# Quadratic Relations

### By: Harkomal Mann

## Introduction

## Vertex Form

## What is a Vertex?

Vertex is the highest point or lowest point on a graphh. The vertex is where the graph changes direction. It is also the location where the axis of symmetry and parabola meet (h,k).

## Axis of Symmetry

## Optimal Value

## Quadratic Relations

## Identifying Transformation in Vertex Form

For a simple parabola with a vertex that is at (0,0) is y=x² which is used to make a new graph that has a corresponding new equation. This equation could be written in vertex form. Which is

y=a(x−h)2+k

- a represents the direction of opening and compression or stretch (it is a stretch if the number is a whole number, if number is a decimal or fraction then it would be a compression).
- k is the vertical translation (moves parabola up or down depending on it's sign. If negative it moves down, and if positive, it moves up).
- h is the horizontal translation (moves parabola left or right depending on it's sign)

## The Zeros and Y-intercepts

## Graphing Vertex Form

## Word Problem

The height, h, in meters, of a soccer ball, t seconds after being thrown can be modeled by the equation:* h= -0.10 (t - 4)*²*10*

a) What is the soccer ball's height 3 seconds after it is kicked?

The ball reached it's maximum height at 3 seconds because of the x value of the vertex.

## Additional Information

- to find the y-intercept, set x=0 and solve for y
- to solve, set y=0 and solve for x or expand and simplify to get the standard form, then use the quadratic formula

## Factored Form

## MULTIPLYING BINOMIALS & SPECIAL PRODUCTS

FOIL (First, Outside, Inside, Last)

An example would be:

## Special Cases

## Difference of Squares

## Common Factoring

4x+2

First: Find the GCF which is 2

Second: Write the solution with brackets 4x/2 + 2/2

Equations with exponents still have the same steps

An example would be:

For example 24 has factors : 1 2 3 4 24 12 8 6

When factoring polynomial expressions, we need to examine both the numerical coefficients and variables to find the greatest common factor.

Factoring is the opposite of expanding

## Quadratic Equations in Factored Form

The zeros, the x-intercepts, the roots, the solution (to "solve")

- Value of a gives the shape and direction of opening
- The value of r and s give the two x-intercepts (r,0) and (s, 0)
- To find the vertex, use the zeros to find the AOS and sub this x value into the given equation and solve for y

## Simple Factoring

Algebra tiles example of factoring

A quadratic in standard form can factor to get you factored form

x^2+bx+c (x-r)(x-r)

Standard Form to Factored Form

Where r+s=b sum

and rs=c product

and r and s are the integers

Step 1: Look at signs of b and c in the given expression (x^2+bx+c)

- If b and c are positive, then both r and s are positive
- If b is negative and c is positive, then both r and s are negative
- If c is negative and one of r and s is negative

Step 2: Find the "product and sum"

- Find the two numbers whose product is c
- Find two numbers which numbers sum b

Ex: x^2+4x-5

(x+5)(x-1)

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

## Complex Factoring

Complex trinomials have a coefficient that is other than 1 in front of the x^2 term.

Where a can never be 1.

## Difference of Squares and Perfect Squares

How to factor 36x^-4

Recall: (a+b)(a-b)= a^2-b^2Apply this concept to the question

To find a and b, simplify square root and first and last term.

## Perfect Square Trinomials

**(1) the first and last terms are perfect squares****(2) the middle term is twice the product of the square roots of the first and last terms **

## Finding Zeros

An example of a simple trinomial...

## Difference of Squares Trinomial

## Complex Trinomial

## Key Information

- the value of a gives you the shape and direction of opening
- the value of r and s give you the x-intercepts
- to find the y-intercept, set x=0 and solve for y

## Graphing

## Word Problem

The height of a rock thrown from a walkway over a lagoon can be approximated by the formula h=-5r^2+20t+60 where t is the time in seconds, and h is the height in meters.

Write the formula in factored form

## Standard Form

## Maximum and Minimum Value

**If the "a" value is a number and/or a negative number?**

**Step 1: Group the x terms together **

**Step 2: ****Common factor only the constant terms.**

**Step 3: Complete the square**

**Step 4: ****Write the trinomial as a binomial squared**

## Quadratic Formula

Down below is the quadratic formula:

## Word Problem

Ax+By=C

1X+0.50Y=200

The value of a gives you the shape and direction of opening and the value of c is the y-intercept.