# All About Quadratics

### Summary of the Quadratics Unit By Alexia Brown

## Table of Contents

__1. Vertex Form__

- Axis of Symmetry
- Optimal Value
- Transformations(Translations Vertical or Horizontal, Vert stretch, Reflection)
- X - Intercepts or Zeros ( sub y = 0 Solve)
- Step pattern

__2. Factored Form__

- Zeroes and X - Intercept ( r and s)
- Axis of symmetry
- Optimal Value ( sub in)

__3. Standard Form__

- Zeroes (Quadratics formula)
- Completing the squares
- Simple Trinomials
- Complex Trinomials
- Perfect Squares
- Difference of Squares

## Vertex Form

## Optimal Value

To find the optimal value of this equation

y=0.5x^2+2x+3

To find the optimal value, we need to find the vertex.

In order to find the vertex, we first need to find the x-coordinate of the vertex.

To find the x-coordinate of the vertex, use this formula x= -b/2a

So the x-coordinate of the vertex is x=-2. Note: this means that the axis of symmetry is also x=-2.

So the y-coordinate of the vertex is y=1.

So the vertex is .

Since the max/min occurs at the vertex as the 'y' value, this means that the min is y=1. So the optimal value is is y=1.

## Video on How to Graph parabolas

## Step Pattern

OVER 1 (right or left) from the vertex point, UP 1² = 1 from the vertex point

OVER 2 (right or left) from the vertex point, UP 2² = 4 from the vertex point

OVER 3 (right or left) from the vertex point, UP 3² = 9 from the vertex point

OVER 4 (right or left) from the vertex point, UP 4² = 16 from the vertex point

and so on ...

where the "Up" numbers are the sequence of "Perfect Square" numbers ...

but always counting from the vertex each time.

## Transformations ( Translations Vertical or Horizontal, Vertically stretched, reflection

## Factored Form

## Optimal Value

The optimal value is what determines the maximum and minimum in factored form

## Factored form to Standard form

To convert the equation of a quadratic relation from

factored form to standard form, we will expand,

regroup, then simplify the factored form.

## Factored form to Vertex form

We know that the x-coordinate of the vertex occurs at the average of the 2 zeroes of the Quadratic Function. This function has zeroes at x = -3 and x = 1. So the x-coordinate of the vertex is (-3 + 1)/2 = (-2)/2 = -1.

To get the y-value of the vertex, sub in x = -1 into the original equation to get:

y = -2(x + 3)(x - 1) = -2(-1 + 3)(-1 - 1) = -2(2)(-2) = 8

Therefore, y = -2(x + 3)(x - 1) has it's vertex at (-1, 8).

## Standard Form

## Quadratic formula

h= 4x^2 + 5t +15

- First step is to determine your a,b and c value. a= 4, b= 5 and c= 15
- Sub in the values into the quadratics formula -4±√ 5^2 - 4(4)(15)/2(8)
- solve -4 ± √10 + 240/ 8
- -4 ± √ 230/8
- -4 ± √15.16/8
- -4 + 15.16/8 = 1.45
- -4 - 15.16/8 = -2.39

## Completing the squares

Steps

Now we can solve a Quadratic Equation in 5 steps:

- Step 1 Divide all terms by (the coefficient of x2).
- Step 2 Move the number term (c/a) to the right side of the equation.
- Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
- Now you have something like this (x + p)2 = q, which can be solved rather easily:
- Step 4 Take the square root on both sides of the equation.
- Step 5 Subtract the number that remains on the left side of the equation to find x.

## Simple trinomials

Example: x^2 + 4x - 32

- Step one you get the 32 and you divide it by 2 then you square root it you should get positive 4
- Step two You find a number times 4 to give you 32 which is 8
- Step three The equation you have now should be - 4 + 8 = 4 then (-4) (8) = -32
- The last equation should be (x - 4)(x + 8) you got the x because you expanded it

## Difference of squares

The factors of (a^2 - b^2)

(a + b) and (a - b)

Factor: x^2 - 9

Both x^2 and 9 are perfect squares. Since subtraction is occurring between these squares, this expression is the difference of two squares.

What times itself will give x^2 ? The answer is x.

What times itself will give 9 ? The answer is 3.

These answers could also be negative values, but positive values will make our work easier.

The factors are (x + 3) and (x - 3).

Answer: (x + 3) (x - 3) or (x - 3) (x + 3)

## Graphing From Standard form

- Factor the equation.
- Set each bracket equal to 0 to determine the x - intercepts roots and zeroes.
- Find the axis of symmetry and the x - intercepts.
- Sub in the A.O.S into either the standard equation.
- Graph using the x , y and the vertex.

## Dicriminant

Discriminant is a number inside the square root (b²-4ac) of the quadratic formula. It shows how many solutions a quadratic equation has.

Discriminant: b²-4ac

- D>0, 2 x-intercepts
- D<0, No x-intercepts
- D=0, 1 x-intercept