KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 05 · Lesson 04

Tangent, Cotangent & Their Asymptotes

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Tangent, Cotangent & Their Asymptotes

U05 · L04
1

For y=4sin(3x)−2, state the amplitude.

Triggraph visualInterpret and construct periodic functions by tracking amplitude, period, frequency, phase, midline, asymptotes, and contextual meaning. xy
Answer: 4
Reasoning: Amplitude is |A|.
2

For y=4sin(3x)−2, state the period.

Triggraph visualInterpret and construct periodic functions by tracking amplitude, period, frequency, phase, midline, asymptotes, and contextual meaning. xy
Answer: 2π/3
Reasoning: Period is 2π/|B|.
3

For y=2cos(x−π/4)+1, state the phase shift.

Triggraph visualInterpret and construct periodic functions by tracking amplitude, period, frequency, phase, midline, asymptotes, and contextual meaning. xy
Answer: π/4 right
Reasoning: The graph is shifted by C in x−C.
4

State the midline of y=−3sin(2x)+5.

Triggraph visualInterpret and construct periodic functions by tracking amplitude, period, frequency, phase, midline, asymptotes, and contextual meaning. xy
Answer: y=5
Reasoning: The vertical shift D is the midline.
5

State one vertical asymptote of y=tan x.

Triggraph visualInterpret and construct periodic functions by tracking amplitude, period, frequency, phase, midline, asymptotes, and contextual meaning. xy
Answer: x=π/2
Reasoning: Tangent is undefined where cos x=0.
6

A periodic quantity ranges from 8 to 20. Find amplitude and midline.

Triggraph visualInterpret and construct periodic functions by tracking amplitude, period, frequency, phase, midline, asymptotes, and contextual meaning. xy
Answer: Amplitude 6; midline 14
Reasoning: Half-range is 6 and average of extrema is 14.
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