Sine and Cosine Graphs

Develop amplitude and range, period and frequency factor.

Lesson 1 of 42 parts Practice Flashcards

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Amplitude and Range

Objective: Explain amplitude and range, apply the rule, and justify relevant restrictions.

For y=A sin(Bx)+D or y=A cos(Bx)+D, with B≠0, amplitude is |A| and the range is [D−|A|,D+|A|]. A negative A reverses the graph about its midline but does not produce a negative amplitude. The midline is y=D. Amplitude measures maximum deviation from the midline, rather than the full distance between maximum and minimum.

Formulas

$Amplitude=|A|; range=[D−|A|,D+|A|]$

Worked examples

Try itFor y=−10 sin(5x)+3, find the amplitude.Show answer

Step 1: Amplitude is the absolute value of the coefficient outside sine. Step 2: Take |−10|. Step 3: The amplitude is 10; the negative sign reflects the graph.

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Amplitude and Range

Objective: Explain amplitude and range, apply the rule, and justify relevant restrictions.

Formulas

$Amplitude=|A|; range=[D−|A|,D+|A|]$

Worked examples

Try itFor y=−10 sin(5x)+3, find the amplitude.Show answer

Step 1: Amplitude is the absolute value of the coefficient outside sine. Step 2: Take |−10|. Step 3: The amplitude is 10; the negative sign reflects the graph.

Period and Frequency Factor

Objective: Explain period and frequency factor, apply the rule, and justify relevant restrictions.

For sine and cosine with nonzero input multiplier B, the period is 2π/|B|. This is the smallest positive horizontal distance that restores the argument modulo 2π. A larger |B| compresses the graph horizontally. In a time model, frequency in cycles per unit time is 1/T, whereas angular frequency is |B| in radians per unit time; they are different quantities.

Formulas

$T=2π/|B|; f=1/T$

Worked examples

Try itFor y=3 cos(7x)+5, give the period as a coefficient of π.Show answer

Step 1: Cosine repeats every 2π in its argument. Step 2: Divide 2π by |7|. Step 3: The period is 2/7π.

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