# Quadratics Relations

### By: Simran Heer

## Table of Contents

**Introduction:**

- Second Differences
- Properties of Parabolas
- Transformations In Vertex Form

**Types of Equations:**

- Factored Form: a(x-r)(x-s)
- Standard Form: ax² + bx + c
- Vertex Form: a(x-h)² + k

**Word Problems:**

- 2 examples

**Reflection:**

- Test

## Second Differences

## Properties of Parabolas

- Vertex: where the graph changes direction (x, y)
- Axis of Symmetry: the line which passes when x=0, and divides the parabola into two equal parts
- Optimal Value: the y coordinate, either the highest (max) or lowest (min) point
- X-intercepts: the points which cross the x axis. There could be 2 solutions, 1 solution, or no solution.
- Y-intercepts: when the graph crosses the y axis

## Types of Equations

## Factored Form: a(x-r)(x-s)

**When using these equations, you deal with factoring and solving for the zeros. A zero of a parabola is another way to say x-intercept, and in order to find the solutions, you need to set y=0.**

To graph a parabola in factored form, you need to:

Find the Axis Of Symmetry:

- Find the zeros
- Solve by adding both zeros, then dividing by 2. This will give you the AOS

How to find the Vertex:

- Substitute the value of the axis of symmetry in the factored equation. Solve to find y.

## Expanding Factored to Standard

You use the method of expanding when you want to go from factored to standard form. You multiply each term in the brackets, and then collect the like terms.

## Common Factoring

## Simple Factoring: x² + bx + c = (x + r)(x + s)

## Special Cases

(a - b)(a + b), and a polynomial of the form a² x² +- 2 a b x + b is a perfect square and can be factored as (a x +- b)².

## Factor by Grouping

## Vertex Form: a(x - h) + k

- The (-h) moves the vertex of the parabola left or right , when the (h) is negative it moves to the right and when its positive it moves to the left.
- The (k) moves the vertex of the parabola up or down, when the (k) is negative it moves down and moves up when its positive.
- The (a) vertical stretch or compression. The parabola is reflected upon the axis if a the (a) is negative.

When solving for the x-intercepts set y=0 and solve the equation. There could be 2 solutions, 1 solution, and no solution.

The vertex is given by the equation, where (h, k) is the vertex.

## Standard Form: ax² + bx + c

- (a) gives the shape and direction of opening of the parabola
- (c) is the y-intercept

You can solve for the x-intercepts using factoring, completing the square, or the **quadratic formula**.

The Quadratic Formula solves for the x-intercepts

**Discriminant:**is a part of the quadratic formula which determines how many solutions there are to an equation.

Find the X-Intercepts:

- Use the quadratic formula to get the zeros.

- Add the two solutions together and then divide by 2 to get the axis of symmetry.

- Substitute the (x) value you solved for above, and solve for y.
- The axis symmetry will be the (x) value in the vertex
- Vertex: (h,k) h=x value, k=y value

**Completing the Square:**from standard form to vertex form, you need to complete the square. Completing the square is very simple. You need to take the (b) value and divide it by 2, once you get that answer you square it. This video shows you the exact steps you need to make a perfect square trinomial.

## Word Problems

## Link Between Equations and Graphs

**Equations and Parabolas:**All three forms of equations, describe a parabola. It gives you information about different parts of the parabola, you either have to solve for it, or just by looking at it you can get some details. Equations are a written version of a graphed parabola, the graphed parabola is a visual for the equation. In each equation you have to solve for x-intercepts, and the vertex, this allows you to plot 3 points of the parabola.

**Discriminant and X-intercepts: **Discriminant's tells you how many solutions you have to an equation or parabola. This helps us find the exact number of x-intercepts.