5.1: Amplitude, Period, and Midline | Precalculus

How it works

In a transformed sinusoid, the absolute outside coefficient gives amplitude, the vertical shift gives the midline, and the absolute inside frequency determines period. These interpretations use a sine or cosine model with a nonzero frequency.

y=A\sin(Bx)+D,\quad\text{amplitude}=|A|,\quad T=\frac{2\pi}{|B|}

Worked examples

Example 1

Find amplitude, period, and midline of $\displaystyle y=3\sin(2x)+1$ .

Amplitude is $\displaystyle 3$ , period is $\displaystyle \pi$ , and midline is $\displaystyle y=1$ .

Example 2

A sinusoid has maximum $\displaystyle 9$ and minimum $\displaystyle 1$ . Find amplitude and midline.

Amplitude is $\displaystyle (9-1)/2=4$ and midline is $\displaystyle y=(9+1)/2=5$ .

Problems

Problem 5.1.1

For $\displaystyle y=2\sin x+1$ , find amplitude, period, midline, and range.

Sinusoid with maximum 3, minimum negative 1, midline 1, and period 2 pi. — Read the vertical extremes and distance between repeats.

Show worked solution

Amplitude $\displaystyle 2$ , period $\displaystyle 2\pi$ , midline $\displaystyle y=1$ , and range $\displaystyle [-1,3]$ .

Sinusoid with maximum 3, minimum negative 1, midline 1, and period 2 pi. The solution adds labeled relationships. — Amplitude = 2, midline = 1, period = 2 pi, range [-1,3].

Problem 5.1.2

Find the period of $\displaystyle y=-4\cos(3x)$ .

Show worked solution

The period is $\displaystyle 2\pi/3$ . The negative sign reflects the graph but does not change the period.

Problem 5.1.3

A sine model has period $\displaystyle 8$ . Find a positive frequency coefficient.

Show worked solution

$\displaystyle B=2\pi/8=\pi/4$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 5.1.1

Problem 5.1.2

Problem 5.1.3

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