Parent Functions and
Transformations
Section 1.1 beginning on page 3
The Big Ideas
In this section we will learn about….
o Families of functions
• A family of functions is a group of functions that share the same key
characteristics.
• The parent function is the most basic function in the family.
o Transformations
• Functions in the same family are transformations of the parent function.
• Changes to the parent function create specific changes to the graph of the
function, these changes are consistent through the different families of
functions.
Core Vocabulary
Previously Learned:
New:
Function
Parent Function
Domain
Transformation
Range
Translation
Slope
Reflection
Scatter Plot
Vertical Stretch
Vertical Shrink
Basic Parent Functions
Constant:
Linear:
Parent Function:
𝑓 𝑥 =1
Absolute Value:
Parent Function:
𝑓 𝑥 = 𝑥
Domain:
All Real Numbers
Domain:
All Real Numbers
Range:
𝑦=1
Range:
𝑦≥0
Parent Function:
𝑓 𝑥 =𝑥
Quadratic:
Parent Function:
𝑓 𝑥 = 𝑥2
Domain:
All Real Numbers
Domain:
All Real Numbers
Range:
All Real Numbers
Range:
𝑦≥0
Identifying a Function Family
Example 1: Identify the function family to which 𝑓 belongs. Compare the graph of
𝑓 to the graph of its parent function.
The graph of 𝑓 is V shaped, so
𝑓 is an absolute value function.
The graph of 𝑓 is shifted up
and is narrower than the
graph of the parent function.
The domain of 𝑓 is all real
numbers. (same as the parent
function)
𝑔 𝑥 = 𝑥
𝑓 𝑥 =2 𝑥 +2
The range of 𝒇 is 𝒚 ≥ 𝟏. (the
range of the parent function
is 𝑦 ≥ 0)
Identifying a Function Family
Example 1: Identify the function family to which 𝑓 belongs. Compare the graph of
𝑓 to the graph of its parent function.
The graph of 𝑓 is U shaped, so
𝑓 is a quadratic function.
The graph of 𝑓 is shifted right
and is widerthan the graph of
the parent function.
The domain of 𝑓 is all real
numbers. (same as the parent
function)
𝑔 𝑥 = 𝑥2
𝑓 𝑥 =
1
(𝑥 − 3)2
4
The range of 𝒇 is 𝒚 ≥ 𝟎.
(same as the parent function)
Describing Transformations
A transformation changes the size, shape, position, or orientation of a graph.
A translation is a transformation that shifts a graph horizontally and/or vertically but
does not chance its size, shape, or orientation.
A reflection is a transformation that flips a graph over a line called the line of reflection.
A reflected point is the same distance from the line of reflection as the original point
but on the opposite side.
Another way to transform a function is to multiply all of the y-coordinates by the same
positive factor (other than 1). When the factor is greater than 1, the transformation is a
vertical stretch. When the factor is greater than 0 and less than 1, it is a vertical shrink
(also known as a vertical compression).
*** we will learn about horizontal stretches and shrinks in the next section.
Describing Transformations
Example 2: Graph 𝑔 𝑥 = 𝑥 − 4 and its parent function. Then describe the transformation.
The graph of 𝑔 is a linear
function.
𝑓 𝑥 =𝑥
The graph of 𝑔 is 4 units
below the graph of the parent
linear function 𝑓.
The graph of 𝑔 𝑥 = 𝑥 − 4 is
a vertical translation 4 units
down of the graph of the
parent linear function.
Describing Transformations
Example 3: Graph 𝑔 𝑥 = −𝑥 2 and its parent function. Then describe the transformation.
The graph of 𝑔 is a quadratic function. 𝑓 𝑥 = 𝑥 2
Use a table of values to graph each function:
𝒙
𝒚 = 𝒙𝟐
𝒚 = −𝒙𝟐
-2
4
−4
-1
0
1
0
−1
0
1
1
−1
2
4
−4
The graph of g is the graph of the parent function
flipped over the x-axis
The graph of 𝑔 𝑥 = −𝑥 2 is a reflection in the x-axis of the graph of the parent quadratic function.
Graphing and Describing Stretches and
Shrinks
Example 4 a : Graph 𝑔 𝑥 = 2 𝑥 and its parent function. Then describe the transformation.
The graph of 𝑔 is an absolute value function.
𝑓 𝑥 = 𝑥
Use a table of values to graph each function:
𝒙
𝒚= 𝒙
𝒚=𝟐𝒙
-2
2
4
-1
0
1
0
2
0
1
1
2
2
2
4
The y-coordinate of each point on g is two times the y-coordinate of the corresponding point on the
parent function.
The graph of 𝑔 𝑥 = 2 𝑥 is a vertical stretch of the graph of the parent absolute value function.
Graphing and Describing Stretches and
Shrinks
1
Example 4 b : Graph 𝑔 𝑥 = 2 𝑥 2 and its parent function. Then describe the transformation.
The graph of 𝑔 is a quadratic function.
𝑓 𝑥 = 𝑥2
Use a table of values to graph each function:
𝒙
𝒚 = 𝒙𝟐
-2
4
𝟏 𝟐
𝒙
𝟐
2
-1
0
1
0
0.5
0
1
1
0.5
2
4
2
𝒚=
The y-coordinate of each point on g is one-half of the y-coordinate of the corresponding point on the
parent function.
1
The graph of 𝑔 𝑥 = 2 𝑥 2 is a vertical shrink of the graph of the parent quadratic function.
Combinations of Transformations
Example 5: Use a graphing calculator to graph 𝑔 𝑥 = − 𝑥 + 5 − 3 and its parent
function. Then describe the transformations.
The function g is an absolute
value function.
The graph shows that
𝑔 𝑥 = − 𝑥 + 5 − 3 is a
reflection in the x-axis …
… followed by a translation 5
units left and 3 units down of
the graph of the parent absolute
value function.
Combinations of Transformations
Use a graphing calculator to graph the function and its parent function. Then describe
the transformation.
1
8) ℎ 𝑥 = 4 𝑥 + 5
The function h is a
linear function.
The graph shows that
h is a reflection in the
x-axis followed by a
vertical translation 5
units up and a vertical
shrink.
Combinations of Transformations
Use a graphing calculator to graph the function and its parent function. Then describe
the transformation.
9) 𝑑 𝑥 = 3(𝑥 − 5)2 −1
The function d is a
quadratic function.
The graph shows that
h is translation 5 units
right and 1 unit down,
and a vertical stretch.
Modeling With Mathematics
Example 6: The table shows the height y of a dirt bike x seconds after jumping off a
ramp. What type of function can you use to model the data? Estimate the height
after 1.75 seconds.
You can model this data
Time
Height
with a quadratic function.
(seconds), (feet), y
x
0
8
0.5
20
1
24
1.5
20
2
8
The graph shows that the height is
about 15 feet after 1.75 seconds.
𝑥 = 1.75
(1.75, 15)
Modeling With Mathematics
The table shows the amount of fuel in a chainsaw over time. What time of function
can you sue to model the data? When with the tank be empty?
Time (minutes), x
0
Fuel remaining (fluid ounces), y 15
10
20
30
40
12
9
6
3
The tank will be
empty when y=0.
The tank will be empty
after 50 minutes.
You can model the data
with a linear function.
Use the graph to
predict the value of x.
You may have also
detected a pattern in
the table of values to
help you predict the
value of x.