# Quadratics

### Tavleen Mann

## Background Check! What are Quadratics?

A quadratic relationship is always symmetrical. Same on the left, and same on the right. The graph for a quadratic relation has a *parabola*. The parabola can either open up, creating a smiley face if it is positive, or open down, creating a sad face if it is negative.

## Terms used in Quadratics

**The maximum or minimum point on the graph. It i the point where the graph changes direction. Labelled as (x,y).**

__Vertex:____ Minimum/Maximum Value or Optimal Value:__ The highest or lowest point on the parabola, the y-value on the vertex. Labelled as (y=#).

__ Axis of Symmetry:__ A line that initially "cuts" your parabola in half, the x-value on the vertex. Labelled as (x=#).

__ Y-Intercept:__ Where the parabola meets the y-axis. Labelled as (0,#).

__ X-Intercept(s):__ Where the parabola meets the x-axis. There can be 0,1 or 2 intercepts. Also called "zeros". Labelled as (#,0).

## In order for a relation to be quadratic, they need to have the same second differences.

## Vertex Form

## Step Pattern

## Transformations

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

vertex equation

how to find a

## Factored Form

## Multiplying Binomials

## Common Factoring

__Step 1:__ Find a GCF that all terms have in common

__Step 2:__ Divide the expression by the GCF which means the number will now be placed outside of the brackets.

__Step 3:__ Simplify and leave the GCF outside of the bracket for your final answer.

## Factoring Trinomials by Grouping

## Factoring Simple Trinomials

**x**

**²+bx+c**

In order to change a simple trinomial into factored form, we need to:

__Step 1:__ Make sure that the format of the equation is in the equation written above. Ex. x²+5x+6.

__Step 2:__ Find 2 factors that multiply to give the sum of c. Ex. 2 and 3 multiply to give us 6, which is the c value, which means these factors work.

__Step 3:__ Make sure the factors add to equal the b value. Ex. 2 and 3 add to give us 5, which is the b value, so these factors work.

__Step 4:__ Write the equation out in factor form with the factors put into the brackets, in a order that gives us the answer of the simple trinomial we first started with. Ex. (w+2)(w+3) works since it gives us the original simple trinomial we started with.

## Factoring Complex Trinomials

Ex: 8x²+4x-5 becomes (4x+5)(2x-1).

## Perfect Squares

## Difference of Squares

## Word Problems

## Standard Form

- A way to write a quadratic relation
- y=ax²+bx+c
- An example in standard form is
**y=3x²+15x+18**

## The Quadratic formula is...

## How to find the zeros using the quadratic formula

## Discriminants

**b²-4ac**- A discriminant is the equation inside of the square root
- The discriminant tells us how many solutions there are, if there are any
- If the discriminant is negative, then we know there will be no solutions as a negative number cannot be square rooted
- If the discriminant is 0, then there is only 1 solution, since a 0 would not change if you either add or subtract
- If the discriminant is greater than 0, there will be 2 solutions, like the example above

## Axis of Symmetry

- To find the axis of symmetry, we use the equation outside of the discriminant in the quadratic equation

## Optimal Value

- Substituting the axis of symmetry found into the original equation, we get y, which is the optimal value.

Refer to the example above, for the answer below

## Completing the Square

- Completing the square means converting from Standard form to Vertex form
- y=ax²+bx+c to y=a(x-h)²+k

## Reflection

*a lot*of help, I understood them. Something I found really easy from the start was common factoring. I understood it really well and did well with it.