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Trig F07 O’Brien How to Graph y a tan (bx c) d and y a cot (bx - c) d 1. Identify a, b, c, and d. 2. 5. Determine if there has been an x-axis reflection: if a is negative, there has been an x-axis reflection. π Find the period: period = . b 1 Find the x-increment: x-increment = period . 4 Find the phase shift by setting the argument, bx – c, equal to zero and solving for x: c c c c P.S. = right (if we see bx – c and is positive) or P.S. = left (if we see bx + c and is negative). b b b b Find the vertical shift: |d| up if d is positive or |d| down if d is negative. 6. Identify the left asymptote, the three key points, and the right asymptote. 3a. b. 4. cotangent: To find the left asymptote, let x = phase shift, x LA: x c b (x of LA + x-increment, d + a) c . b (x1 + x-inc, d) (x2 + x-inc, d – a) RA: x = x3 + x-inc Note: The left asymptote of a tangent function is NOT x = phase shift. π tangent: To find the left asymptote, set the argument, bx – c, equal to and solve for x. 2 c π LA: x (x of LA + x-increment, d – a) (x1 + x-inc, d) (x2 + x-inc, d + a) RA: x = x3 + x-inc b 2b 7. Plot the asymptotes and the three key points. Connect the points with a smooth, continuous curve. Extend the graph to two full periods. Example 1: y = –3 cot (2x + π) +1 1. a = –3 2. There is an x-axis reflection. 3. period = 4. phase shift = 5. vertical shift = 1 up 6. LA: x b=2 π 2 c = –π d=1 x-increment = π left 2 π 4π 2 8 1 π π 4 2 8 2x π 0 2x π x 3π , 2 8 2π , 1 8 π , 4 8 π 4π 2 8 RA: x = 0 7. 1 Trig F07 O’Brien 1 Example 2: y = 2 tan x π – 3 4 b= 1 4 c=π d = –3 1. a=2 2. There is no x-axis reflection. 3. period = π 1 4π x-increment = 4 4. phase shift = 4π right 5. vertical shift = 3 down 6. LA: 1 4π π 4 1 xπ 0 4 1 π xπ x 4π 2π x 2 π 4 2 x 4π 0 (3π, –5) x 4π (4π, –3) (5π, –1) RA: x = 6π 7. 2