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Lecture 3

# Lecture 3 - 1.3.pdf

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McGill University

Mathematics & Statistics (Sci)

MATH 140

Ewa Duma

Summer

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MATH140 - Lecture 3 Notes
Transformation of Functions:
By applying certain transformations to the graph of a given function we can obtain the
graphs of certain related functions. This will give us the ability to sketch the graphs of
many functions quickly by hand.
Example 1: Vertical and Horizontal Shifts
Suppose that c > 0. To obtain the graph of:
• y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) - c, shift the graph of y = f(x) a distance c units downward
•
• y = f(x - c), shift the graph of y = f(x) a distance c units to the right
38 |||| CHAPTER 1 FUNCTIONS AND MODELS
• y = f(x+ c), shift the graph of y = f(x) a distance c units to the left
y y
y=ƒ+c y=cƒ
(c>1)
y=f(x+c) c y =ƒ y=f(x-c) y=f(_x)
y=ƒ
c c y= ƒ
c
0 c x 0 x
y=ƒ-c
y=_ƒ
FIGURE 1 FIGURE 2
In order to consider stretching and reﬂecting transformation, we have to consider c > 1,ting the graph of ƒ
the the graph of y = cf(x) is the graph of y = f(x) stretched by a factor of c in the vertical
direction. The graph of y = -f(x) is the graph of y = f(x) reﬂected about the x-axisng transformations. If c ▯ 1, then the
graph of y ▯ cf▯x▯ is the graph of y ▯ f▯x▯ stretched by a factor of c in the vertical
because the point (x, y) is replaceddirection (because each y-coordinate is multiplied by the same number c). The graph of
y ▯ ▯f▯x▯ is the graph of y ▯ f▯x▯ reﬂected about the x-axis because the point ▯x, y▯ is
Example 2: Vertical and Horizontal Sreplaced by the point ▯x, ▯y▯. (See Figure 2 and the following chart, where the results of
other stretching, compressing, and reﬂecting transformations are also given.)
Suppose that x > 1. To obtain the graph of:
VERTICAL AND HORIZONTAL STRETCHING AND REFLECTING Suppose c ▯ 1. To
• y = cf(x), stretch the graph of y = f(x) vertically by a factor of c
y ▯ cf▯x▯, stretch the graph of y ▯ f▯x▯ vertically by a factor of c
• y = (1/c)f(x), compress the graph of y = f(x) vertically by a factor of c
• y = f(cx), compress the graph of y = f(x) horizontally by a factor of c of y ▯ f▯x▯ vertically by a factor of c
• y = f(x/c), stretch the graph of y = f(x) horizontally by a factor of cy ▯ f▯x▯ horizontally by a factor of c
y ▯ f▯x▯c▯, stretch the graph of y ▯ f▯x▯ horizontally by a factor of c
y ▯ ▯f▯x▯, reflect the graph of y ▯ f▯x▯ about the x-axis
y ▯ f▯▯x▯, reflect the graph of y ▯ f▯x▯ about the y-axis
Figure 3 illustrates these stretching transformations when applied to the cosine function 42 |||| CHAPTER 1 FUNCTIONS AND MODELS
EXAMPLE 6 Iff▯x▯ ▯ x2 and t▯x▯ ▯ x ▯ 3, ﬁnd the composite functionf ▯ t
and t ▯ f
SOLUTIONWe have
• y = -f(x), reﬂec▯f ▯ t▯▯x▯ ▯ f▯t▯x▯▯ ▯ f▯x ▯ 3▯ ▯ ▯x ▯ 3▯
38 |||| CHAPTER 1 FUNCTIONS AND MODELS • y = f(-x), reﬂect the graph of y = f(x) about the y-axis
2 2
▯t ▯ f▯▯x▯ ▯ t▯f▯x▯▯ ▯ t▯x ▯ ▯ x ▯ 3
y y
| NOTE You can see from Example 6 that, in general, f ▯ t ▯ t ▯ fRemember, the
y=ƒ+c notation f ▯ teans that the functiont=is applied ﬁrst and thef is applied second. In
Example 6, f ▯ ts the function that ﬁrst subtracts 3 and then sqt ▯ fis the function
that ﬁrst squares and then subtracts 3.
y=f(_x)
y=f(x+c) c y =ƒ y=f(x-c) V EXAMPLE 7 Iff▯x▯ ▯ s x and t▯x▯ ▯s 2 ▯ x , ﬁnd each function and its domain.
(a) f ▯ t

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