# Quadratics 101!

### By: Harkirat Singh Bassi

## 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

**How can you tell if a relation is linear or non linear?**

A relation is quadratic if the degree of the equation is 2, in other words the highest exponent in the equation is 2.

Another way to find out is by checking the finite differences. If the first differences are constant than its linear. If the first differences are not constant and the second differences are then that mean the relation is 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**

The axis of symmetry can be found in an equation in the vertex form when looking at the "h" value. (Shown as x=-1(h))

***The value of -1(h) is the axis of symmetry. Opposite Signs***

Examples:

Equation:

1. y=2(x-6)^2+7

2. y=5(x+4)^2+21

AOS:

1. x=6

2. x=-4

**Optimal Value**

The optimal value is the "k" in vertex form. It is the maximum or minimum value on a parabola. (Shown as y=k)

Examples:

Equation:

1. y=2(x-6)^2+7

2. y=5(x+4)^2-21

Optimal Value:

1. y=7

2. y=-21

**Vertex**

The vertex of the parabola can be found by taking the coordinates of the axis of symmetry and optimal value. (Shown as (x,y) )

Example:

Axis of Symmetry

x=3

Optimal Value

y=5

Vertex Coordinates

(3,5)

**Transformations**

Horizontal Transformation - Moves the parabola left and right

-If the h value is less than zero, then it will move to the right

-if it is more than zero, then it will move left.

Vertical Transformation - Moves the parabola up and down

-If the k value is greater than 0, the parabola will move up

-If the k value is less than 0, the parabola will move down

Vertical Stretch

Increasing the value of a will vertically expand the graph (make the arms move closer together

Decreasing the value of a will vertically compress the graph (make the arms spread out wider).

Reflection - Determines whether the direction of opening is upwards or downwards

If the "a" value is greater than 0, direction of opening is upwards

If the "a" value is less than 0, direction of opening is downwards

**Step Pattern**

The step pattern shows how steep the parabola gets and use it to find next points after vertex is found. The step pattern is y=x^2. So it is over 1 and up 1, over 2 and up 4, etc. The "a" can change the steepness, have to multiply the "a" by the 1 to find next point.

**Zeroes**

To find the zeroes in vertex form, you need to sub in y=0 and isolate for x.

## Factored Form

__Factored Form of a quadratic looks like y=a(x-s)(x-t). __

**Zeroes**

The x-intercepts are at (s, 0) and (t, 0) and they will be opposite. for example, if "s" was 2 and "t" was 3 in the equation ((y=a(x-2)(x-3)), then the zeroes would be at (2,0) and (3,0) because they are opposite of "-2" and "-3" .

Zeros & X-intercepts

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

x - 3 = 0 x - 4 = 0

x = 3 x = 4

**Axis Of Symmetry**

To find the axis of symmetry, need to sub in x-intercepts into x=s+t/2.

Axis of Symmetry

x = 3 + 4 / 2

x = 3.5

**Optimal Value**

To get the optimal value, need to get the "x" from AOS and sub into the equation, y=a(x-s)(x-t), this will give the y value and then use the x and y coordinates to get vertex.

Optimal Value

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

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

y = 12.25 - 14 - 10.5 -12

y = -24.25

**Expanding and Simplifying**

When expanding from factored form, first the "a" will be multiplied to the "x" and "s" in first bracket and then "x" from the first bracket must be multiplied to other "x" in second bracket. Then the "x" from first bracket is multiplied to "t." After, the "s" is multiplied with "x" from second paragraph and then "t." At the end collect like terms.

## Quadratic Formula

## Standard Form

**Standard Form of a quadratic looks like y=ax^2+bx+c**

**Zeroes**

To find the zeroes of standard form, will have to use quadratic formula.

**Axis Of Symmetry**

After the x intercepts are found using the quadratics formula, then add the two together and divide the result by two and it will give the "x" value.

**Optimal Value**

To get the optimal value in standard form, need to sub in "x" from AOS into the equation and solve. This will give the y coordinate for the vertex.

__Factoring__Next we will factor to convert from standard form to factored form.

**Common Factoring**

Common factoring is finding what is common between the two equations, then divide the whole thing by what is common. Then with what is left will go in bracket and what was used to divide, will now be used to multiply with what is in brackets.

**Simple Trinomial Factoring**

A simple trinomial is when you have three different terms, but the "a" value is equal to one in standard form equation, y=ax^2+bx+c. This will become factored form at the end. Two numbers must multiply to "c" and add to "b".

**Complex Trinomial Factoring **

Use complex trinomial factoring when the "a" value is higher than one in standard form. Instead of putting an "x" at the beginning, put what are the multiples of the "a". For example if it was 3x^2+1x+1, then we would do (3x+1)(x+1) because factors of 3 are 1 and 3 so we use those if it has more then guess with both that mutiply to the number.

**Perfect Squares Factoring**

Check if you can factor "a" and "c" value in standard form y=ax^2+bx+c. If can factor those two terms, then can factor a perfect square. When factoring perfect square, it looks like this: (a+b)^2 which can also be a^2+ 2ab + b^2.

**Difference Of Squares Factoring**

Difference of square is when there are squares at beginning and end but they have no middle term. After factored, one bracket should be adding two terms and other should be subtracting.

## Word Problems

**The First Example**

**Second Example**

**Third Example**