1. PARENT FUNCTIONS
Module 2 Lesson 4

2. 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 graphthesefunctions, wewill usea
tableof values to createpoints onthegraph.

3. Parent Functions we will
explore
xy =
Name Parent Function
Constant Function
f(x) = a numberLinear
f(x) = xAbsolute Value
f(x) = |x|Quadratic
f(x) = x2
Cubic
f(x)= x3
Square Root
Cubic Root
Exponential
x
3
)( xxf =
xxf =)(

4. 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.

5. Examples
Even Functions Odd Functions

6. Parent Functions Menu (must be done from Slide Show
View)
Constant Function Cubic Root
Linear (Identity) Exponential
Absolute Value
Square Root
Quadratic Cubic
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.

7. 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}.
( ),−∞ ∞

8. Example of a
Constant Function
Domain: ( ),−∞ ∞
Range: 2y =
f(x) = 2
x – intercept:
y – intercept: ( )0, 2
None

9. Graphing a
Constant Function
Graph Description: Horizontal Line
x Y
-2 2
-1 2
0 2
1 2
2 2
f(x) = 2

10. 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 .( ),−∞ ∞

11. Example of a
Linear Function
( ),−∞ ∞
f(x) = x + 1
Domain: ( ),−∞ ∞
Range:x – intercept:
y – intercept:
(-1, 0)
(0, 1)

12. x y
-2 -1
-1 0
0 1
1 2
2 3
Graph Description: Diagonal Line
f(x) = x + 1
Graphing a
Linear Function

13. Absolute Value
Function
f(x) = │x│
All absolute value functions are V shaped. They will have one y-
intercept 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.
( ),−∞ ∞

14. Example of an
Absolute Value
Function
f(x) = │x +1│-1
Domain: ( ),−∞ ∞
Range:x – intercepts: (0, 0) & (-2, 0)
y – intercept: ( )0, 0
),1[ ∞−

15. x y
-2 -2
-1 -1
0 0
1 1
2 2
Graph Description: “V” - shaped
f(x) = │x+1 │-1
Graphing an
Absolute Value
Function

16. f(x) = x 2
Quadratic Function
All quadratic functions are U shaped. They will have one y-
intercept 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.
( ),−∞ ∞

17. f(x) = x 2
-1
Example of a
Quadratic Function
Domain: ( ),−∞ ∞
Range: ),1[ ∞−x – intercepts: (-1,0) & (1, 0)
y – intercept: (0, -1)

18. f(x) = x 2
-1
x y
-2 3
-1 0
0 -1
1 0
2 3
Graph Description: “U” - shaped
Graphing a
Quadratic Function

19. 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 .
( ),−∞ ∞

20. Example of a
Cubic Function
f(x) = -x 3
+1
Domain: ( ),−∞ ∞
Range:x – intercept: (1, 0)
y – intercept: (0, 1)
( ),−∞ ∞

21. x y
-2 9
-1 2
0 1
1 0
2 -7
Graph Description: Squiggle, Swivel
f(x) = -x 3
+1
Graphing a
Cubic Function

22. f(x) = x
Square Root
Function
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. .

23. f(x) = x + 2
Example of a
Square Root
Function
Domain:
Range:x – intercept: none
y – intercept: (0, 2) ),0[ ∞
),2[ ∞

24. x y
0 2
1 3
4 4
9 5
16 6
Graph Description: Horizontal ½ of a Parabola
f(x) = x + 2
Graphing a
Square Root
Function

25. Cubic Root Function
3
)( xxf =
All cubic functions are S shaped. They will have 1, 2, or 3 y-
intercepts and can have 1 x-intercept.
The domain and range will be {All Real numbers}, or .( ),−∞ ∞

26. Example of a
Cubic Root Function
3
2)( += xxf
Domain:
Range:x – intercept:
y – intercept: ),( ∞−∞
),( ∞−∞
)2,0( 3
)0,2(−

27. x y
-2 0
0 1.26
1 1
6 2
Graph Description: S shaped
3
2)( += xxf
Graphing a
Cubic Root
Function

28. Exponential Function
( ) x
f x b=
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.
),( ∞−∞

29. f(x) = 2x
Example of an
Exponential Function
Domain:
Range:x – intercept: none
y – intercept: ),( ∞−∞
),0( ∞
)1,0(

30. f(x) = 2x
x y
-2 0.25
-1 0.5
0 1
1 2
2 4
Graph Description: Backwards “L” Curves
Graphing an
Exponential
Function