Class Notes (810,294)
MATH 140 (141)
Ewa Duma (13)
Lecture 3

# Lecture 3 - 1.3.pdf

2 Pages
83 Views

School
McGill University
Department
Mathematics & Statistics (Sci)
Course
MATH 140
Professor
Ewa Duma
Semester
Summer

Description
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
More Less

Related notes for MATH 140

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.