2. Preliminaries 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2
0.1
THE REAL NUMBERS AND THE CARTESIAN PLANE
0.2
LINES AND FUNCTIONS
0.3
GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS
0.4
TRIGONOMETRIC FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
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3. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.1
Suppose that f and g are functions with domains D1 and D2, respectively. The functions f + g, f − g and f · g are defined by
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4. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.1
Combinations of Functions
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5. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.1
Combinations of Functions
The domain of f is the entire real line and the domain of g is the set of all x ≥ 1.
The domain of both ( f + g) and (3 f − g) is
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6. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.1
Combinations of Functions
The domain of f is the entire real line and the domain of g is the set of all x ≥ 1.
The domain is
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7. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.2
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8. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.2
Finding the Composition of Two Functions
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9. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.2
Finding the Composition of Two Functions
Note that for x to be in the domain of g, we must have
x ≥ 2. The domain of f is the whole real line, so this places no further restrictions on the domain of f ◦ g.
The domain of ( f ◦ g) is
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10. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.2
Finding the Composition of Two Functions
The resulting square root requires x2 − 1 ≥ 0 or |x| ≥ 1. Since the “inside” function f is defined for all x, the domain of g ◦ f is
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11. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.4
Vertical Translation of a Graph
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12. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.4
Vertical Translation of a Graph
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13. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
Vertical Translations
In general, the graph of y = f (x) + c is the same as the graph of y = f (x) shifted up (if c > 0) or down (if c < 0) by |c| units.
We usually refer to f (x) + c as a vertical translation (up or down, by |c| units).
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14. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.5
A Horizontal Translation
Compare and contrast the graphs of
y = x2 and y = (x − 1)2.
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15. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.5
A Horizontal Translation
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16. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.5
A Horizontal Translation
To avoid confusion on which way to translate the graph of y = f (x), focus on what makes the argument (the quantity inside the parentheses) zero.
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17. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
Horizontal Translations
In general, for c > 0, the graph of y = f (x − c) is the same as the graph of y = f (x) shifted c units to the right.
Likewise (again, for c > 0), you get the graph of
y = f (x + c) by moving the graph of y = f (x) to the left c units.
We usually refer to f (x − c) and f (x + c) as horizontal translations (to the right and left, respectively, by c units).
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18. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.7
Comparing Some Related Graphs
Compare and contrast the graphs of
y = x2 − 1, y = 4(x2 − 1) and y = (4x)2 − 1.
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19. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.7
Comparing Some Related Graphs
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20. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
5.7
Comparing Some Related Graphs
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21. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
Scaling the x- and y-axes
Based on Example 5.7, notice that to obtain a graph of
y = cf (x) for some constant c > 0, you can take the graph of y = f (x) and multiply the scale on the y-axis by c.
To obtain a graph of y = f (cx), for some constant c > 0, you can take the graph of y = f (x) and multiply the scale on the x-axis by 1/c.
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22. TRANSFORMATIONS OF FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
Summary
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Slide 22