# Quadratics

### All you need to know about parabolas!

## By: Krishna Patel

## Overview of the Unit

## Quadratics 1

· Intro to Quadratics

· Different parts of a parabola

· Vertex Form

· Finding equation and analyzing vertex form

· Factored form

## Quadratics 2

· Multiplying Binomials

· Common Factors and Simple Grouping

· Factoring Trinomials with leading 1

· Factoring Trinomials without leading 1

· Factor Special Trinomials

· Solving by factoring

## Quadratics 3

· Completing the square

· Solving from vertex form

· Quadratic Formula + Discrimnant

· More Quadratic and Discriminant

## Quadratics 1

## Introducing the Parabola

- Parabolas can open up or down
- The zero is also known as the x-intercepts or "roots"
- The axis of symmetry divides the parabola into two equal halves
- The vertex is the point where the axis of symmetry and the parabola meet displaying its maximum or minimum value
- The optimal value is the value of the y co-ordinate

**Zeros:**also known as the x- intercepts are found on the x axis (vertical axis)

**Vertex: **Is always located half way between the zeroes, and it is where the parabola meets, there can either be a minimum of a parabola or a maximum.

**Optimal Value:** It is also known as the y coordinate and is the highest point of the parabola

**Axis of Symmetry:** The axis of symmetry goes through the x axis and is also known as the x value

## Vertex Form

## Graphing Vertex Form

a= vertical compression or stretch

k= changes the y coordinate (up or down)

h= changes the x coordinate (left or right)

Always remember to describe a graph using SRT

Example: y= 2(x-3)^2+5

S: vertical stretch by a factor of 2

R; vertical reflection

T: Horizontal Translation to the right by 3 units (-3)

Vertical Translation up by 5 units (5)

## Finite Differences

If the first differences are constant it is linear, if the second differences are constant it is a quadratic relation.

## 1) Graphing using Step Pattern

**Ex. y= 1/2(x-3)^2-5**

Step 2: Plot the vertex of the equation (3, -5)

Step 2: 1,1 and 2,4 are the steps you use for step pattern

Step 2: Multiply the a value of the equation (1/2) by the y values of (1-1) and (2-4)

Step 3: Move accordingly

## 2) Graphing using Mapping Notation

- Write a mapping formula
- Complete a table of values for y= x^2
- Determine the transformed "key" points
- Sketch the base y= x^2
- Sketch the new graph
- State possible x and y values

**Practice Questions (try these on your own):**

Graph the following using mapping notation and step pattern:

1) y= -0.5 (x+4)^2 -3

2) h= -4 (t-6)^2 + 19

3) A parabola has an equation of y= 6 (x+3)^2 + 4

a) Graph this equation

b) State the direction of opening

c) Write the x intercepts

## Finding Equations and Analyzing Vertex Form

Step 1: y= a (x-h)^2 + k

Step 2 : y= a (x-2)^2 + 6

Step 3: 3= a (5-2)^2 + 6

Step 4: 3= a (3)^2 + 6

Step 5: 3= a (9) + 6

Step 6: 3= 9a+ 6

Step 7: 3-6 = 9a

Step 8: -3= 9a

Step 9: -3/9 = a

Step 9 (simplify the fraction): -1/3 = a

Therefore the equation is:** -1/3 (x-2)^2 + 6**

**For more practice there are similar questions in the math workbook (chapter 4.4 page 42) **

## Factored Form

Find x-intercepts by setting y to 0 and the vertex is between the two x intercepts. For factored form the a is the stretch or compression, its a stretch if its a whole number and a compression if its a fraction.

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

## Quadratics 2

## Multiplying Binomials

**Foil:**

- F: First
- O: Outer
- I: Inner
- L: Last

Example: (x+1) (x+3)

- F: (x) (x) = x^2
- O: (x) (2)= 2x
- I: (1) (x) = x
- L: (1) (3) = 3

x^2 + 2x + x + 3

x^2 + 3x + 3

**Distributing:**

Example 1:

(r-8) (r+3)

(multiply the first r by the second bracket, and the -8 with the second bracket)

r^2 + 3r - 8r -24

r^2 - 5r - 24

Example 2:

5(x+4) (x+6)

(distribute 5 to the first bracket only)

(5x+ 20) (x+6)

5x^2 + 30x+ 20x + 120

5x^2+ 50x+ 120

**Squaring Binomials: **

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

Example:

(x+4)^2

= (x)^2 + 3(x) (4)+ (4)^2

= x^2 + 8x + 16 (the answer is called **perfect square trinomial**)

**Product of Sum and Differences:**

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

Example:

(x+5) (x-5)

= (x)^2 - (5)^2

= x^2- 25 (the answer is called **differences of squares**)

## Common Factoring and Simple Grouping

2x^2------> **2. x** . x

4x------> 2. **2. x **

GCF: 2x

2x 2x^2+ 4x divided by 2

2x (x+2)

## Factoring Simple and Complex Trinomials

**Simple:**Simple Trinomials are relations without a coefficient in front of the first value. We factor simple trinomials by using product and sum and decomposition

**Product and Sum:**

Example 1:

Factor x^2=7x+12

(Find a number that multiples to 12 and adds to give 7)

P= 12 (4 times 3)

S= 7 (4 plus 3)

Therefore: (x+3) (x+7)

**Decomposition:**

Example 1:

Factor: x^2+7x+12

x^2+**7x**=12

x^2+ **4x+ 3x**+ 12

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

(x+4) (x+3)

## Complex Trinomials

**Chart:**

Example: 6x^2 + 11x + 3

The first 2 numbers should multiply to 6, and the other 2 should multipy to give 3

When you multiply across and add the numbers they should add to 11.

3x---------1 ------> (3 times 3 = 9)

2x ---------3 ------> (2 times 1 = 2)

9+2 = 11

(3x+ 1) (2x+ 3)

**Decomposition:**

6x^2+ 11x+ 3

Sum= 11 Product= 6 x 3= 18 (9, 2)

6x^2+ 9x+2x+3

3 (2x+3) + 1 (2x+3)

(2x+3) (3x+1)

## Factor Special Trinomials

**Difference of Squares**

- a^2- b^2= (a+b) (a-b)
- Both a and be should be perfect squares

Example:

81x^2 - 64y^2

(9x)^2 (8y)^2

= (9x+8y) (9x-8y)

Example 2:

144y^2 - 169

(12y)^2 (13)^2

= (12y+13) (12y-13)

**Perfect Square Trinomials**

- The first and last terms are perfect squares, and the middle term is twice the product of the square roots
- a^2- 2ab+b^2 = (a+b)^2

Example 1:

16m^2 + 24m+ 9

(4m)^2+ 2(4m)(3)+ (3)^2

(4m+3)^2

This website helped me understand Perfect Square, I liked how they explained everything they did step by step and also gave reasons to why they're doing something in a specific way.

## Solving by Factoring

**Example 1**:

(x+ 3) (x+4)= 0

x+ 3= 0 (x=** -3**)

or

x-4= 0 (x= **-4**)

therefore the x intercepts are x= -3 and x= -4

**Example 2:**

x^2+ 5x+6 = 0

P= 6 (3 times 2)

S= 5 (3 plus 2)

(x+3) (x+2) = 0

x+3= 0 (x= **3**)

or

x+2= 0 (x= **2**)

Therefore the x intercepts are x= 3 and x= 2

**Example 3 (it gets harder!)**:

2x^2+ 14x+ 12= 0

(factor out the 2)

2 (x^2+ 7x+ 6)

P= 6 (6 times 1)

S= 7 (6 plus 1)

(x+6) (x+1)= 0

x+6= 0 (x= **-6**)

or

x+1= 0 (x= **-1)**

Therefore the x intercepts are x= -6 and x= -1

## Quadratics 3

## Completing the Square

Simple Practice

1) x^2 +** 2**x + _____ (Answer= **2**/2^2 = 1)

2) x^2 +** 8**x +1____ (Answer= **8**/2^2= 16)

## Quadratic Formula and Discriminant

If the answer is greater than 0 there will be two solutions (roots)

If the answer is less than 0 there will be no solution (roots)

If the answer is 0 there will be 1 solution (root)

## Quadratic Formula

## Optimization

Sorry if the video is a little blurry

## Reflection

**Below are some useful YouTube links that are not mine but I have found them helpful throughout this unit**