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Writing Trig Equations from Graph Features

A free Trigonometry lesson from the “Graphs of Trig Functions” unit, with a worked example and practice problems including step-by-step solutions.

This lesson works in reverse: instead of reading features off a graph, you start with the features (amplitude, period, midline, phase shift, or a max/min) and build the equation y = a*sin(b(x-h))+d or the cosine version. Amplitude gives a, the period gives b through b = 2pi/period, the midline gives d, and the horizontal shift gives h. Max/min pairs convert to amplitude and midline using a = (max-min)/2 and d = (max+min)/2.

What you'll learn

Why it matters: Engineers and scientists who measure a repeating signal (tides, daylight hours, an AC voltage, a heartbeat) record its high, low, and timing, then write a sine equation to predict future values.

Worked example

Problem. A sinusoid has amplitude 4, period 2pi, midline y = 3, and no phase shift. Write it as a sine equation.

  1. Amplitude 4 gives a = 4; period 2pi gives b = 2pi/2pi = 1.
  2. Midline y = 3 gives d = 3; no phase shift means h = 0.
  3. Assemble: y = 4sin(1(x-0)) + 3 = 4sin(x) + 3.

Answer: y = 4sin(x) + 3

Practice problems

1. Write a sine equation with amplitude 2, b = 1, and midline y = 0 (no phase shift).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Write a sine equation with amplitude 2, b = 1, and midline y = 0 (no phase shift).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Amplitude 2 gives a = 2 and b = 1.
  4. Midline y = 0 gives d = 0, and h = 0.
  5. Write y = 2sin(x).
  6. Check the result by substituting or estimating: the response should match y = 2sin(x) and make sense in the original problem.

Answer: y = 2sin(x)

2. Write a cosine equation with amplitude 5, b = 1, and midline y = 2 (no phase shift).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Write a cosine equation with amplitude 5, b = 1, and midline y = 2 (no phase shift).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Amplitude 5 gives a = 5 and b = 1.
  4. Midline y = 2 gives d = 2, and h = 0.
  5. Write y = 5cos(x) + 2.
  6. Check the result by substituting or estimating: the response should match y = 5cos(x) + 2 and make sense in the original problem.

Answer: y = 5cos(x) + 2

3. A sinusoid has amplitude 3 and period 2pi with midline y = 0. Find the value of b.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A sinusoid has amplitude 3 and period 2pi with midline y = 0. Find the value of b.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use b = 2pi/period.
  4. b = 2pi/2pi.
  5. b = 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

4. A sinusoid has amplitude 6 and period pi. Find the value of b.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A sinusoid has amplitude 6 and period pi. Find the value of b.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use b = 2pi/period.
  4. b = 2pi/pi.
  5. b = 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

5. Write a sine equation with amplitude 4, period pi, and midline y = 1 (no phase shift).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Write a sine equation with amplitude 4, period pi, and midline y = 1 (no phase shift).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. a = 4; b = 2pi/pi = 2.
  4. d = 1 and h = 0.
  5. Write y = 4sin(2x) + 1.
  6. Check the result by substituting or estimating: the response should match y = 4sin(2x) + 1 and make sense in the original problem.

Answer: y = 4sin(2x) + 1

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