1. Warm Up:
How does the graph of
compare to ? Sketch both to confirm.

2. Homework Issues???
From Pages # 2ab, 4ac, 5abd, 10ab, 11, 12c

3. McGraw-Hill Ryerson
Pre-Calculus 12
Chapter 1
Function 
Transformations

4. 1.2 Reflections and Stretches 1
Chapter
1.2 Reflections and Stretches 1
Focus On ...
• developing an understanding of the effects
of reflections on the graphs of functions
and their related equations
• developing an understanding of the effects
of vertical and horizontal stretches on the
graphs of functions and their related
equations

10. horizontal

11. Independent Practice: 
Page 28 #1, 2 ( f(x) and h(x) only), 3b, 4bc, 5-8
-Heads up: In-Class Assignment Friday September 16

16. 1.2 Compare the Graphs of y = f(x), y = –f(x), and y = f(–x)
Example 1
Compare the Graphs of y = f(x), y = –f(x), and y = f(–x)
a) Given the graph of y = f(x), graph the functions
y = –f(x) and y = f(–x).
b) How are the graphs of y = –f(x) and y = f(–x)
related to the graph of y = f(x)? 1
a) Use key points on the graph of y = f(x) to create tables of values. 1 2
The image points on the graph of y = –f(x) have the same x-coordinates but 
different y-coordinates. Multiply the y-coordinates of points on the graph of 
y = f(x) by –1. 2
Continue Next Page

17. A B C D E –4 –2 1 3 5 x –4 –2 1 3 5 y = f(x) –3 4 –1(–3) = 3 –1(0) = 0
1.2
Example 1 Continued
Compare the Graphs of y = f(x), y = –f(x), and y = f(–x) 3 3 x
y = f(x) A –4 –3 B –2 C 1 D 3 4 E 5
Graph
Graph
Graph 4 x
y = –f(x) A' –4
–1(–3) = 3 B' –2 C' 1
–1(0) = 0 D' 3
–1(4) = –4 E' 5
–1(–4) = 4 4
Graph
Continue Next Page

18. –4 –2 1 3 5 –3 4 A" –3 B" C" D" 4 E" –4 1.2 x A B C D E x y = f(–x)
Example 1 Continued
Compare the Graphs of y =f(x), y = –f(x), and y = f(–x) 5
The image points on the graph of y = f(–x) have the same y-coordinates 
but different x-coordinates. Multiply the x-coordinates of points on the 
graph of y = f(x) by –1. 5 6 x
y = f(x) A –4 –3 B –2 C 1 D 3 4 E 5 6
Graph
Graph
Graph 7 7 x
y = f(–x)
A"
–1(–4) = 4 –3
B"
–1(–2) = 2
C"
–1(1) = –1
D"
–1(3) = –3 4
E"
–1(5) = –5 –4
Graph
Continue Next Page

19. 1.2 Compare the Graphs of y =f(x), y = –f(x), and y = f(–x)
Example 1 Continued
Compare the Graphs of y =f(x), y = –f(x), and y = f(–x) 8
b) The transformed graphs are congruent to the graph of y = f(x).
The points on the graph of y = f(x) relate to the points on the
graph of y = –f(x) by the mapping (x, y) → (x, –y). The graph
of y = –f(x) is a reflection of the graph of y = f(x) in the x-axis. 8 9
Notice that the point C(1, 0) maps to itself, C'(1, 0).
This point is an invariant point. 9
The points on the graph of y = f(x) relate to the points on the
graph of y = f(–x) by the mapping (x, y) → (–x, y). The graph
of y = f(–x) is a reflection of the graph of y = f(x) in the y-axis. 10 10 11
The point (0, –1) is an invariant point. 11

20. 1.2
Example 1: Your Turn
a) Given the graph of y = f(x), graph the functions y = –f(x) and
y = f(–x).
b) Show the mapping of key points on the graph of y = f(x) to image
points on the graphs of y = –f(x) and y = f(–x).
c) Describe how the graphs of y = –f(x) and y = f(–x) are related to
the graph of y = f(x). State any invariant points.
Answer
c) The functions are mirrors of the original
over the x-axis and y-axis.
The first function has an invariant point
at (–1, 0). The second function has an
invariant point at (0, 2). a)
b) Example points.

21. x 4 2 1.2 8 –6 –2 5 y = f(x) y = g(x) = 2f(x)
Example 2
Graph y = af(x)
Given the graph of y = f(x),
• transform the graph of f(x) to sketch
the graph of g(x)
• describe the transformation
• state any invariant points
• state the domain and range of the
functions
a) g(x) = 2f(x) b) 1
a) Use key points on the graph of y = f(x) to create a table of values. 1 2 2
The image points on the graph of g(x) = 2f(x) have the same
x-coordinates but different y-coordinates. Multiply the y-coordinates
of points on the graph of y = f(x) by 2.
y = g(x)
y = f(x) 3 3 x
y = f(x)
y = g(x) = 2f(x) –6 4 8 –2 2 5
y = f(x)
y = g(x)
Continue Next Page

22. 1.2
Example 2 Continued
Graph y = af(x)
a) g(x) = 2f(x) 4
Since a = 2, the points on the graph of y = g(x) relate to the points
on the graph of y = f(x) by the mapping (x, y) → (x, 2y). Therefore,
each point on the graph of g(x) is twice as far from the x-axis as the
corresponding point on the graph of f(x). The graph of g(x) = 2f(x) is
a vertical stretch of the graph of y = f(x) about the x-axis by a factor
of 2. 4 5
The invariant points are (–2, 0) and (2, 0). 5 6
For f(x), the domain is {x | ­–6 ≤ x ≤ 6, x ∈ R}, or [–6, 6],
and the range is {y | 0 ≤ y ≤ 4, y ∈ R}, or [0, 4]. 6
For g(x), the domain is {x | –6 ≤ x ≤ 6, x ∈ R}, or [–6, 6],
and the range is {y | 0 ≤ y ≤ 8, y ∈ R}, or [0, 8]. 7 7 7
Continue Next Page

23. x –6 –2 5 y = f(x) 4 1.2 y = g(x) = f(x) 2 1 b)
Example 2 Continued 8
b) The image points on the graph of have the same .
of points on the graph of y = f (x) by
x-coordinates but different y-coordinates. Multiply the y-coordinates
Graph y = af(x) b) 8 9 9 x
y = f(x)
y = g(x) = f(x) –6 4 2 –2 1 5 1
y = g(x)
y = f(x) 2
y = f(x)
y = g(x)
Continue Next Page

24. 1.2
Example 2 Continued
Since a = , the points on the graph of y = g(x) relate to the points
on the graph of y = f(x) by the mapping (x, y) → (x, y). Therefore,
each point on the graph of g(x) is one half as far from the x-axis as the
corresponding point on the graph of f(x). The graph of g(x) = f(x) is
a vertical stretch of the graph of y = f(x) about the x-axis by a factor
of . 10
Graph y = af(x) a) 10 11
The invariant points are (­–2, 0) and (2, 0). 11 11 12
For f(x), the domain is {x | ­–6 ≤ x ≤ 6, x ∈ R}, or [­–6, 6],
and the range is {y | 0 ≤ y ≤ 4, y ∈ R}, or [0, 4]. 12
For g(x), the domain is {x | ­–6 ≤ x ≤ 6, x ∈ R}, or [­–6, 6],
and the range is {y | 0 ≤ y ≤ 2, y ∈ R}, or [0, 2]. 13 13 13

25. Example 2: Your Turn 1.2 Given the function f(x) = x2,
Answer a) a)
The transformation is a vertical stretch 

of 4. The domain is {x | x ∈ R}. The 
range is {y | y ≥ 0, y ∈ R}. There is an 
invariant point at the origin.
Example 2: Your Turn
Given the function f(x) = x2,
• transform the graph of f(x) to sketch the graph of g(x)
• describe the transformation
• state any invariant points
• state the domain and range of the functions
a) g(x) = 4f(x)
b) g(x) = f(x)
Answer b) b)
The transformation is a vertical stretch 

of . The domain is {x | x ∈ R}. The 
range is {y | y ≥ 0, y ∈ R}. There is an 
invariant point at the vertex (0, 0).

26. 1.2 Graph y = f(bx) Given the graph of y = f(x),
Example 3
Graph y = f(bx)
Given the graph of y = f(x),
• transform the graph of f(x) to sketch
the graph of g(x)
• describe the transformation
• state any invariant points
• state the domain and range
of the functions
a) g(x) = f(2x)
b) g(x) = f( x) 2 1 1
a) Use key points on the graph of y = f(x) to create a table of values. 1 2
The image points on the graph of g(x) = f(2x) have the same
y-coordinates but different x-coordinates. Multiply the x-coordinates
of points on the graph of y = f(x) by . . 2
Continue Next Page

27. x –4 –2 x –2 –1 1 y = f(x) 4 2 y = f(2x) 4 2 1.2 Graph y = f (bx)
Example 3 Continued
Graph y = f (bx)
a) g(x) = f(2x) 3 3 x
y = f(x) –4 4 –2 2
y = f(x)
y = f(x)
y =g(x) 4 4 x
y = f(2x) –2 4 –1 2 1
y =g(x)
Continue Next Page

28. 1.2
Example 3 Continued
Since b = 2, the points on the graph of y = g(x) relate to the points
on the graph of y = f(x) by the mapping (x, y) → ( x, y). Therefore,
each point on the graph of g(x) is one half as far from the y-axis as the
corresponding point on the graph of f(x). The graph of g(x) = f(2x) is
a horizontal stretch of the graph of y = f(x) about the y-axis by a factor
of . 5
Graph y = f(bx)
a) g(x) = f(2x) 5
The invariant point is (0, 2). 6 6 7
For f(x), the domain is {x | ­–4 ≤ x ≤ 4, x ∈ R}, or [­–4, 4],
and the range is {y | 0 ≤ y ≤ 4, y ∈ R}, or [0, 4]. 7
For g(x), the domain is {x | ­–2 ≤ x ≤ 2, x ∈ R}, or [­–2, 2],
and the range is {y | 0 ≤ y ≤ 4, y ∈ R}, or [0, 4]. 4 8 4 8
Continue Next Page

29. b)
1.2
Example 3 Continued
b) The image points on the graph of have the same .
y-coordinates but different x-coordinates. Multiply the x-coordinates
of points on the graph of y = f(x) by 2. 9
Graph y = af(x) 9
y = g(x)
y = f(x) 10 10 x
y = f(x) –6 4 –2 2 5 2 1
y = g(x) = f( x)
y = f(x)
y = g(x)
Continue Next Page

30. 1.2 Graph y = f(bx) a) g(x) = f( x)
Example 3 Continued
Graph y = f(bx)
a) g(x) = f( x) 2 1
Since b = ½, the points on the graph of y = g(x) relate to the points
on the graph of y = f(x) by the mapping (x, y) → (2x, y). Therefore,
each point on the graph of g(x) is twice as far from the y-axis as the
corresponding point on the graph of f(x). The graph of g(x) = f(½x) is
a horizontal stretch of the graph of y = f(x) about the y-axis by a factor
of 2. 11 11
The invariant point is (0, 2). 12 12
For f(x), the domain is {x | ­–4 ≤ x ≤ 4, x ∈ R}, or [­–4, 4],
and the range is {y | 0 ≤ y ≤ 4, y ∈ R}, or [0, 4]. 13 13 14
For g(x), the domain is {x | ­–8 ≤ x ≤ 8, x ∈ R}, or [­–8, 8],
and the range is {y | 0 ≤ y ≤ 4, y ∈ R}, or [0, 4]. 14

31. Example 3: Your Turn 1.2 Given the function f(x) = x2,
Answer a)
stretch of . The domain is {x | x ∈ R}. 3 1
The range is {y | y ≥ 0, y ∈ R}.
The transformation is a horizontal a)
Example 3: Your Turn
Given the function f(x) = x2,
• transform the graph of f(x) to sketch the graph of g(x)
• describe the transformation
• state any invariant points
• state the domain and range of the functions
a) g(x) = f(3x)
b) g(x) = f( x) 4 1
Answer b)
The transformation is a horizontal 

stretch of 4. The domain is {x | x ∈ R}. 
The range is {y | y ≥ 0, y ∈ R}. b)

32. 1.2 Write the Equation of a Transformed Function
Example 4
Write the Equation of a Transformed Function
The graph of the function y = f(x) has been transformed by either a
stretch or a reflection. Write the equation of the transformed graph, g(x). 1
a) Notice that the V-shape has changed, so the graph has been
transformed by a stretch.
Since the original function is f(x) = |x|, a stretch can be
described in two ways. 1
Choose key points on the graph of y = f(x) and determine their
image points on the graph of the transformed function, g(x). 2 2
Continue Next Page

33. 1.2
Example 4 Continued 3 3 3 3
Case 1
Check for a pattern in the y-coordinates. x
y = f(x)
y = g(x) –6 6 18 –4 4 12 –2 2
The transformation can be described by the mapping (x, y) → (x, 3y).
This is of the form y = af(x), indicating that there is a vertical stretch
about the x-axis by a factor of 3. The equation of the transformed
function is g(x) = 3f(x) or g(x) = 3|x|. 4 4
Continue Next Page

34. 1.2
Example 4 Continued
Case 2
Check for a pattern in the x-coordinates. 5 5 5 6 6 x
y = f(x)
–12 12 –6 6 x
y = g(x) –4 12 –2 6 2 4 7
The transformation can be described by the mapping (x, y) → ( x, y) .
This is of the form y = f(bx), indicating that there is a horizontal
stretch about the y-axis by a factor of . The equation of the
transformed function is g(x) = f(3x) or g(x) = |3x|. 7
Continue Next Page

35. 1.2
Example 4 Continued
b) Notice that the shape of the graph has not changed, so the graph has
been transformed by a reflection.
Choose key points on the graph of f(x) = |x| and determine their
image points on the graph of the transformed function, g(x). 8 8 9 9 x
y = f(x)
y = g(x) –4 4 –2 2
The transformation can be described by the mapping (x, y) → (x, –y).
This is of the form y = –f(x), indicating a reflection in the x-axis. The
equation of the transformed function is g(x) = –|x|. 10 10

36. Example 4: Your Turn 1.2 g(x) = 4x2 or g(x) = (2x)2
The graph of the function y = f(x) has been transformed.
Write the equation of the transformed graph, g(x).
Answer
g(x) = 4x2 or g(x) = (2x)2