**Quadratic Functions**

**1.1 Identify Functions**

1.2 Domain and Range

1.3 Analyse Quadratic Functions

1.4 Stretches of Functions

1.5 Translations of Functions

1.6 Sketch Graphs Using Transformations

1.2 Domain and Range

1.3 Analyse Quadratic Functions

1.4 Stretches of Functions

1.5 Translations of Functions

1.6 Sketch Graphs Using Transformations

**1.2 Domain and Range**

The

Domain

of a function is the set of all

x-values

of a function. We represent the Domain by using the variable "

D

"

The

Range

of a function is the set of all

y-values

of a function. We represent the Range by using the variable "

R

"

Signs you need to know

I

or

1.1 - Identify Functions

A

function

is a set of ordered pairs in which no two ordered pairs have the same x-coordinate. This means that

each x can only produce one possible y.

Function can be identified by:

a) A table of values

b) a graph

c) an equation

d) a mapping diagram

y=x^2

a) Table of Values

b) Graph

c) Equation

d) Table of Values

Vertical Line Test

When a graph is given, visualize a

vertical

line moving across the graph. If the vertical line intersects the graph in

more than one point

, than the graph is

not

a function.

Mapping Diagrams

If a mapping diagram represents a function, it is

not possible to have two or more arrows starting at the same x column.

Non- function

Function

To represent functions, we use notations such as

f(x) and g(x).

The notation f(x) is read "f of x". It means that the expression that follows contains

x

as a

variable

.

Ex: f(4) means sub 4 in for every x of the equation and solve for y.

f(x)=(x)^2-4(x)+6

f(4)=(4)^2 - 4(4)+6

Note:

Domain and Range can be found as...

A set of points

(0,7.2),(1,9.0),(2,15.1),(3,25.2), (4,39.6)

A graphed function

A word problem

A clothing store sold between 50 and 75 shirts at 15$ each.

State the domain and range.

**1.3 Analyzing Quadratic Functions**

A quadratic function is a function in the form y=ax^2 +bx+c.

The graph of these functions is called a parabola. In a parabola, a, b and c are real numbers and a can't be equal to 0.

*If a=0, it is a line, not a parabola.

Parabolas have a vertex, axis of symmetry, maximum/minimum value and a direction of opening.

Vertex

The vertex is the point of the parabola with the greatest/smallest y coordinate.

Axis of Symmetry

The vertex of this

parabolais located at (0,0).

The axis of symmetry is a vertical line that goes through the x coordinate of the vertex.

Vertex: (2, 1)

The equation of the axis of symmetry is x=2.

Direction of opening and maximum/minimum value

The direction of opening is the way the parabola opens (up or down).

**1.5 Translations of Functions**

**A**

translation

is a slide of shift, which moves a graph right or left, up or down.

The shape and size of the graph are not affected by the changed translation

translation

is a slide of shift, which moves a graph right or left, up or down.

The shape and size of the graph are not affected by the changed translation

The laws for h

if

h

is

positive

, the parabola is translated horizontally

h

units to the

right.

If h is

negative

, the parabola is translated

h

units to the

left.

*important note*

:

h

flips its signs

(x-h) changes and mean

h

is positive

(x+h) changes and means

h

is negative

y=x^2 + 4

y=(x-3)^2 + 4

The laws for k

If

k

is

positive

, the parabola is vertically translated

up

by

k

units.

If

k

is

negative,

the parabola is vertically translated

down

by

k

units.

y=x^2 - 12

If the parabola opens up, it has a minimum value. If it opens down, it has a maximum value.

You can determine

the direction of opening using a chart and the value of the second differences.

The constant value of the second differences is positive so the parabola opens up.

If the constant value was negative, the parabola would open down.

**1.6 Sketch Graphs using Transformations**

**To sketch a graph using transformations, you can use the step pattern.**

**Start by plotting the vertex and then apply the stretch/compression factor to the step pattern by multiplying by**

a

.

a

.

**The original step pattern is 1, 3, 5. If**

a

is 3, the new step patten will be 3, 9, 15.

a

is 3, the new step patten will be 3, 9, 15.

Tips for graphing: -Vertical stretch/compression first

-Reflection second

-Translations last

**What transformations occurred to the function y=x^2?**

**What is the equation of this transformed function?**

**Find the vertex, axis of symmetry and the direction of opening of the function below.**

Element symbol

**1.4 - Stretches of Functions**

**A stretch function is a transformation that can**

vertically stretched

or

compressed

or

reflected

in the

x-axis

.

vertically stretched

or

compressed

or

reflected

in the

x-axis

.

If "a" is

negative

, the parabola is

reflected in the x-axis

Ex:

If a >1 or a <-1, the parabola is vertically

stretched

by a factor of "a"

Stretch

:

The

blue

parabola has been

stretched into the

red

y=3x^2

y=x^2

Compression:

If a is -1 <a <1, the parabola is vertically

compressed

by a factor of "a"

The

red

parabola has been compressed into the

blue

parabola

y=x^2

y=0.2x^2