# Quadratic Relations

### All you need to know about Quadratics!

## Intro to Quadratics

## Identifying Quadratic Relations in Different Forms/Formulas

**Vertex Form: a(x-h)²+k**

What does each variable represent?

- a is the vertical stretch (Either opens up or down).

- h moves the parabola left or right (The x coordinate & is always the opposite of the sign given for example -4 would be +4 & +3 is -3).

- k moves the parabola up or down (The y coordinate).

- The "²" sign is what makes the parabola a U-Shaped line.

e.g. y=(x-1)²+3

Three types of transformations:

- Vertical Stretch
- Vertical Compression
- Vertical Reflection

Optimal Value: Highest point or Lowest point of the parabola, can be found on y-axis (Represents y coordinate).

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

What does each variable represent?

Axis of Symmetry: The mid-point of the equation

Optimal Value: The max or min of the equation

e.g. y=1(x+3)(x-9)

**Standard Form: y=ax²+bx+c**

Also solved as ax²+bx+c = 0

The vertex has the x-coordinate x = -b/2a

The y-coordinate of the vertex is the maximum or minimum value of the function.

The y-intercept of the equation is c.

e.g. y= 2x²+4x+10

## Finding the x-intercepts in Vertex Form

a) y= -4(x+2)²+16

To find the y-intercept, set x=0

y= -4[(0)+2]²+16

y= -4(2)²+16

y= 0

Therefore y-intercept is (0,0)

To find the x-intercepts, set y=0

0= -4(x+2)²+16

-isolate for x, BEDMAS backwards

(Bring the 16 to the left side)

-16= -4(x+)² (now subtract 16)

-16/-4=-4(x+2)²/-4 (now divide -4)

√4=(x+2)² (now square root)

x+2= +/- √4

x= +/- √4-2 (now subtract 2)

(Now you are going to find the x-intercepts)

x= + √4-2 or x= - √4-2

x=2-2 x=-2-2

x=0 & x=-4

Vertex: (3,4)

Step pattern:

2x1=2 & 2x4=8

Point one: over 1 and up 2

Point two: over 2 and up 8

## Determine the Equation for Vertex form from a Graph

The parabola opens upward, so we know "a" will be positive.

y=a(x-h)²+k

y=a(x-2)²-1

The parabola passes through point (0,3). Substitute x=0 and y=3 and solve from there.

3=a(0-2)²-1

3=a(-2)²-1

3=a(4)-1 (Bring -1 to the other side and change the sign)

3+1=4a

4=4a

4/4=4/4a (Divide to cancel the "4" from the "a" value)

1=a

The equation is y=1(x-2)²-1

## Graphing Factored Form

__How to Find each Variable:__

- X Value:

**+3**)(x

**-9**)

There are two x-intercepts in factored form, here they are -3 & +9.

Finding the x value a.k.a the axis of symmetry: Add the x values and divide by 2

Ex. x= -3 and x= 9

-3+9/2

x= 3

- Y Value:

To find the Y value a.k.a the optimal value: Sub the x value in the original equation

Ex. y=0.5(x+3)(x-9)

y=0.5(3+3)(3-9)

y=0.5(6)(-6)

y= -18

Therefore, the vertex for this graph is (3,-18).

## Common Factoring

Examples:

1) Factor the number 6

6= 3x2

2) Factor (2x+6)

- You have to see what is common in this equation.

So you should get that 2 is common because it can be multiplied to 6 and 2x

Finally answer would be 2(x+3)

3) 8x+6

- Find whats common or find GCF (greatest common factor)

GCF= 2

- Just write down the solution with brackets

Finally answer: 2(4x+3)

4) 12x3y-6x2y+18x2y

GCF= 6x2y

Solution:

= 6x2y(2x-1+3)

= 6x2y(2x+2)

## Trinomials

__What is a "trinomial"?__

EX. 1) x²+3x+4x

Ex. 2) 2x²+8x+9

Ex. 3) 3x²-5x+4

## Simple Trinomial

- Simple Trinomials are trinomials that have an "a" value of 1
- Many polynomials, such as 12x²+7x+2, can be written as the product of two binomials of the form (x + r) and (x + s)
- When factoring a polynomial of the form ax²+bx+c (when a = 1), we want to find:

(1) 2 numbers that ADD to give b

(2) 2 numbers that MULTIPLY to give c

To factor these trinomials, use a table to help you until you become familiar with the procedure

Example. Factor: x²-8x+12

=(x-6)(x-2)

Examples:

Factor the following trinomials using the procedure above.

a) m²-13m+26

13x2=26

13+2= -13

- not possible

b) h²-7h-18

-9x2= -18

-9+2= -7

=(h-9)(h+2)

## Complex Trinomials

ax²+bx+c

(where a=1)

Can you factor the following?

3x²+17+10

- No common factor

- a=3

Example 1: Binomial Common Factoring

Factor: 3x(z-2)+2(z-2)

(Hint: a binomial can be the common factor)

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

Example 2: Factor by graphing

Factor: (df+ef)+(dg+eg)

(There is no common factor in all terms)

(We can graph terms together that have a common factor)

= f(d+e)(f+g)

## How to Solve Complex Trinomials

A) factor

3x²+8x+5

Product of a multiplied c

= 3 x 5

=15

3x5=15

3+5=8

* Rewrite the middle terms with the 2 factors

3x²+3x+5x+5

Not 3x²+5x+3x+5

* Factor by grouping

(3x²+3x)+(5x+5)

= 3x(x1)+5(x+1)

* Think of the binomial (x+1) as

= (x+1)(3x+5)

**EXAMPLES OF COMPLEX TRINOMIAL**

Example 1) 5x²-14x+8

P: 5x8= 40

-10 x (-4)=40

-10 + (-4)= -14

= (5x2-10x)(-4x+8)

= 5x(x-2)-4(x-2)

= (x-2)(5x-4)

Example 2) 8a²+10x-3

p; 8 x (-3)= -24

12x(-2)= -24

12+(-2)= 10

=(8a2-2a)+(12a-3)

=2a(4a-1)+3(4a-1)

=(4a-1)(2a+3)

**HOW TO GRAPH COMPLEX TRINOMIAL**

When solving an equation that requires factoring, ONE SIDE MUST EQUAL ZERO

Example 1: 5x²-14x+8

Step 1: Factor

P: 5x8= 40

-10 x (-4)=40

-10 + (-4)= -14

= (5x2-10x)(-4x+8)

= 5x(x-2)-4(x-2)

= (x-2)(5x-4)

Step 2: Find x-intercepts

(Now to solve for for x, you must let each bracket t zero)

First x-intercept

x-2=0

x=2

Second x-intercept

5x-4=0

5x=4 (Now divide 5 to find x)

5x/5=4/5

x=4/5

Example 2: 8a²+10x-3

Step 1: Factor

p; 8 x (-3)= -24

12x(-2)= -24

12+(-2)= 10

=(8a²-2a)+(12a-3)

=2a(4a-1)+3(4a-1)

=(4a-1)(2a+3)

Step 2: Find x-intercepts

First x-intercept

4a-1=0

4a=1

4a/4=1/4

a=1/4

Second x-intercept

2a+3=0

2a= -3

2a/2= -3/2

a=-3/2

## How to go from Factored form to Standard form

Expand and Simplify:

y=(x+6)(x+4)

y=x²+4x+6x+24

y=x²+10x+24

Therefore the standard form of y=(x+6)(x+4) is y=x²+10x+24

Example 1) y=(x+9)(x+4)

Expand and Simplify:

y=(x+9)(x+4)

y=x²+4x+9x+36

y=x²+10x+36

Therefore the standard form of y=(x+9)(x+4) is y=x²+10x+36

## Graphing Standard Form

Check to see if the coefficient in front of the x² value can be divided equally and if you can divide the equation by that number

Now you should get a number like y=2(x2-2x-8) for example

You should now break the numbers in the bracket to make two numbers that multiply each other ex. y=2(x-4)(x+2) now find the x-intercepts, find the axis of symmetry & find the optimal value

And there you should get a parabola

EXAMPLES OF GRAPHING STANDARD FORM

Example 1) y=2x²-4x-16

Part 1

Make the equation into factored form

y=2x²-4x-16

y=2(x2-2x-8)

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

Part 2

Now graph it in the factored form

a) State the zeros

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

0=2(x-4)(x+2)

x-4=0 and x+2=0

x=4 and x= -2

b) State the axis of symmetry

x=4+(-2)/2

x= 2/2

x= 1

c) State the optimal value

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

y=2(1-4)(1+2)

y=2(-3)(3)

y=2(-9)

y= -18

d) State the direction of opening

The direction of opening is up

Example 2) y=2x²+8x-24

Part 1

Make the equation into factored form

y=2x²+8x-24

y=2(x2+4x-12)

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

Part 2

Now graph it in the factored form

a) State the zeros

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

=2(x+6)(x-2)

x+6=0 and x-2=0

x= -6 and x= 2

b) State the axis of symmetry

x= -6+2/2

x= -4/2

x= -2

c) State the optimal value

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

y=2(-2+6)(-2-2)

y=2(4)(-4)

y=2(-16)

y= -32

d) State the direction of opening

The direction of opening is up

## Standard Form Word Problem Example

## Perfect Squares

For example, (x + 1) × (x + 1) = x2 + x + x + 1 = x2 + 2x + 1 and x2 + 2x + 1 is a perfect square trinomial

EXAMPLES OF PERFECT SQUARES

Example 1) 9x²+24xy+16y²

=(3x)²+2(3x)(4y)+(4y)²

= (3x+4y)(3x+4y)

=(3x+4y)²

Example 2) 4m²-20mn+25n²

= (2m)²-2(2m)(-5n)+(-5n)²

= (2m-5n)(2m-5n)

= (2m-5n)²

Example 3) 49a²+42ab+9b²

= (7a)²+2(7a)(3b)+(3b)²

= (7a+3b)(7a+3b)

= (7a+3b)²

## Differences of Squares

For example, (x+8)(x+8) = x²- 64 and x²+64 is a difference of square trinomial

EXAMPLES OF DIFFERENCE OF SQUARE

Example 1) 9x²-49

= (3x)²-(7)²

= (3x+7)(3x-7)

Example 2) 81a²-4

= (9a)²-(2)²

= (9a+2)(9a-2)

Example 3) 4h²-49

= (2h)²-(7)²

= (2h-7)(2h+7)

Example 4) 144y²-169

= (12y)²-(13)

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

## Completing the Square

## Examples of Completing the Square

## Quadratic Formula

## How to solve for x-intercepts using quadratic formula

## Word Problems for the Quadratic Formula

## Axis of Symmetry H= -B/2A

WHEN CAN WE USE AXIS OF SYMMETRY:

You can use Axis of Symmetry when you are asked to find the vertex in a standard form. You have to use the Axis of Symmetry when you are using the quadratic formula and the discriminants values are negative.

## Steps to find the Vertex in the Axis of Symmetry in Quadratic Equation

## Connections between topics

## Standard Form - How to go from Standard Form to Vertex Form - How to go from Standard Form to Factored Form | ## Vertex Form- How to go form Vertex Form to Standard Form - How to go from Vertex Form to factored Form | ## Factored Form- How to go from Factored Form to Standard Form - How to go from Factored From to Vertex Form |

## Standard Form

- How to go from Standard Form to Factored Form

## Vertex Form

- How to go form Vertex Form to Standard Form

- How to go from Vertex Form to factored Form