PreCalc 2 · Trigonometric Functions
Name the parent (tan through the origin, cot falling through π2). Then read period, stretch, and shifts.
Which graph is it?
Through (0, 0) going up, asymptotes at −π2 and π2 → tangent.
Asymptotes on the y-axis and at π, falling through π2 → cotangent.
Period
Find the distance between neighboring asymptotes. That distance is the period. Then B = πperiod.
Phase shift
See how far the middle crossing (or the asymptote pair) has slid from the parent graph. That is C.
Vertical stretch and shift
A stretches the steepness. D slides the crossing up or down. Tan and cot have no max or min.
This is a parent graph. Is it tangent or cotangent? Write the equation.
It goes through (0, 0) and rises.
Asymptotes at −π2 and π2.
y = tan x
Answer: y = tan x
This is a parent graph. Is it tangent or cotangent? Write the equation.
Asymptotes at 0 and π.
It falls through (π2, 0).
y = cot x
Answer: y = cot x
The graph goes through (0, 0) but falls. Write the equation.
Through the origin, going down → flipped tangent.
y = −tan x
Answer: y = −tan x
Neighboring asymptotes of a tangent graph are π2 apart, and it still crosses (0, 0) going up. Write the equation.
Period π is the parent period.
y = tan x
Answer: y = tan x
A cotangent graph has parent asymptotes and is flipped (it rises through π2). Write the equation.
Flipped cotangent.
y = −cot x
Answer: y = −cot x
Asymptotes of a tangent graph are π4 apart (half the parent period). It still crosses the origin going up. Write the equation.
Period = π2, so B = 2.
y = tan(2x)
Answer: y = tan(2x)
A tangent graph has period π2 and crosses (0, 0) going up. Write the equation.
Period = π2.
B = ππ2 = 2.
y = tan(2x)
Answer: y = tan(2x)
A cotangent graph has period π3. Write a parent-shaped equation (no phase shift).
B = ππ3 = 3.
y = cot(3x)
Answer: y = cot(3x)
Tangent, period π, crossing at (0, 2) going up. Write the equation.
D = 2.
y = tan x + 2
Answer: y = tan x + 2
Tangent, period π, flipped, through the origin. Write the equation.
y = −tan x
Answer: y = −tan x
Cotangent, period π2, no other changes. Write the equation.
y = cot(2x)
Answer: y = cot(2x)
Tangent stretched by 3, parent period, through the origin going up. Write the equation.
A = 3.
y = 3 tan x
Answer: y = 3 tan x
Tangent, period π2, shifted right π8, through the shifted origin going up. Write the equation.
B = 2.
C = π8.
y = tan(2(x − π8))
Answer: y = tan(2(x − π8))
For a tangent graph with period π and phase shift left π4, write the equation.
y = tan(x + π4)
Answer: y = tan(x + π4)
Cotangent, period π2, shifted down 1. Write the equation.
y = cot(2x) − 1
Answer: y = cot(2x) − 1
Flipped tangent, period π2, no phase shift. Write the equation.
y = −tan(2x)
Answer: y = −tan(2x)
Tangent, period π, right π6, up 1. Write the equation.
y = tan(x − π6) + 1
Answer: y = tan(x − π6) + 1
Cotangent, period π2, left π4. Write the equation.
y = cot(2(x + π4))
Answer: y = cot(2(x + π4))
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