# Quadratics

### By Ramanveer Brar

## Table of Contents

**Vocabulary**

**Linear or Quadratic**

**Parabolas**

Graphing Parabolas Video

**Factoring**

**Types of Forms**

**All About Vertex Form**

The Axis of Symmetry and Optimal Value

Transformations

Zeros & X-intercepts

Step pattern

**All About Factored Form**

Zeros & X-intercepts

Axis of Symmetry

Optimal Value

**All About Standard Form**

Zeros

Axis of Symmetry

Optimal Value

Completing the Square to Turn to Vertex Form

Factoring to Turn to Factored Form

- Common
- Simple Trinomial
- Complex Trinomial
- Perfect Squares
- Difference of Squares

**Word Problems**

**Making Connections**

**Reflection**

## Vocabulary

**Non-linear relation:**a relationship between two variables that does not follow a straight line when graphed.

**Curve of best fit: **a smooth curve drawn to approximate the general path or trend in a scatter plot.

**Parabola: **the graph of a quadratic relation, which is U-shaped and symmetrical

**Vertex:** The point on a parabola where the curve changes direction. The maximum point if the parabola opens down. The minimum point if the parabola opens up.

Axis of symmetry: the line that divides a figure into two congruent parts.

**Zero: **a value of x for which a relation has a value of 0. Corresponds to an x-intercept of the graph of the relation.

**x-intercept: **the x-coordinate of the point where a line or curve crosses the x-axis. At this point y=0

**Perfect square trinomial: **a trinomial that is the result of squaring a binomial.

**Difference of squares:** an expression that involves the subtraction of two squares.

## Linear or Quadratic

## Parabolas

- Parabolas can open up or down
- The zero of a parabola is where the graph crosses the x-axis
- "Zeros" can also be called "x-intercepts" or "roots"
- The axis of symmetry divides the parabola into two halves
- The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value.
- The optimal value is the y co-ordinate of the vertex
- The y-intercept of a parabola is where the graph crosses the y-axis

To graph parabolas you can use Desmos. It is a quick substitute to drawing a parabola onto graph paper.

## Factoring

## Types of Forms

Vertex form - **a(x-h)²+k**

**a(x-r)(x-s)**

Standard form -

**ax²+bx+c**

## Vertex Form

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

Axis of Symmetry **(x=h)**

Optimal Value** (y=k)**

**Transformations**

*Orientation*

If the a value is greater than zero the parabola opens up.

If the a value is less than zero the parabola opens down.

*Shape*

If the a value is zero the parabola is vertically compressed.

If the a value is greater or less than one the parabola is vertically stretched.

**Zeros & x-intercepts**

**Step Pattern**

The step pattern is the rate at which the parabola goes up/down.

Ex. up 1, up 3, up 5, up 7, each time it goes up more but to the side by 1. This is just one example.

## Factored Form

y = ( x - 3 ) ( x - 4 )

x - 3 = 0 x - 4 = 0

x = 3 x = 4

Axis of Symmetry

x = 3 + 4 / 2

x = 3.5

Optimal Value

y = ( x - 3 ) ( x - 4 )

y = ( 3.5 - 3 ) ( 3.5 - 4 )

y = 12.25 - 14 - 10.5 -12

y = -24.25