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Pre-Calc Exam Notes 111

# Pre-Calc Exam Notes 111 - the horizontal line that divides...

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Properties of Graphs of Trigonometric Functions Section 5.2 111 Example5.5 The period of y = cos 3 x is 2 π 3 and the period of y = cos 1 2 x is 4 π . The graphs of both functions are shown in Figure 5.2.2: x y 0 1 1 π 6 π 3 π 2 2 π 3 5 π 6 π 7 π 6 4 π 3 3 π 2 5 π 3 11 π 6 2 π 13 π 6 7 π 3 5 π 2 8 π 3 17 π 6 3 π 19 π 6 10 π 3 7 π 2 11 π 3 23 π 6 4 π y = cos 1 2 x y = cos 3 x Figure 5.2.2 Graph of y = cos 3 x and y = cos 1 2 x We know that 1 sin x 1 and 1 cos x 1 for all x . Thus, for a constant A negationslash= 0, −| A | ≤ A sin x ≤ | A | and −| A | ≤ A cos x ≤ | A | for all x . In this case, we call | A | the amplitude of the functions y = A sin x and y = A cos x . In general, the amplitude of a periodic curve f ( x ) is half the difference of the largest and smallest values that f ( x ) can take: Amplitude of f ( x ) = (maximum of f ( x )) (minimum of f ( x )) 2 In other words, the amplitude is the distance from either the top or bottom of the curve to
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Unformatted text preview: the horizontal line that divides the curve in half, as in Figure 5.2.3. x y | A | −| A | π 4 π 2 3 π 4 π 5 π 4 3 π 2 7 π 4 2 π 2 | A | | A | | A | Figure 5.2.3 Amplitude = max − min 2 = | A |− ( −| A | ) 2 =| A | Not all periodic curves have an amplitude. For example, tan x has neither a maximum nor a minimum, so its amplitude is unde±ned. Likewise, cot x , csc x , and sec x do not have an amplitude. Since the amplitude involves vertical distances, it has no effect on the period of a function, and vice versa....
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