# Vertex Form

### Amour Ahmed

## LEARNING GOALS

- We can use the 1st and 2nd differences from a data table to determine if a equation is linear, quadratic or neither.
- We can use a table of values to create a parabola
- We can graph a basic quadratic equation without using technology

## An Intro To Quadratics

## Different Directions of Parabola Parabola's can have different directions depending on the equation. They can either go up or down. | ## Parts Of A Parabola- Vertex
- Y- Intercept
- X-Intercept
- Axis Of Symmetry
| ## What Is A Parabola ?A parabola is the graph of a quadratic equation/function. It an be either in the shape of the letter 'U' or it can be in the shape of the inverted letter of 'U'. |

## Different Directions of Parabola

## Key Features Of Quadratic Relations | ## What Do We Know About Parabolas ?- A parabola can either open up or down.
- The zeros of a parabola is where it crosses the x-axis
- The Axis of symmetry divides the parabola into two equal parts
- The vertex of a parabola is the point where the AOS and the parabola meet.
- The vertex is a y-co-ordinate which is optimal value
- The y-intercept of a parabola is where the graph crosses the y-axis
- The optimal value is the value of the y co-ordinate of the vertex
- The vertex is the point where the parabola is at its maximum or minimum value.
| ## Different Forms and Types of Quadratic EquationsA quadratic equation is any equation having the form ax2 + bx + c = 0. This is where a, b and c are known values. a can't be 0."x" is the variable or unknown (we don't know it yet). If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and are known as the quadratic coefficient, the linear coefficient and the constant. |

## What Do We Know About Parabolas ?

- A parabola can either open up or down.
- The zeros of a parabola is where it crosses the x-axis
- The Axis of symmetry divides the parabola into two equal parts
- The vertex of a parabola is the point where the AOS and the parabola meet.
- The vertex is a y-co-ordinate which is optimal value
- The y-intercept of a parabola is where the graph crosses the y-axis
- The optimal value is the value of the y co-ordinate of the vertex
- The vertex is the point where the parabola is at its maximum or minimum value.

## Different Forms and Types of Quadratic Equations

A quadratic equation is any equation having the form ax2 + bx + c = 0. This is where a, b and c are known values. a can't be 0."x" is the variable or unknown (we don't know it yet). If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and are known as the quadratic coefficient, the linear coefficient and the constant.

## First DifferencesFirst differences are the changes in y-values for consecutive x-values in a table. On this table the first differences | ## Second DifferencesSecond differences are changes in the consecutive first differences. On this table the first differences |

## First Differences

First differences are the changes in y-values for consecutive x-values in a table. On this table the first differences **are constant** by - 3, this shows that the graph will be **Linear**.

## How Can You Tell If A Line Is Linear or Non-Linear (Quadratic) ?

Well....

**Linear Relationships**: Points that follow a straight line are known as linear relationships.

**Quadratic Relationship**: If points follow a curved line (parabola), the relation is quadratic.

Here are examples you can try to guess...

## What Can You Tell From A Vertex Form Equation ?Vertex form gives us the value of the axis of symmetry and the optimal value. The h values gives the axis of symmetry. The value of k gives away the optimal value. Vertex form also gives away the stretch with the factor that represents a. If the value of a is greater than 1, the parabola is vertically stretched. If the value of a is less then 1, then the parabola is horizontally compressed. | ## Example Here's a more realistic example if the other one is a bit complicated. | ## Formula |

## What Can You Tell From A Vertex Form Equation ?

Vertex form gives us the value of the axis of symmetry and the optimal value. The h values gives the axis of symmetry. The value of k gives away the optimal value. Vertex form also gives away the stretch with the factor that represents a. If the value of a is greater than 1, the parabola is vertically stretched. If the value of a is less then 1, then the parabola is horizontally compressed.

## Something Called The Mapping Formula ?!

The general mapping rule for quadratics is

(x,y) → (x + h, ay + k)

The variables a, h and k can be found in the turning point formula: y = a(x - h)² + k

Example: y=4x2+16

To get y = 4x² + 16 to turning point formula, you must complete the square

y = 4x² + 16

Divide each side by 4 to get rid of any factors before x²

y/4 = x² + 4

y/4 = (x + 0)² - 0 + 4

y = 4(x + 0)² + 16

Therefore: a = 4, h = 0, k = 16

Substitute these values into (x,y) → (x + h, ay + k)

(x,y) → (x + 0, 4y + 16)

Therefore: (x,y) → (x, 4y + 16)

## How To Write Quadratic Equations Given The Vertex And A Point

Just as quadratic equations can map a parobla, the parobla points can help write a corresponding quadratic equation. In vertex form, y=a(x-h)^2+k, the variables h and k are coordinats for the parobla's vertex. The x point sign much change though.

First substitute the vertex's coordinates for h and k in the vertex form equation. For example let use a vertex of (2,3). The equation would be y=a(x-2)^2+3.

Then substitute the point's coordinate for x and y in th equation. In this case w can use points (3,8). The new equation would be 8=a(3-2)^2+3.

After solve the equation for a. In this example solving for a would result in a becoming 5.

Lastly subsitute the values of a into the first equation. The final answer should be y=5(x-2)^2+3.