# Quadratics

### How could we defend our castle?

## Background Info

__Name__

Quadratic is a Latin term. It is originated from quadratus, the past participle of quadrare which means "to make square."

__Who invented the Quadratic Formula ???__

Euclid, a Greek Mathematician, created an intellectual concept during 300 BC. Both Pythagoras and Euclid created a well based procedure to find multiple solutions of the quadratic equation.

## Basic Quadratic Terminology

A symmetrical open plane cure (a U-Shape).

2. Vertex

The highest point/tip

Format: The X coordinate comes first. The Y coordinate comes second. (X,Y)

3. Axis of Symmetry (AOS)

A line of symmetry for a graph. The axis of symmetry is considered as a mirror to split images on either side.

4. Optimal Value (Maxima/Minima)

The largest and smallest value/point on the parabola. (y=___)

5. X-Intercept

The point which the graph crosses the x-axis. (x,0)

6. Y-Intercept

The point which the graph graph crosses the y-axis. (0,y) Another way of saying X-Intercept is Zeros.

## Vertex Form

## Axis of Symmetry

Vertex Form Equation: y = a (x__-h__)^2 + k

Vertex: (__h__,k)

Axis of Symmetry: x = h

Small Note: The Axis of Symmetry is a vertical line, so it has to be written: x = __

## Optimal Value (Maxima/Minima)

Vertex Form Equation: y = a(x-h)^2 + k

Vertex: (h,__k__)

Optimal Value: y = k

Small Note: The Optimal Value is a horizontal line, so it has to be written: y = __

## Transformations

1. Vertical Translation

2. Horizontal Translation

3. Vertical Stretch

4. Reflection

## X-Intercepts/Zeros & Y-Intercept

__X-Intercept/Zeros__

To find the X-intercept/ Zeros, you have to understand that the coordinate is (x,0). Since we know this, to find the X-Intercept, the Y value in the equation of the parabola must be 0 to solve for the X value.

* y* = a (x-h)^2 + k

Y-Intercept

To find the Y-intercept, you have to understand that the coordinate is (0,y). Since we know this, to find the Y-Intercept, the X value in the equation of the parabola must be 0 to solve for the Y value.

## Step Pattern

## Factored Form

## X-Intercept/Zeros

__For Instant:__

y = (x + 13) y = (x-6)

0 = x + 13 0 = x -6

-13 = x 6 = x

## Axis of Symmetry

Formula:

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

Final Formula:

x = r+s divided by 2

For Instant:

y = (x-r) (x-s)

x = (x-6) (x-8)

x = 6 + 8 divided by 2

x = 7

## Optimal Value

For Instant:

y = (x-r) (x-s)

x = (x-6) (x-8)

x = 6 + 8 divided by 2

x = 7

Substitute

y = (7-6) (7-8)

y = (1) (-1)

y = -1

Conclusion

The Optimal Value is y = -1. Also the vertex is (7,-1)

## Standard Form

## Zeroes

Formula:

ax^2 + bx + c

Converted Formula:

b + - square root b^2 - 4(a)(c) divided by 2(a)

For Instant:

Figure out the following X-Intercepts for the Standard Form Equation 3^2 + 12x + 6

Procedures;

1st Step: Substitute the Standard Equation into the Quadratic Formula

-12 +- square root (12)^2 - 4(3)(6) divided by 2(3)

2nd Step: Continue solving it

12 +- square root 144 - 72 divided by 6 // -12 +- square root 72 divided by 6 // -12 +-

8.49 divided by 6 //

Final Step: @ the end you will have two different X-Intercepts

1st X-Intercept:-0.585 // 2nd X-Intercept: -20.49

## Axis of Symmetry

-b/2(a)

For Instant:

When finding the AOS for the given Stand Form Equation y = 4x^2 + 12x + 8, you need:

1st: Make the value of B negative (y = 4x^2 + 12x + 8)

Turn 12 into -12

2nd: You must divide the new negative B value by 2(a)

-12 divided by 2(3)

3rd: Finally, you solve everything

-12 divided by 6 = -2

In Conclusion, the AOS of this equation is (-2)

## Optimal Value (Maxima/Minima)

For Instant:

Using the value we've received for x, substitute that in and solve.

(4(-2)^2 + 12(-2) + 8)

8^2 - 24 + 8

64 - 24 +8

48

In Conclusion, your Optimal Value is 48.

## Completing the square to turn to vertex form

## Factoring

## Types of Factoring Equations

2. Simple Trinomial

3. Complex Trinomial

4. Perfect Square

5. Differences of Squares