## Basics of Parabolas

· May open upwards or downwards

· The zero is where the graph crosses the x-axis

· Zeroes are also called x-intercepts

· The axis of symmetry (AOS) divides the parabola into two equal halves

· Vertex is where the AOS and parabola meet. It is the parabola's maximum or minimum value

· Optimal value is y co-ordinate of the vertex

· Y-intercept is where the graph crosses y-axis

Vertex Form

y=a(x-h)+k

Factored Form

y=a(x-r)(x-s)

Standard Form

y=ax^2+bx+c

## VERTEX FORM

An equation in vertex form can be used to solve for:

- The optimal value (The y value of the vertex)

-The axis of symmetry (The x value of the vertex)

- Stretch or Compression of parabola

- Direction of opening

- x intercepts/zeroes

In this form:

x=h is the axis of symmetry

y=k is the optimal value
If "a" is positive then the parabola opens upwards like a regular "U".

If "a" is negative, then the graph opens downwards like an upside down "U".

If "a" < 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways.

If |a| > 1, the graph of the graph becomes narrower (The effect is the opposite of |a| < 1).

Changing a Quadratic Function into Vertex Form

## The Step Pattern

The first step in graphing a parabola is to understand the step pattern:

With the equation y=x^2, the pattern is (1,1),(2,4),(3,9).

In the step pattern the x value increases at a constant rate of 1. In the equation y=x^2, the x value is squared to get the y value, which gives you the pattern 1,4,9 etc.

If the a value in the equation is more than 1, for example y=4x^2, the y values in the step pattern are multiplied by the a value. In the case, the points would become (1,4),(2,16),(3,36).

## Finding the Vertex

To find the vertex in an equation in vertex form, all you need to do is find the h and k values in the equation.

If there is no h or k value in the equation, then the value is 0.

In the equation y=x^2, there is no h or k value, meaning the vertex is at (0,0).

## Finding x intercepts

To find the x intercepts with an equation in vertex form put y=0 and solve the equation.

## FACTORED FORM

An equation in factored form can be used to solve for:

-The x intercepts (r and s values)

-The axis of symmetry (x value of vertex)

-The optimal value (y value of vertex)

## Finding the x intercepts

The x intercepts are the r and s values in a factored form equation. Just like the h value in vertex form, the r and s values are negative, making their values opposite. For example if the equation is y=(x+5)(x-6), the x intercepts would be (-5,0) and (6,0).

Quadratic Relations of the Form y = a(x r)(x s)

## The optimal value

To find the optimal value, you to find the axis of symmetry. Once you have that, you put it into to your equation as the value of x and solve.

## The axis of symmetry

To find the axis of symmetry, all you need to do is add the r and s values and divide by 2 as shown in the image above.

## An equation in standard form can be used to solve for:

-The x intercepts

-The axis of symmetry

-The optimal value

A standard form equation can also be converted to vertex form by completing the square.

And from Vertex to Standard by expanding.
Movie on 2014 12 12 at 23 01

## Common Factoring

Used when the numbers have a common factor.

## Simple Trinomial

The factors of c have a sum of b

## Complex Trinomial

Factors of a and c expand to get b. Used with trial and error.
How to Factor Trinomials - Step By Step Tutorial

## Perfect Square

a and b can both be square rooted.
factor perfect square trinomial

## Difference of Squares

The variables a and b can be square rooted, b value must be negative, meaning one factor of b will be positive and one will be negative
Factoring the Difference of Two Squares - Ex 1