertex Form

Consider a parabola:

y = 3(x-1)^2+2

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


Translations:

The parabola will open up , it is vertically compressed by a factor of 3. It is horizontally translated to the right by 1 unit and vertically translated up by 2 units.


Vertex Form:

  • The "a" determines weather it is vertically compressed or stretched (narrow or wide). If the "a" value is greater than 1, the parabola will be vertically stretched and if the value of "a" is less than 1 the parabola will be vertically compressed.
  • The "h" value will gives us the horizontal translation of right or left.
  • The "k" value will give us the vertical translation of up or down.
  • The sign in front of the "a" value will determine if the parabola open ups or down, If the sign is a positive it will open up, and if the sign is a negative, it will open down.
  • Remember that the "h" value will change signs when brought out from the bracket, if it is a negative in the bracket it will become a positive outside the bracket, therefore will move to the right, and vice-versa.
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Writing Equations from Given Transformations

If given the transformation of a parabola, for example:

The graph of y = x^2 is opening downward, stretched vertically by a factor of 3, translated horizontally to the left by 4 units and translated vertically down by 5 units.


The equation would become:

y = -3(x+4)^2 - 5

Graphing Transformations of Quadratics Relations

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Mapping Notation

Mapping notation is used to accurately graph any quadratic relation.


In general, to go from the graph of y = x^2 to y = a(x-h)^2 + k you can use the mapping notation:

(x,y) = x + h, ay + 1


Example:

a) y =5(x-4)^2 + 2 will become x + 4, 5y + 2

b) y =(x+7)^2 will become x - 7, y

c) y =(x-3)^2 + 5 will become x + 3, y +5

4.4 Graphing Quadratics in Vertex Form

Graph using Step Pattern and Mapping Notation

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Untitled

Word Problems for Quadratics in Vertex Form

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Finding x and y intercepts in Vertex Form

November 30, 2015

Mini Test #1

5.1 Multiplying Binomials

Simplifying polynomials includes both collecting like terms and use of the distributive property. It is important to be sure to use BED MAS at all times. Exponent rules must be used for both multiplication and division when simplifying.


When multiplying binomials there are two rules you must follow

  1. Distributive Property
  2. Collect like terms


Example 1:

(x+3)(x+4)

  1. We will use the distributive property to multiply the first two terms
  2. Then we will multiply the two inner terms with the outer terms and add the outcome, in this case the outer term is 1 so you would simply add 3 and 4
  3. Then we will multiply the two inner terms with each other

x^2 + 4x + 3x + 12

  1. Lastly we will add the like terms together

x^2 + 7x + 12


Reminder - Multiply coefficient + add power

2x^2 (2x)

4x^3


Example 2:

2x(3x-4) + (x+3) - (2x-7)

(6x^2-8x) + (x+3) - (2x-7)

(6x^2-7x+3) - (2x-7)

6x^2 - 7x - 2x + 3 + 7

6x^2 - 9x + 10

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5.2 Special Products

The formula for factoring a special product is a^2 + 2ab + b^2 = (a+b)^2 and (a-b)^2


Example:

(3y+7x) = a^2 + 2ab + b^2

(3y)^2 + 2(3y)(7x) + (7x)^2

9y^2 + 42xy^2 + 49x^2


5.3 Common Factoring

Factoring is the opposite of expanding, the three methods of factoring are:

  1. Finding the GCF
  2. Common Factoring
  3. Grouping


1. When factoring by GCF, you simply find the greatest common factor of the terms , for example the terms 5c + 10d both have 5 in common, so you divide both terms by 5. You always keep the common factor outside of the bracket and place everything else inside of the bracket, SO 5(c+2d).


2. Common Factoring is when the two binomials are exactly the same, for example

5x(3x+2) + 4(3x+2), since the two binomials are the same you will put them as one bracket and place the remaining into another bracket so, (3x+2)(5x+4).


3. When grouping factors you group a set of two like terms into brackets

  • ax + ay + 2x + 2y
  • (ax+ay) + (2x+2y)

Then you will remove the like term to the outside of the bracket

  • a(x+y) + 2(x+y)

Now since you can see that there is a common binomial you follow the rules for Common factoring

  • (a+2)(x+y)

5.4 Factoring Simple Trinomials

The equation used for factoring simple trinomials is ax^2 + bx + c


Where x is a variable and "a", "b", "c" are constants.


You can factor a quadratic in standard form to get factored form:

x^2 + bx + c = (x+r)(x-s)

STD FORM Factored Form


Where "r+s" is" b" and " rs" is "c"


Step 1: Find the product and sum

  • find the two numbers whose product is "c"
  • find the two numbers whose sum is "b"


Example 1: x^2 + 6x + 5

  1. Find the two numbers whose product when multiplied is 5 (1)(5) = 5
  2. Find the two numbers when added together equals to 6 5+1 = 6
  3. The factored form will be (x+5)(x+6)


Look at the signs of "b" and "c" to make it easier to figure out the sign of the two numbers

  • if "b" and "c" are positive, both "r" and "s"
  • if "b" is negative and "c" is positive , both "r" and "s" are negative
  • if "c" is negative , only ONE of "r" or "s" is negative
  • if both "c" and "b" are negative, only ONE of "r" or "s" is negative


Example 2:

n^2 + 6n + 8

(4)(2) = 8

4 + 2 = 6

(n+4)(n+2)

5.5 Factoring Complex Trinomials

Remember we are trying to break down the middle term so that it equates to 4 terms at which we can proceed to factor by grouping.

  1. Always look for a common factor
  2. To factor ax^2 + bx + c, find the two integers whose product is "ac" and whose sum is "b", similar to simple trinomial factoring


Example 1:

3x^2 + 8x + 4


(3)(4) = 12

(2)(6) = 12

2 + 6 = 8


Break up the middle term

(3x+6x) + (2x+4)


Factor by Grouping

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


Collect the binomials

(3x+2)(x+2)


You follow the same steps when factoring a trinomial with two variables.

6.2 Solving Quadratics in Factored Form

When solving quadratics in factored form or better known as finding the zeros you must follow the following steps:


  1. Make one side equal to zero
  2. Set each bracket equal to zero
  3. Solve for "x"


Example 1:

x^2 + 9x + 14 = 0

x^2 + 7x + 2x + 14 = 0

x(x+7) + 2(x+7) = 0

(x+2)(x+7) = 0

x+2=0 x+7= 0

x= -2 x= -7

(-2,0) (-7,0)

6.3 Graphing Quadratics in Factored Form

Using the example from above, you can than proceed find the axis of symmetry by adding the two x intercepts together and dividing by two

(-2)+(-7) / 2

= -4

The -4 is the "x" value


The AOS can used to find the optimal value (y intercept)

y = (x+7)(x+1)

y = (-4+7)(-4+1)

y = (3)(-3)

y = -9

This is the "y" value


Now since you have the "x" and "y" values you can put them together to make the vertex, so the vertex in this case would be (-4,-9).


At the end of this you have 4 points:

  1. (0,7)
  2. (-7,0)
  3. (-1,0)
  4. (-4,-9)

You can use the points above to plot your graph

Mini Test #2

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6.1 Maximum and Minimum Values

Since not all quadratic functions are written in Vertex Form, when functions are written in standard form you can covert them into vertex form using a process called completing the square.


Completing the Square Steps:

  1. Group the x terms
  2. Divide the coefficient of the middle term by 2, square it, then add and subtract that number inside the brackets
  3. remove the subtracted term from the brackets
  4. factor the brackets as a perfect square trinomial


Example 1:

y = x^2 - 6x + 4

y = (x^2 -6x) + 4

y = (x^2 - 6x + 9 - 9) + 4

y= (x^2 - 6x + 9) - 9 + 4

y = (x-3)^2 -5


Vertex = (3,-5)

AOS = 3

Max or Min value = min -5

Values that "x" may take = all real numbers

Values that "y" may take = y greater than -5


When there is a number in front of the "a" value you must first divide the middle term by that number and place then proceed with the steps. You must also multiply the square outside of the brackets.

When there is a negative in front of the "a" value you must multiply the middle term by the negative, if it is already a negative it will become a positive and if it is a positive it will become a negative. You must also multiply the square outside of the bracket by the negative.

6.4 the Quadratic Formula

The quadratic can be useful when a quadratic function cannot be factored.

The discriminant

All quadratic equations of the from ax^2 + bx + c can be solved using the quadratic equation. But you cannot figure out how many solutions a quadratic equation has.


In order to figure that out you must find the discriminant. The discriminant is the value under the square root in the quadratic formula.


The formula to find the discriminant is D = b^2 - 4ac


When the equation gives you a positive number or a number greater than 0, there will be 2 solutions (x intercepts).

When the equation gives you a negative number or a number less than 0, there will no solutions (x intercepts).

When the equation gives a number equal to 0, there will be 1 solution (x intercepts).

6.5 Word Problems in Standard Form

November 30, 2015

Mini Test #3

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Reflection

This unit was quite interesting for me. Although many students question why we must learn things that seem useless at the time, learning about parabolas has been a real eye opener. Before this unit i was living ignorant to the vast majority of things in the world around me, i just recently noticed that the M in the McDonalds sign uses two parabolas or that the curves in your windows are parabolas, now many can say that's useless information but its amazing how everything in the world is connected, you cannot learn one subject and not expect there to be bits of another tied into it. For example in math, communication counts for a part of your overall mark and communication is basically English. So similar to that math is related to the world, obviously in a bigger picture and when you learn something like quadratics you should always except to encounter it at some point in your life and i am glad that i have the opportunity to get an education and learn about such information. As for the unit itself, i must say it was a struggle and required long nights of studying and preparation but in the end i am satisfied with how much effort i put in and i have learnt a very important life lesson from the hard work i put into this unit, which is that nothing in life comes easy, however perseverance will lead can lead you to victory. Although i did not achieve the nest marks i believe i tried and never gave up which for me means that i am winner.

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