Name That Graph…. Parent Graphs or Base Graphs
Quadratic
Linear
f ( x)  x
f ( x)  x
Absolute Value
Square Root
f ( x)  x
f ( x)  x
Cubic
Exponential
f ( x)  x
2
f ( x)  2
3
Math 30-1
x
1
Identify the function equation that is different from the rest.
yx
2
y  x3
y x
y x
yx
y x3
In mathematics, transformations refer to a manipulation of
the graph of a function or relation such as a translation, a
reflection or a stretch. The result of a transformation can be
called the image graph or image function.
A transformation is indicated in a function equation by
including a parameter in the parent function.
y  af  b  x  h    k
Math 30-1
2
y  f ( x  h)
1.1A Families of Functions
Represents a horizontal translation
h > 0, the translation is to the right h < 0, the translation is to the left
h = 3, shift three units right
 x, y 
  x  3, y 
y  f ( x  3)
Notice
Replacement
y = f (x)
h
y = f (x - h)
h = -2, shift two units left
 x, y    x  2, y 
y  f  x   2  
y  f ( x  2)
y = f (x + h)
h
With a horizontal translation, the domain may be affected,
Math 30-1 the range stays the same. 3
Consider the graph of y = f(x).
Could f(x) be written as
an equation?
Yes, Piecewise function.
Graph y = f(x + 1)
Horizontal shift one unit left
Key Points Mapping
(x, y)
(x-1, y)
(-2, 0)
(-3, 0)
(-1, 3)
(-2, 3)
(2, 3)
(1, 3)
(5, 1)
(4, 1)
Domain of image in Interval Notation
 3, 4
Range of image in Set Builder Notation
 y | 0  y  3
Set Notation assumes Real Numbers
Math 30-1
4
Write the equation of the function after a horizontal translation
of 3 units left.
Function Notation
y  f (x  3)
Specific
yx
2
y  x  3
2
Coordinates in Mapping Notation
(x, y)
(x  3, y)
x is replaced by
(x - 3) in ordered pair
(3,9)
(0,9)
x is replaced by
(x + 3) in function equation
y  x2  2x  1
y   x  3  2  x  3  1
2
y  x 2  8 x  16
Coordinates in Mapping Notation
(x, y)
 3,16 
Math 30-1
(x  3, y)
 0,16 
5
Given f ( x)  2 x  3
Write the equation of the function after a horizontal
translation of 4 units to the right.
x is replaced by (x – 4) in the function
equation
y  f  x  4
y  2 x4 3
Math 30-1
6
Write the equation of the
transformed function, g(x).
Horizontal shift two units right
Replace x with (x – 2) in function equation
y  f ( x)  y  f ( x  2)
Mapping Notation:
 x, y    x  2, y 
Domain in Interval Notation:
Original
Image
 2, 2
0, 4
Math 30-1
7
Textbook p. 12 – 15
Low: 1c, 2bc, 3a,
Medium: 7
High: 12, 18 b
Math 30-1
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1. The graph of f(x – 2) is the graph of f(x) translated…
 2 units up
 2 units left
 2 units down
x 2 units right

2. How does the graph of f(x + 5) compare to the graph of f(x)? The graph will
translate…
 5 units up
x 5 units left

 5 units down
 5 units right
3. In general, the transformation of f(x) → f(x – h) translates the graph…
x h units horizontally

 h units vertically
4. The _______ of the original points are affected.
x x-values

 y–values
Move It
Math 30-1
9
Investigating Vertical Translations
Sketch the basic absolute value graph and list domain and range.
y=|x|
Do : x 
Ra :  y | y  0
Suppose the graph was vertically
translated three units down.
Predict the equation of the image
graph.
Test your prediction using a point on
the original graph, translating three
units down, then verifying in your
equation.
Math 30-1
10
y = | x | Vertically Translated three units down.
Key Points
(x, y)
→
(x, y – 3)
(0, 0)
→ (0, – 3)
(1, 1)
→ (1, – 2)
(-2, 2)
→ (-2, – 1)
 x, y    x, y  3 
 y  3 
f ( x)
y  x 3
Replacement
Math 30-1
11
y  k  f ( x)
1.1A Families of Functions
y  f ( x)  k
Represents a vertical translation
k > 0, the translation is up
k = 4, shift four units up
 x, y 
  x, y  4 
y  4  f ( x)
y  f ( x)  4
y – k = f (x)
k
Notice
Replacement
k < 0, the translation is down
k = -5, shift five units down
 x, y    x, y  5 
y  5  f  x
y  f ( x)  5
y = f (x)
k
y + k = f (x)
With a vertical translation, the domain stays the same,
Math 30-1 the range may change.
12
Horizontal and Vertical Translations
Given the graph of
f(x) = x3, write the
function equation
for each
transformed graph.
y = (x + 3)3
(y - 3) = x 3
y = x3
y = (x - 3)3
(y + 3) = x 3
Math 30-1
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1.1B.7
Horizontal and Vertical Translations
Describe the translation
that has been applied to
the graph of f (x)
to obtain the graph of g(x).
Horizontally translated 4 units right
Vertically translated 9 units down
 x, y    x  4, y  9 
Determine the equation of the
translated function in the form
y - k = f (x - h)
The prime ( ‘ ) is used
to indicate the image
point
y  9  f ( x  4)
Is the order of the transformations important in this situation?
Math 30-1
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Write the Equation of g(x) Using Replacement
The graph of f(x) = | x | is translated 4 units down and 3 units
to the left. Determine the image function equation, g(x).
g(x) = | (x + 3) | - 4
The graph of f(x) = (x - 2)3 - 5 is translated 3 units down and
5 units to the left. Determine the image function equation, h(x).
h(x) = ((x + 5) - 2)3 – 5 - 3
h(x) = (x + 3)3 - 8
Given the functions f(x) = |x – 2| + 3 and g(x) = |x + 2| +1, the
transformations that will transform y = f(x) to become y = g(x)
are a translation of
A. 4 units left and 2 units down
B. 4 units right and 2 units up
C. 1 unit left and 3 units up
D. 2 units left and 4 units down
Math 30-1
15
Consider f (x) = x + 4
Jessica said that the function
f(x) = x has been vertically
translation 4 units up.
Mark said that the function f(x)
has been horizontal translation
4 units to the left.
Who is correct? Explain why.
Math 30-1
16
Textbook p. 12 – 15
Low: 1abde, 2ad, 3, 4,
Medium:5, 6, 8 – 11,
High: 13, 15, 17 – 19
Math 30-1
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