Mission Quadratics

Sanmeet Sandher

Table of Contents

- What is Quadratics?

- What is a Parabola?


  1. Vertex Form
  • Learning Goals
  • Summary of Unit
  • Graphing

- First and Second Differences

- X and Y value chart

- Mapping Notation

  • How to solve Word Problems
- Finding "a" when the vertex and point are given
2. Factored Form
  • Learning Goals
  • Summary of Unit
  • How to factor an equation
  • Expanding and simplifying
  • Types of Factoring
-Monomial Factoring

- Binomial Factoring

- Factoring by grouping

- Simple Trinomial factoring

- Complex Trinomal factoring

- Special Product: Difference of squares

- Special Product: Perfect square trinomial


  • Graphing the parabola
  • Video: Word problem

3. Standard Form


  • Learning goals
  • Summary of the unit
  • Completing the square
  • Quadratic Formula
  • Find the x-intercept and vertex form
  • Graph
  • Word Problem

How are the 3 forms connected?

What is Quadratics?

In quadratics, there are 3 forms of equation: vertex,factored, and standard

- Vertex: y=a(x-h)²+k

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

- Standard: y=ax²+bx+c

All these equations can create a parabola to make on a graph

Parabola

What is a Parabola?


  • The graph of a quadratic relation is called the parabola.
  • Parabola can open up or down
  • The zero of the parabola is where the graph crosses the x-axis
  • "zeros" can also be called x-intercepts or roots
  • The axis of symmetry divides the parabola into 2 equal halves
  • The vertex of the parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value
  • The y-intercept of a parabola is where the graph crosses the y-axis

Vertex Form

Learning Goals


  1. Will understand the first and second differences
  2. You will learn how to solve the word problems
  3. You will learn how to find "a" when the vertex and the point are given

Summary of the unit


  • To solve word problems/problems you have to use the vertex form which is y=a(x-h)²+k
  • y=a(x-h)²+k --> "h" is for the horizontal translation
  • y=a(x-h)²+k --> "k" is for the vertical translation
  • y=a(x-h)²+k --> "a" tells you the direction of opening and compression or stretch (Scroll down to Image 1 for an example)
  • The vertex of (h,k): Axis of symmetry (x=h) and optimal value (y=k)
  • Optimal Value: The minimum/maximum value of the objective function
  • Step pattern: X and Y value chart (Scroll down to Image 2 for an example)
  • When you are told to find the y-intercept, set x=0 and solve for "y"
  • -When you are solving for "x", set y=0

Graphing

First and Second Differences (Linear, Quadratic or Neither)


If you are given a table like the following, first make sure that the x is increasing by the same value, in this case it is increasing by one. When finding the first difference, it is easier if you subtract going up so 7-6, 8-7, and so on. If the first difference is the same you don't need to find a second difference because now we know that our relation is linear, if our first difference wasn't the same but our second difference was, than our relation would've been quadratic, if the first and second difference are totally different than our relation was neither linear or quadratic.




X and Y chart (step pattern)

https://www.youtube.com/watch?v=SFc4F0iUF3U


Map Notation

https://www.youtube.com/watch?v=DQtW4wTF72g

How to solve a Word Problem

Questions you are usually asked:


What is the maximum height of the ball?

- To solve this you have to find the vertex, the y-value is the maximum height, so in the equation y=a(x-h)²+k, y=k


At what time did the ball reach the maximum height?

- To solve this you have to find the vertex and the x-value is the time when the max height occurred, so in the equation y=a(x-h)²+k, x=h


What was the initial height of the ball?

- To solve this you have to set x=0 and solve for y (the height of the ball when time=0) or you can simply find the y-intercept


How long was the ball in the air?

- To solve this you have to set y=0 and solve for x (the time when the ball hit the ground, when height=0)


When did the ball hit the ground?

- To know when the ball hit the ground, you have to see when the height was 0 and when the height is 0 that means that was when the ball hit the ground


What is the height of the ball at 3 seconds?


- To solve this you have to sub the x-value into the equation y=a(x-h)²+k, so you can solve for y



How to find "a" when the vertex and point are given

Ex: Find the Vertex Form of a Quadratic Function Given the Vertex and a Point

Factored Form

Learning Goals

1. You will learn how to factor an equation

2. You will learn about the types of factoring

3. You will learn how to graph a parabola with a given equation

4. You will know how to solve a word problem using factored form

Summary of the unit

  • y=a(x-r)(x-s)
  • The value of 'a' = the shape and direction of opening of parabola
  • If a is negative than the parabola is going downwards and if it is positive than the parabola is going upwards
  • The value of 'r' and 's' = the x-intercepts
  • axis of symmetry: x= (r+s) and than divide it by 2. If you sub the x value into equation you will find the optimal value
  • to find the y-intercept, set x=0 and than solve to find y
  • Monomial Factoring: Finding the greatest common factor(GCF) of the coefficients and variables. Dividing each term by the GCF
  • Binomial Factoring: If binomials are same than you will be considering that as a binomial common factor
  • Factoring by grouping (4 terms): factor groups of two terms with a common facotr to produce a binomial common factor
  • Simple trinomial factoring: ax²+bx+c , where x is a variable and a,b,c are the constants, a is not 0. A simple Trinomial is where a=1
  • Complex trinomial factoring: ax²+bx+c, where a is does not equal to 1
  • Special Product - Difference of Squares: product of sum and difference. When two terms are being subtracted and both of them are perfect squares. ex: a²-b²=(a+b)(a-b)
  • Special Product - Perfect Square Trinomial: When the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last term

How to factor an equation?

  • Factoring a polynomial is the opposite of expanding a polynomial
  • Expanding is multiplying to remove the brackets
  • Factoring is finding common multiples to create a bracket
  • Finding the GCF (greatest common factor) is the first method to factor a polynomial

expanding ⇢

x(x+3)=x²+3x

⇠ factoring

Expanding and Simplifying

Types of Factoring

Monomial Factoring

  • Steps
  1. Find GCF of coefficients and variables
  2. Divide each term by GCF
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Binomial Factoring

  • If there are two binomials that are exactly the same, than consider that as a binomial common factor
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Factor by Grouping

  • Factor groups are two terms with a common factor to produce a binomial common factor
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Simple Trinomial Factoring

  • ax²+bx+c, where x is a variable and a,b,c are constants, where a does not equal 0. A simple trinomial is where a=1
  • When given a quadratic in standard form, you can factor it to get factored form
  • Standard form: x²+bx+c
  • Factored form: (x+r)(x+s), where 'r+s' is 'b', and 'rs' is 'c', and 'r' and 's' are integers
  • Steps:
1. Find the "product and sum

- Find two numbers whose product is 'c'

- Find two numbers whose sum is 'b'

* The two numbers found for product and sum have to be the same *

Example: x²+12x+27

- Find two numbers whose product is 27

9 x 3=27

- Find two number whose sum is 12

9 + 3=12

x²+bx+c = (x+r) (x+s)

x²+12x+27 = (x+9) (x+3)


2. Look at the signs of 'b' and 'c' in the given expression x²+bx+c

- If 'b' and 'c' are positive, both 'r' and 's' are positive

Ex: x²+7x+12 --> (x+4) (x+3) {4,3 are the numbers the give the product of 12 and sum of 7}

- If 'b' is negative and 'c' is positive, both 'r' and 's' are negative

Ex: x²+29x+28 --> (x-28) (x-1)

- If 'c' is negative, one of 'r' or 's' is negative

Ex: x²+3x-18 --> (x-3) (x+6)

- If 'b' and 'c' both are negative, one of 'r' or 's' is negative

Ex: x²-5x-24 --> (x-8) (x+3)

Complex Trinomial Factoring

  • Factoring quadratics of the from ax²+bx+c when a does not equal 1
  1. Always look at the common factor first when factoring a trinomial
  2. To factor ax²+bx+c, find two integers whose product is 'ac' and whose sum is 'b'
  3. Then check the middle term and factor by grouping

* Not all quadratic expression can be factored *


Ex: 3x²+14x+8

- To decompose or break up the x term, find 2 numbers whose product is 'ac' and whose sum is 'b'

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Difference of Squares

  • factoring a difference of squares: a²-b²=(a+b) (a-b)
  • Product of sum and difference
  • Two terms are being subtracted and both of them are perfect squares

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Perfect Square Trinomial


  • In a perfect square trinomial, the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms
  • Check the middle term. It should be twice the product of the first and second terms
  • (a+b)² = a²+2ab+b² OR (a-b)² = a²-2ab-b²
  • Factoring: a²+2ab+b² = (a+b)² OR a²-2ab+b² = (a-b)²
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Graphing

  1. Graph the following parabola y=x²-10x+24
Solving Word Problems Using Factored Form

Standard Form

Learning Goals

1. You will learn how to complete the square and find the vertex

2. You will learn how to use the quadratic formula

3. You will learn how to solve word problems using the quadratic formula

Summary of the unit

  • Equation: y=ax²+bx+c
  • For a quadratic equation in the form y=ax²+bx+c=0, where a does not equal to 0 use the quadratic formula
  • Quadratic Formula: -b+√b²-4ac/2a
  • The quadratic formula is used when a certain quadratic cannot be factored. This formula was developed by completing the square and solving the quadratic ax²+bx+c=0
  • The value of a gives you the shape and direction of opening
  • The value of c is the y-intercept
  • Solve using the quadratic formula to get the x-intercepts
  • Complete the square to get the vertex, MAX or MIN?
  • Vertex form: y=a(x-h)²+k
  • Squaring a binomal =perfect square trinomial: (a+b)² = a²+2ab+b²
  • All quadratic equations of the form ax²+bx+c=0, can be solved using the quadratic formula
  • Using the discriminant: - If d<0 than there is no x-intercept

- If d>0 than there are 2 x-intercepts

- If d=0 than there is only 1 x-intercept

Completing the Square

This technique is best used for equations of the form x²+bx+c=0,


(x+2)²=x²+4x+4

(x-3)²=x²-6x+9


  • How to make perfect square if it's not given:

We look at a few perfect square situations with blanks. Ex: (x+_)²+10x+25

- In order to make this (x+5)², the last blank has to be 25. However, we need to understand the relationship between the 10 and the 25. How can we mathematically manipulate the 10 to make it equal to 25


  • 2 Steps:
  1. Divide 10 by 2. The result is 5
  2. Square the 5. 5² is 25

Another Example: x²+8x+__=(x+__)²

-Take the half of 8 to get 4. Square the 4 to get 16

x²+8x+16=(x+4)² (4)²=16

*The number inside the bracket, the 4 in this case, is always half of the middle term of the trinomial, which is the 8.*

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  • In the example above, using the completing the square formula, the vertex was found! how? 5 Steps:


  1. Putting a bracket around the first 2 term (x²,8x)
  2. Finding the half of the middle term, 8, and squaring the half, 4, to get 16
  3. Add and Subtract the value inside the bracket
  4. Take the last digit inside the bracket outside the bracket, so in this case, -16
  5. Write the equation as squaring a binomial and solve to get the number outside the brackets, -16+5=-11


  • How did I get (x+4)²-11

- 4 is the half of 8 and when it is squared it gives us 16.

- (-11) when -16 was taken outside the bracket and added with 5 the answer of those two numbers is -11

*y=ax²+bx+c*




MAX OR MIN??

The vertex of that equation is (-4,-11), you take the half and the number outside you bracket, which also are your b and c, if your b is positive inside the bracket it will negative once it is out and the number outside the bracket will stay they same. -11 is the y-intercept


  • Since both of the numbers are negative, that means the parabola will be opening upwards (MIN)
  • If the numbers are positive that means it is MAX, which means the paradola will be opening downwards

Quadratic Formula

This method always gives exact solutions, but in some cases the other methods are easier to use. -b+√b²-4ac/2a

  • To use this formula you first have to make sure you equation is in the form of ax²+bx+c than you can easily find your a,b, &c. Since you already have you values, you just have to sub them in!
  • The + is a symbol which means 'plus or minus' so the formula means that you can either add the -b with the √b²-4ac √b²-4ac


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How did I find my x-intercepts?

  1. You find your a,b,c
  2. You sub in the value of a,b,c in the quadratic formula
  3. You solve, but you don't find the square root of the number until you have separated it into adding and subtracting
  4. You also do not divide the number at the bottom until the very end
  5. Once you found your exact root, you add and subtract
  6. You than find the square root and divide the bottom number and once you have done that you should have the two x-intercepts

Finding the x-intercepts and vertex form

How to find the x-intercepts using the quadratic formula and the vertex form using completed square for: y=x²-6x+5

Also how to graph it using the information provided

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Solving Word Problems: Completed Square

How are the 3 forms connected?

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Reflection

The quadratics unit was a simple unit to follow if you did well practice and understood the methods. At first I did struggle with the unit but with practice it seemed to just get easier since I knew what I was doing. When learning about the 3 forms of quadratics I was very interested because it was very interesting to learn something new and something that will relate back to what you have already completed in your previous units.


In the quadratics unit, I found myself more comfortable with using standard form because I understood how to solve the problems and really liked it. Since I understood standard form I did not really struggle with completing the square. Completing the square was something that was confusing at first but as you work on problems that require completing the square you do not find it as confusing and understand the material better.


Something that I didn't seem to get a good understanding of was vertex form. I found that vertex form was probably the most confusing forms out of all three and that is because every time i will begin to understand something, something else requiring the skills that I didn't really have while using vertex form would follow along, but as I was completing the first part of this assignment I found where I was getting confused and understood what I was doing wrong after carefully reading the notes and completing the problems.


Solving word problems was fairly easy for the most part since all that was needed was to read the problem and solve it using what we had learned. However, I did sometimes get confused while writing out the 'let' statements, but other than that, the problems were understandable.


I did enjoy the quadratics unit despite some of the struggles that occurred, I feel that if I had a better understanding of vertex form I would've done very well in this unit. Many sections of this unit are fairly easy but a great amount of practice is needed.

Assessment