650 CHAPTER 6 The Circular Functions and Their Graphs Connecting Graphs with Equations 6.5 Exercises CONCEPT PREVIEW Fill in the blank to correctly complete each sentence. 1. The least positive value x for which tan x = 0 is . 2. The least positive value x for which cot x = 0 is . 3. Between any two successive vertical asymptotes, the graph of y = tan x . (increases / decreases) NOTE Because the circular functions are periodic, there are infinitely many equations that correspond to each graph in Example 6. Confirm that both y = -1 - cot1-2x2 and y = -1 - tan a2x - p 2b are equations for the graph in Example 6(b). When writing the equation from a graph, it is practical to write the simplest form. Therefore, we choose values of b where b 70 and write the function without a phase shift when possible. EXAMPLE 6 Determining Equations for Graphs Determine an equation for each graph. (a) x y 2 –1 –2 0 p 2 –p 2 (b) x y 1 –2 –1 –3 0 p 2 p 4 SOLUTION (a) This graph is that of y = tan x but reflected across the x-axis and stretched vertically by a factor of 2. Therefore, an equation for this graph is y = -2 tan x. Vertical stretch x-axis reflection (b) This is the graph of a cotangent function, but the period is p 2 rather than p. Therefore, the coefficient of x is 2. This graph is vertically translated down 1 unit compared to the graph of y = cot 2 x. An equation for this graph is y = -1 + cot 2 x. Period is p 2 . Vertical translation down 1 unit S Now Try Exercises 39 and 43.
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