Graphs of Sine and Cosine

Explore amplitude and midline, period, phase and vertical shifts.

Lesson 3 of 53 parts Practice Flashcards

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Amplitude and Midline

Objective: Explain amplitude and midline and use it to solve or interpret a problem.

For y=A sin(B(x−C))+D, amplitude is |A| and the midline is y=D when A≠0 and B≠0. Distinguish peak-to-peak height from amplitude. No, it is twice amplitude.

Worked examples

Try itGraphs of Sine and Cosine — practice 1.1: For y=3sin(5x)+7, find amplitude.Show answer

Step 1: Amplitude is the absolute value of sine coefficient. Step 2: |3|=3. Step 3: The amplitude is 3.

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Amplitude and Midline

Objective: Explain amplitude and midline and use it to solve or interpret a problem.

For y=A sin(B(x−C))+D, amplitude is |A| and the midline is y=D when A≠0 and B≠0. Distinguish peak-to-peak height from amplitude. No, it is twice amplitude.

Worked examples

Try itGraphs of Sine and Cosine — practice 1.1: For y=3sin(5x)+7, find amplitude.Show answer

Step 1: Amplitude is the absolute value of sine coefficient. Step 2: |3|=3. Step 3: The amplitude is 3.

Period

Objective: Explain period and use it to solve or interpret a problem.

Sine and cosine have base period 2π; the rule y=sin(Bx) has period 2π/|B| for B≠0. Include the input angular unit. No, it changes vertical scale only.

Worked examples

Try itGraphs of Sine and Cosine — practice 2.1: For y=sin(8x), find period as the coefficient of π.Show answer

Step 1: Sine has base period 2π. Step 2: Divide by frequency multiplier 8. Step 3: Period is 1/4π.

Phase and Vertical Shifts

Objective: Explain phase and vertical shifts and use it to solve or interpret a problem.

C moves the graph horizontally and D vertically in y=A sin(B(x−C))+D. Write the inside as B(x−C) before reading the phase shift. An unfactored inside constant is not directly the shift amount.

Worked examples

Try itGraphs of Sine and Cosine — practice 3.1: For y=sin(x−11), find horizontal shift in radians.Show answer

Step 1: Set inside x−11=0. Step 2: This occurs at x=11. Step 3: The graph shifts right 11 radians.

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