Transcript Module 2 Lesson 4 Notes

PARENT FUNCTIONS
Module 2 Lesson 4
What is a parent function?
We use the term ‘parent function’ to describe a family of
graphs. The parent function gives a graph all of the unique
properties and then we use transformations to move the graph
around the plane.
You have already seen this with lines. The parent function for a lines
is y = x. To move the line around the Cartesian plane, we change the
coefficient of x (the slope) and add or subtract a constant (the y
intercept), to create the family of lines, or y = mx + b.
To graph these functions, we will use a
table of values to create points on the graph.
Parent Functions we will explore
Name
Parent Function
Constant Function
f(x) = a number
Linear
f(x) = x
Absolute Value
f(x) = |x|
Quadratic
f(x) = x2
Cubic
y x
f(x)= x3
Square Root
f ( x)  x
Cubic Root
f ( x)  3 x
Exponential
f(x) = bx where b is a rational number
Symmetry
Many parent functions have a form of symmetry.
There are two types of symmetry:
 Symmetry over the y-axis
 Symmetry over the origin.
If a function is symmetric over the y-axis, then it is an
even function.
If a function is symmetric over the origin, then it is an odd
function.
Examples
Even Functions
Odd Functions
Parent Functions Menu
(must be done from Slide Show View)
1) Click on this Button to View All of the Parent Functions.
or
2) Click on a Specific Function Below for the Properties.
3) Use the
button to return to this menu.
Constant Function
Linear (Identity)
Absolute Value
Quadratic
Cubic Root
Exponential
Square Root
Cubic
Constant Function
f(x) = a
where a = any #
All constant functions are line with a slope equal to 0. They will
have one y-intercept and either NO x-intercepts or they will be the
x-axis.
The domain will be {All Real numbers}, also written as  ,   or
from negative infinity to positive infinity. The range will be {a}.
Example of a
Constant Function
f(x) = 2
y – intercept:  0, 2 
Domain:
x – intercept: None
Range:
 , 
y2
Graphing a
Constant Function
f(x) = 2
x
-2
-1
0
1
2
Y
2
2
2
2
2
Graph Description: Horizontal Line
Linear Function
f(x) = x
All linear functions are lines. They will have one y-intercept and
one x-intercept.
The domain and range will be {All Real numbers}, or
 ,  .
Example of a
Linear Function
f(x) = x + 1
y – intercept: (0, 1)
Domain:
x – intercept: (-1, 0) Range:
 , 
 , 
Graphing a
Linear Function
f(x) = x + 1
x y
-2 -1
-1 0
0 1
1 2
2 3
Graph Description: Diagonal Line
Absolute Value
Function
f(x) = │x│
All absolute value functions are V shaped. They will have one yintercept and can have 0, 1, or 2 x-intercepts.
The domain will be {All Real numbers}, also written as  ,   or
from negative infinity to positive infinity. The range will begin at the
vertex and then go to positive infinity or negative infinity.
Example of an
Absolute Value
Function
f(x) = │x +1│-1
y – intercept:  0, 0
Domain:  ,  
x – intercepts: (0, 0) & (-2, 0)
Range:
[1, )
Graphing an
Absolute Value
Function
f(x) = │x+1 │-1
x
-2
-1
0
1
2
y
-2
-1
0
1
2
Graph Description: “V” - shaped
Quadratic Function
f(x) = x 2
All quadratic functions are U shaped. They will have one yintercept and can have 0, 1, or 2 x-intercepts.
The domain will be {All Real numbers}, also written as  ,   or
from negative infinity to positive infinity. The range will begin at the
vertex and then go to positive infinity or negative infinity.
Example of a
Quadratic Function
f(x) = x 2 -1
y – intercept: (0, -1)
Domain:  ,  
x – intercepts: (-1,0) & (1, 0)
Range: [1, )
Graphing a
Quadratic Function
f(x) = x 2 -1
x
y
-2
3
-1
0
0
-1
1
0
2
3
Graph Description: “U” - shaped
Cubic Function
f(x) = x 3
All cubic functions are S shaped. They will have one y-intercept
and can have 0, 1, 2, or 3 x-intercepts.
, 

The domain and range will be {All Real numbers}, or
.
Example of a
Cubic Function
f(x) = -x 3 +1
y – intercept: (0, 1)
Domain:  ,  
x – intercept: (1, 0)
Range:
 , 
Graphing a
Cubic Function
f(x) = -x 3 +1
x
-2
-1
0
1
2
y
9
2
1
0
-7
Graph Description: Squiggle, Swivel
Square Root
Function
f(x) = x
All square root functions are shaped like half of a U. They can
have 1 or no y-intercepts and can have 1 or no x-intercepts.
The domain and range will be from the vertex to an infinity.
.
Example of a
Square Root
Function
f(x) = x + 2
y – intercept: (0, 2)
Domain: [0, )
x – intercept: none
Range:
[2, )
Graphing a
Square Root
Function
f(x) = x + 2
x
0
1
4
9
16
y
2
3
4
5
6
Graph Description: Horizontal ½ of a Parabola
Cubic Root Function
f ( x)  3 x
All cubic functions are S shaped. They will have 1, 2, or 3 yintercepts and can have 1 x-intercept.
The domain and range will be {All Real numbers}, or  ,  .
Example of a
Cubic Root Function
f ( x)  3 x  2
y – intercept: (0, 3 2 )
Domain: (, )
x – intercept: (2,0)
Range: (, )
Graphing a
Cubic Root
Function
f ( x)  3 x  2
x
-2
0
1
6
y
0
1.26
1
2
Graph Description: S shaped
Exponential Function
f ( x)  b
x
Where b = rational number
All exponential functions are boomerang shaped. They will
have 1 y-intercept and can have no or 1 x-intercept. They will also
have an asymptote.
The domain will always be (, ) , but the range will vary
depending on where the curve is on the graph, but will always go
an infinity.
Example of an
Exponential Function
f(x) = 2x
y – intercept: (0,1)
Domain: (, )
x – intercept: none
Range:
(0, )
Graphing an
Exponential
Function
f(x) = 2x
x
-2
y
0.25
-1
0
0.5
1
1
2
2
4
Graph Description: Backwards “L” Curves