Quadratic Relations

By Anudit

Topics

1. MATH TERMINOLOGY


2. Intoduction
  • What is Quadratics & Why is it important
  • Introduction to parabolas
3. Vertex Form
  • Axis of Symmetry
  • Optimal Values
  • Transformations
  • Step Patterns
  • Graphing
4. Factored Form
  • X-Intercepts or Zeroes
  • Axis of Symmetry
  • Optimal Value
  • Word Problems
  • Graphing
5. Standard Form
  • Quadratic Formulas
  • Word Problems

6. Reflection

- Test

- Reflection Paragraph

Math Terminology Overview

AXIS OF SYMMETRY

A line that divides a figure into two congruent parts


BINOMIAL

A polynomial that has two terms 3x+4 is a binomial


COEFFICIENT

The number by which a variable is multiplied. Ex. in the term 8y, the coefficient is 8


COMPLETING THE SQUARE

A process for expressing y=ax^2+bx+c in the form y=a(x-h)^2+k


DIFFERENCE OS SQUARES

An expression of the form a^2-b^2 that involves the subtraction of two squares


DISTRIBUTIVE PROPERTY

a(x+y)=ax+ay


EXPAND

Multiply, often using the distributive property


FACTOR

A number and /or variable that divides evenly into a specific product


FIRST DIFFERENCES

Differences between consecutive y-values in a table of values with evenly spaced x-values


GCF

The greatest number and/or variable that is a factor of two or more numbers or terms


INDEPENDENT VARIABLE

In a relation, the variable that you need to know first. its value determines the value of the dependant variable. On the coordinate grid, the values of the dependant variable are the horizontal axis---in d=85t, t is the independent variable


INTERCEPT

The distance from the origin of the xy-plane to the point at which a line or curve crosses a given axis


LINE OF SYMMETRY

A line that divides a shape into two congruent shapes that are reflections of each other in the line


LINEAR EQUATION

An equation that relates two variables so that ordered pairs satisfying the equation form a straight-line pattern on a graph


LINEAR RELATION

A relation between two variables that appears as a straight line when graphed on the coordinate plane


MAXIMUM POINT

the point on the graph of a non-linear relation, such as a parabola, at which the curve changes from increasing to decreasing


MINIMUM POINT

The point on the graph of a non-linear relation, such as a parabola, at which the curve changes from decreasing to increasing


NON-LINEAR RELATION

A relationship between two variables that does not follow a straight line when graphed


PARABOLA

The graph of a quadratic rotation, which is U-shaped ans symmetrical


PERFECT SQUARE

A number this can be expressed as the product of two identical factors---36 is a perfect square, because 36=6x6


PERFECT SQUARE TRINOMIAL

A trinomial of the form a^2+2ab+b^2 or a^2-2ab+b^2 that is the result of squaring a binomial


POLYNOMIAL

An algebraic expression formed by adding or subtracting terms


QUADRATIC EQUATION

An equation in the form ax^2+bx+c=0, where a,b and c are real numbers


QUADRATIC EXPRESSION

A second-degree polynomial---33x^2+5x-1 is a quadratic expression


QUADRATIC FORMULA

A formula for determining the roots of a quadratic equation of the form ax^2+bx+c=0


QUADRATIC RELATION

A relation whose equation is in the form y=ax^2+bx+c,where a,b and c are real numbers


VERTEX(OF A PARABOLA)

The point on the parabola where the curve changes direction. It is a maximum point is the parabola opens downward and minimum point if the parabola opens upward

1. INTRODUCTION

What is Quadratics & Why is it important?

Quadratics originates from the word "quad" meaning square like (x²) in quadratics the first term is squared making it quadratic. They are often used to graph flight paths of different objects travelling in the shape of an arc. For example, the flight path of an arrow, or a fireball. In order to determine if a relationship is quadratic or not, you can simply look at a table of value; if the first differences are unequal, but the second differences are equal, it is a quadratic relation.

Parabolas

Parabola's are everywhere, you just never noticed it before. They are in places such as the arch of the letter "U" and the three point line on a basketball court. A parabola is a curve where any point is at an equal distance from a fixed point and a fixed straight line. For example, When you shoot a basketball, the ball arcs up and comes down again which creates the shape of a parabola. Some important features include:

2. VERTEX FORM

What is it?

Vertex form is normally expressed as y=a(x-h)²+k. If the a value is negative, the parabola will open downwards; if the a value is positive, the parabola will open upwards. If the a value is less than 1, the parabola will be compressed, meaning the graph of the parabola will widen. However, if the a value is greater than 1, the parabola will be stretched, meaning the graph of the parabola will become more narrow.

Vertex Form

This is the equation used for vertex form with a simple parabola demonstrating how the a value affects the direction opening.


a(x-h)²+k

Axis of Symmetry

The dotted line in this diagram represent the axis of symmetry. It shows what the x value of the vertex is by passing through the point.


AOS

Optimal Value

The green arrow points out the maximum value in this parabola, & the yellow arrow is pointing at the lowest value in the parabola. These two points are called the "Optimal Value" . The optimal value is the highest or lowest point on the parabola.


VERTEX

Transformations

In the vertex form equation y= a(x-h)^2+k there are four different transformations.

Each variable in the equation is in control of a transformation.

If "a" is negative the parabola will have a vertical reflection, the parabola will open downward

  • "a" determines whether the parabola will have a vertical stretch or compression and the direction of opening
  • "h" represents the horizontal translation
  • "k" represents the vertical translation
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Step Pattern

This is a way to plot points from a parabola on a graph. The following video demonstrates the process.
Graphing Quadratics Using Step Patterns

Types of Equations

1. Factored form: a(x-r)(x-s)

2. Vertex form: a(x-h)²+k

3. Standard Form: ax²+bx+c

2. FACTORED FORM

Y=A(X-R)(X-S)

Finding the X-Intercepts

When you are given an equation in factored form, the "r" and "s" values are your X intercepts. Example:

y=(a+b)(r+s)


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

r=3 s=-1


Remember when you take a value out of a bracket the (+)(-) signs flip.

So your X intercepts are (3,-1)

Factored form to Standard form (Expanding)

FACTORED FORM TO STANDARD FORM

COMMON FACTORING

Common Factoring

Multiplying Binomials

MULTIPLYING BINOMINALS

FACTORING SIMPLE TRINOMIALS

https://youtu.be/EmbFarq3YnQ

FACTORING COMPLEX TRINOMIALS

A complex trinomial is where a coefficient more than 1 is placed in front of the 'a' value.

1. First, you have to multiply the 'a' value and the 'c' value together.

2. You have to find two numbers that are a product of that number and also is a sum of the 'b' value.

3. After that you have to remove the value and input our two new numbers

4. Finally, you group the numbers just like above and factor out the GCFs to end up with the answer


EXAMPLE:

PERFECT SQUARES AND DIFFERENCE OF SQUARES

Keep all terms containing x on one side. Move the constant to the right.

Get ready to create a perfect square on the left. Balance the equation.

Take half of the x-term coefficient and square it. Add this value to both sides.

Simplify and write the perfect square on the left.

Take the square root of both sides. Be sure to allow for both plus and minus.

Solve x.



EXAMPLE:


x^2-2x-1 = 0


x^2-2x = 1


x^2-2x+1 = 1+1


x^2-2x+1 = 2


(x-1)^2 = 2


SQRT(x-1)^2 = SQRT 2


x-1 = 1.4


x = 1-1.4 ------------- x = 1+1.4


x = -0.4 ------------ x = 2.4

FACTOR BY GROUPING


Step 1: Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.

Step 3: Factor out the GCF from each of the two groups. In the second group, you have a choice of factoring out a positive or negative number. To determine whether you should factor out a positive or negative number, you need to look at the signs before the second and fourth terms. If the two signs are the same (both positive or both negative) you need to factor out a positive number. If the two signs are different, you must factor out a negative number.

Step 4: If the factors inside of the parenthesis are exactly the same, it is time for the 2 for 1 special. The one thing that the two groups have in common should be what is in parenthesis, so you can factor out what is inside the parenthesis, but only write what is inside the parenthesis once. If what is inside the parenthesis does not match, you need to rearrange the four terms and try again until you get a perfect match. If you have rearranged the problems a couple of times and still have not found a perfect match, then the problem does not factor.

Step 5: Determine if the remaining factors can be factored any further.



Example: y=4x^2+8x+16x+32

Step 1: y=(4x^2+8x)(16x+32)

Step 2:: y=4x(x+4)+16(x+4)

Step 3: y=(4x+16)(x+4)

The Quadratic Formula

EX: -7x^2+2x+9
Quadratic Formula

Using the Quadratic Formula, you can find the two x intercepts of a parabola:


(-1,0) & (1.29,0)


With the two x intercepts, you can find the Axis of Symmetry and using the AOS formula: x= (x+x)/2

x= (-1+1.29)/2

x= 0.15


Now you can find the vertex of the parabola.


-7x^2+2x+9

= -7(0.15^2)+2(-1)+9

= -0.16+7

=6.84


therefore, your vertex = (0.15,6.84)

MORE EXAMPLES

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Graphing Factored Form

Factored form is represented as y= a(x-r)(x-s), where the x's represent the x-intercepts. Factored form is an equation that has been simplified. Where everything multiplies to each other.


Step one: Find the x intercepts. you must set y=0, therefore you will have to take the value of the x-intercepts and carry them to the other side of the equation to isolate x. This will give you the two zeroes.

Step two: Once you plot the two points on the graph, you must find the axis of symmetry. (x=x)/2 = AOS then you can create your axis of symmetry (h-value of the vertex).

Step three: now you must find the optimal value/vertex. It was stated earlier that you must substitute in your axis of symmetry to get this answer. After solving the equation, you will then get your optimal value (k-value of the vertex).

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


x= (-3,0) x= (2,0)


AOS = (-3+2)/2= -0.5


Vertex:

y=(-0.5+3)(-0.5-2)

y= -6.25


therefore your vertex = (-0.5,-6.25)

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ZEROES/X-Intercepts

Zeroes are also known as the x-intercepts of the equation. To figure out the zeroes, we let y to equal 0.


For example:

y= 0.5(x+3)(x-9) the equation will turn into 0= 0.5(x+3)(x-9).


Since everything is being multiplied to make zero, that means either (x+3)=0 or (x-9)=0.

So, it will be x= (-3) because -3+3=0 AND x=9 because 9-9=0


Notice we don't change the 0.5, that's because it is on it's ownand has it's own value.

Finding the Optimal Value/Vertex

VERTEX= (X+X)/2



EXAMPLE:

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


x= (-3,0) x= (2,0)


AOS = (-3+2)/2= -0.5


Vertex:

y=(-0.5+3)(-0.5-2)

y= -6.25


therefore your vertex = (-0.5,-6.25)

FInding the Axis of Symmetry


To find the axis of symmetry or (aos) in factored form, you want to find the zeroes/x-intercepts.


The formula you would want to use looks like: (a+b)/2 where 'a' and 'b' are the x-intercepts.


For example:

Your equation was y= (170-2x) x, you first find the x-intercepts

0= (170-2x)x, which means:

(170-2x)=0 (isolate x) 170=2x 170/2=x x=85

AND x=0 since there's a lone x.

Now with x-intercepts, use formula to find the axis of symmetry.

a+b/2

85+0/2

AOS: x= 42.5

Word Problems

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QUADRATICS MINI TEST 1

Page 4

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Quadratics Reflection

Quadratics is a very interesting a fun unit. Although I struggled at first, i went over my tests and perfected my mistakes. Math is a subject that I have always been struggling with, however, I realized that once you learn how to solve the problems, it can be very easy and fun. Quadratics was very difficult at first, but after coming in for help and doing extra homework, I learned to solve the quadratic equations very easily. In this unit, I learned that quadratics applies to real life situations. For example, Parabolas are literally everywhere, and I learned that there is a greater meaning behind all parabolas because of what I learned in Math class this unit. Also, quadratics can be used in business when a person is trying to find the revenue and many other careers. I learned many different ways to solve quadratics equations (Quadratic formula, Factoring by Grouping, Completing the Square, etc). Overall, this unit was very fun, and I enjoyed learning it with my phenomenal teacher and helpful peers. We all worked together to to help improve one another's marks, and I am really looking forward to continuing the course with them.