Quadratics

By: Sonali Thakkar

Table of Contents

Introduction:


  • Features
  • Parabolas
  • Ways to represent Parabola's
  • Vertex Form
  • Factored Form
  • Standard Form



Vertex form


  1. Investigating
  2. Graphing
  3. Transformations
  4. How to find a equation using a graph



Mini Test #1


Expanding and Factoring:


  • Multiplying Binomials
  • Special Products
  • Common Factoring
  • Factoring simple Trinomials
  • Factoring Complex Trinomials
  • Differences of square Trinomials
  • How to find a equation using a graph


Mini Test #2


  • Finding the Maxima and Minima
  • Graphing using x-Intercept
  • The Quadratic Formula


Test #3


Word Problems

Introduction to Quadratics

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Key Features of a Quadratic Realtion

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Description:

  • the opening of the parabola is up
  • the axis of symmetry is the x value of the vertex
  • the optimal value is the y value of the vertex
  • the zeros are when the parabola touches or goes through the x axis
  • y- intercept is when the the parabola goes through the y axis
  • the vertex is the minimum or maximum point of the parabola

Ways To Represent Parabolas

Table of Values

We have learned previously how to make a table of values with first differences for linear systems. Now, we use second differences to solve for Quadratic Relations.
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Graph

When making Quadratic Graph the line is curved and it is called a Parabola
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Equation

Some quadratic Functions use the equation y= ax^2+b^2+c. when you see a power square in your formula that indicates that this function is quadratic.
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3 Types of Equations

The three types of equations are:

  • Standard Form
  • Vertex Form
  • Factored Form

Standard Form

This is what a standard form equation looks like:
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Vertex Form

This what a vertex form equation looks like:


where h and k represent the vertex. h represents the x value of the vertex. If h is positive in the equation, it is negative on the graph. If h is negative in the equation then, it is positive on the graph.


k represents the y value of the vertex. The value of k does not have any special rules, as you see it on the equation is where you place it on the graph.

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Factored Form

The final equation used in quadratics is factored form.
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Vertex Form

Investigation

The vertex form equation y= a(x-h)+k, where h and k represent represent the vertex, where h represents the axis of symmetry, and k represents the optimal value. a represents the vertical/ horizontal stretch. If the value of a is greater than 1 it means you parabola has a vertical stretch. If the value of a is less than 1 the parabola has a vertical compression. Also, if the value of a is negative it means your parabola's direction of opening is down, and if you parabola is positive, that means your parabola's direction of opening is up.

How to Graph with Vertex Form

Parabola vertex form

Basic Transformations

There are 4 transformations you need to know. and its always easy to start from the beginning of the equation and then move on.

The four transformation are:

  • if your parabola has an opening up or down
  • if your parabola is vertically stretches or compressed
  • how many units is it transforming left or right
  • how many units it is transforming up or down


For example:

y= -2(x-4)-5


The transformations for that equation would be:

  • it has opening down
  • the parabola has a vertical stretch of 2
  • the parabola if transformed 4 units to the right
  • the parabola is transformed 5 units down

How to Find a Equation Using a Graph

Parabola Graph

Unit Test 1

Expanding and Factoring

Expanding Binomials

You can turn a set of binomials into standard form by expanding.


For example, (3x+2)(3x-3)

To expand this binomial, you have to:


  • First, you multiply the outer term with inner term. so you would multiply (3x)(3x)= 9x^2
  • Second, you multiply the outer term with the inner term. (3x)(-3)= -9x
  • Third, you multiply the two inner terms. (2)(3x)= 6x
  • Fourth, you multiply the inner term with the outer term. (2)(-3)= -6
  • Fifth, you group the like terms.
9x^2-9x+6x-6

9x^2-3x-6


  • Finally, you have you standard form equation 9x^2-3x-6
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Special Products

The formula for special products is a^2+2ab+b^2.


(x+4)^2


x is the value a and 4 is the value of b, so now you plug that into the formula:


x^2+ 2(x)(4)+ 4^2

x^2+8x+16

Common Factoring

There are three different methods of factoring:

1) Monomial Common Factor

2) Binomial Common Factor

3) Factor by Grouping


Monomial Common Factor:

i) Find the GCF of coefficients and variables

ii) Divide each term by GCF


Examples:

5c+10d


  • the GCF is 5 therefore you divide both terms by 5.
  • you would rewrite that as


5(c+2d)


8x^2-7x


  • the GCF is x therefore you divide both terms by x.
  • you would rewrite this as
x(8x-7).



Binomial Common Factor:

i) if there are two binomials that are exactly same, consider that as a binomial common factor.


Examples:



  • x(x-2)+2(x-2)
  • x-2 are the same therefore they are the common factor and you would rewrite it as


(x-2)(x+2)


  • the common binomial goes first and what is left comes after.



Factor by Grouping:

i) factor groups of two terms with a common factor to produce a binomial common factor.


Example:



  • ax+ay+2x+2y -ax and ay have the common factor a and 2x and 2y have the common factor 2.
  • (ax+ay)+(2x+2y) - then you add brackets around common terms.
  • a(x+y)+2(x+y) - then you take the common factor outside the brackets. At this you know you are correct if both of the terms inside the brackets are the same.
  • then you set them into a binomial common factor
(x+y)(a+2)
Example 1: Factoring trinomials with a common factor | Algebra II | Khan Academy

Simple Trinomial Factoring

Given a quadratic equation in standard form, you can factor to get factored form.


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


There are 3 steps to factor a simple trinomial.


Step 1: Find the product and sum



  • Find two numbers whose product is c
  • Find two numbers whose sum is b


For example,


x^2+12x+27


  • two numbers whose product is 27 and also have the sum 12 are 3 and 9.


Step 2: Look at the signs of b and c in the given expression


  • If b and c are positive, both r and s are positive.
for example: x^2+7x+12


Both b and c are positive therefore, r and s are positive and the answer is

(x+4)(x+3)



  • If b is negative and c is positive, both r and s are negative.
For example: x^2-29x+28


b is negative and c is positive therefore, both r and s are negative.

(x-28)(x-1)


Step 3:


  • If c is negative, one of r or s is negative.


For example:

x^2+3X-18

c is negative therefore either one of r or s is negative.

(x-3)(x+6)

Factoring Complex Trinomials

Step 1: Always look at the common factor first when factoring trinomial


Step 2: To factor ax^2+bx+c, find two integers whose product is ac and whose sum is b.


Step 3: The check up the middle term and factor by grouping


Example: 3x^2 +8x+4


  • Two integers whose product is (3)(4) and sum is 8
2,6



  • Break up the middle term
(3x^2+2x)+(6x+4)



  • Factor by grouping
x(3x+2)+2(3x+2)


(3x+2)(x+2)

Factoring Complex Trinomials

Differences of Squares

The formula for Differences of Squares is a^2-b^2= (a+b) (a-b)


Example: x^2-16

x^2-4^2

(x+4)(x-4)

Unit Test 2

Finding the Maxima and Minima

To find the maximum or minimum you would use the method completing the squares.

Completing the squares turns a standard form equation (x^2+bx+c) into a vertex form equation (y=a(x+k)^2-h).


Step one: you add brackets around your a and b

Step 2: divide your b value by 2 and square your answer

Step 3: put a positive square in the bracket and a negative square outside the bracket

Step 4: you now have a perfect square trinomial in the brackets, so we turn that into a binomial.

Step 5: solve the values outside the bracket



For example:


y=x^2+8x+3

y=(x^2+8x)+3

y=(x^2+8x+16)-16+3

y=(x+4)^2-16+3

y=(x+4)^2-13


Therefore, the vertex of this equation is (4,-13).

Graphing Using the x-intercepts

When graphing using the x-intercepts you use the method of factoring to find your values of x. then you use your values of x to find your vertex by first finding your axis of symmetry and then finding your optimal value.


y=x^2-6x+8

-4,-2

y=x^2-4x-2x+8

y=(x^2-4x)(-2x+8)

y=x(x-4)-2(x-4)

y= (x-4)(x-2)

x-4=0 x-2=0

x=4 x=2

(4,0) (2,0)

Therefore, the x-intercepts are (4,0) and (2,0)


4+2/2=6/2=3

the AOS is 3. Now you plug in your x value.


y= (x-4)(x-2)

y= (3-4)(3-2)

y=(-1)(1)


y=-1

Therefore the vertex is (3,-1)


Now we can graph this because we know our x-intercepts, our vertex, and the y-intercept. The y-intercept is the value of c in the original equation.

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The Quadratic Formula

Using the quadratic formula will always give you the exact x-intercepts.


  • When given your standard form equation you determine you a value, your b value and c value so there is no confusion.
  • You then plug in all of you variables.
  • First, you square your b and multiply your -4ac
  • Second, multiply your 2a
  • Third, solve inside the square root
  • Fourth, square root that answer
  • Now the equation goes two ways; one you add your -b with your square root divided by your 2a; two you subtract your -b with your square root and divide it with your 2a
  • The two answers you get are your x-intercepts.

Revenue Word Problems

Bryce is selling T-shirts. His regular selling price is $18.50 per T-shirt and he usually sells 216 T-shirts. Bryce finds that for each $0.50 increase in price he will sell 8 less T-shirts.


a) Create an algebraic equation to model Bryce's total sales revenue


Let (18.50+0.50x) represent the cost

Let (216-8x) represent the total number of people


R= (18.50+0.50x)(216-8x)


b) Determine the maximum revenue and the price at which this maximum revenue will occur


R= (18.50+0.50x)(216-8x)

18.50+0.50x=0 216-8x=0

0.50x= -18.50/0.50 -8x= -216/-8

x= -37 x= 27


-37+27/2= -10/2 = -5


R= (18.50+0.50x)(216-8x)

R= (18.50+0.50(-5))(216-8(-5))

R=(18.50-2.5)(216+40)

R=(16)(256)

R= 4096


Therefore the maximum Revenue $4096 will occur at the maximum price of $16

Triangle Word Problems

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Consecutive Integers Word Problem

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Reflection

The unit of quadratics has been a very interesting unit. I had no prier knowledge of quadratics so this was a completely new learning experience. It was hard at some points but I never gave up. This might not help us in the future, but quadratics are all around us. When we play sports for example, basket ball. the arch between your throw and the basket is a parabola. or maybe a bridge we drove through is a parabola. this has been a very fun and challenging experience but it was all worth it because i got to learn something new.

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