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 graph these functions, we will use a table of values to create points on the graph.

3. 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
f(x)= x3
Square Root
Cubic Root
Exponential
f(x) = bx where b is a rational number

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)
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 Cubic Root Linear (Identity) Exponential Absolute Value Square Root Quadratic Cubic

7. f(x) = a where a = any # Constant Function
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. Domain: Range: f(x) = 2 y – intercept: x – intercept: None
Example of a
Constant Function
Domain:
Range:
f(x) = 2
y – intercept:
x – intercept:
None

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

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

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

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

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

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

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

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

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

22. 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

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

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

25. Cubic Root Function
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. y – intercept: Domain: x – intercept: Range: Example of a
Cubic Root Function
y – intercept:
Domain:
x – intercept:
Range:

27. x y -2 1.26 1 6 2 Graph Description: S shaped Graphing a
Cubic Root Function x y -2
1.26 1 6 2
Graph Description: S shaped

28. Exponential Function
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. y – intercept: Domain: x – intercept: none Range: Example of an
Exponential Function
f(x) = 2x
y – intercept:
Domain:
x – intercept: none
Range:

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