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04/21/23 1-9 Parent Functions 1
warm_up #5
• How do you think you did on the last test?
• What parts did you do well in?
• What parts could you have improved upon?
Grade Distribution
1st 3rd 7th A 7 4 6B 16 12 13C 4 6 4D 3 2 4F 0 5 3
No Show 1 2 1Avg 83.64 77.89 80.31
04/21/23 1-9 Parent Functions 3
Introduction to Parent Functions
Section 1-9
04/21/23 1-9 Parent Functions 4
What is a parent function?
• The parent function is the simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function.
04/21/23 1-9 Parent Functions 5
Parent Functions
(–∞, ∞)
[0, ∞)
(0, 0)
y-axis
Constant
f(x) = c
Family
Rule
Graph
Domain
Range
(–∞, ∞)
C
None
y-axis
(–∞, ∞)
(–∞, ∞)
(0, 0)
Origin
(–∞, ∞)
(–∞, ∞)
(0, 0)
Origin
Linear Quadratic Cubic
f(x) = x f(x) = x2 f(x) = x3
Zeros
Symmetry
|x x |x x |x x |x x
|y y c |y y | 0y y |y y
04/21/23 1-9 Parent Functions 6
Parent Functions
Reciprocal
f(x) =
Family
Rule
Graph
Domain
Range
Zeros
Symmetry
1
x
[0, ∞)
[0, ∞)
(0, 0)
None
(–∞, ∞)
[0, ∞)
(0, 0)
y-axis
Squ Root Abs Value
f(x) = √x f(x) = |x|
(–∞, 0) U (0, ∞)
(–∞, 0) U (0, ∞)
None
Origin
| 0x x |x x | 0x x
| 0y y | 0y y | 0y y
04/21/23 1-9 Parent Functions 7
Get some exercise
04/21/23 1-9 Parent Functions 8
Example 1
Identify the parent function for g from its function rule. Graph g(x) on your calculator and describe what transformation of the parent function it represents. Then, identify the domain and range of the transformation.
( ) 3g x x
Linear Function, Down 3
Domain: (–∞, ∞)Range: (–∞, ∞)
04/21/23 1-9 Parent Functions 9
Example 2
Identify the parent function for g from its function rule. Graph g(x) on your calculator and describe what transformation of the parent function it represents. Then, identify the domain and range of the transformation.
2( ) 2g x x
Quadratic Function, Shrinks with scale of 2 OR Horizontal Compression of 1/2
Domain: (–∞, ∞)Range: [0, ∞)
04/21/23 1-9 Parent Functions 10
Example 3
Identify the parent function for g from its function rule. Graph g(x) on your calculator and describe what transformation of the parent function it represents. Then, identify the domain and range of the transformation.
2( )g x x
Quadratic Function, Reflection
Domain: (–∞, ∞)Range: (–∞, 0]
04/21/23 1-9 Parent Functions 11
Your Turn
Identify the parent function for g from its function rule. Graph g(x) on your calculator and describe what transformation of the parent function it represents. Then, identify the domain and range of the transformation.
31( ) 2
2g x x
Cubic Function, Moves 2 units to the Right, Grows by a scale of 1/(Hor.
Stretch by 2 )
Domain: (–∞, ∞)Range: (–∞, ∞)
04/21/23 1-9 Parent Functions 12
Example 4
Identify the parent function for g from its function rule. Graph g(x) on your calculator and describe what transformation of the parent function it represents. Then, identify the domain and range of the transformation.
( ) 4g x xSquare Root
Function, Reflection on x-axis,
Vertical Stretch
Domain: [0, ∞)Range: (–∞, 0]
04/21/23 1-9 Parent Functions 13
Example 5
Determine from the data from this set of ordered pairs, describe the parent function, and the transformation that best approximates the data set.
x y
–2 2
–1 –1
0 –2
1 –1
2 2
Determine the equation and the slope
A) Graph it
B) If the ‘y’ difference and if ‘x’ are consistent:
…differs 1 time: LINEAR
…differs 2 times: QUADRATIC
…differs 3 times: CUBIC
More then two times, it can be EXPONENTIAL, CUBIC, or SQUARE ROOT
04/21/23 1-9 Parent Functions 14
Example 5
Determine from the data from this set of ordered pairs, describe the parent function, and the transformation that best approximates the data set.
2 ( 1) 3
x y
–2 2
–1 –1
0 –2
1 –1
2 2
( 1) ( 2) 1
2 ( 1) 1
( 1) 2 3
Linear
3 (1) 2
(1) ( 1) 2
1 ( 3) 2
Quadratic
Shift of Vertical down shift of 2
04/21/23 1-9 Parent Functions 15
Example 6
Determine from the data from this set of ordered pairs, describe the parent function, and the transformation that best approximates the data set.
x y
–2 –6
–1 1
0 2
1 3
2 10Cubic, Vertical Shift of 2
04/21/23 1-9 Parent Functions 16
Your Turn
Determine from the data from this set of ordered pairs, describe the parent function, and the transformation that best approximates the data set.
x y
0 0
1 1
2 1.414
3 1.732
4 2Square Root, No Shift
04/21/23 1-9 Parent Functions 17
Assignment
Pg 71
3-27 odd, 39A-D
Know the Parent Function Chart