Chapter 6: Q. 103 (page 412)

Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis.

Short Answer

Expert verified

The amplitude is of a sinusoidal graph is used to scale the y-axis and the period scales the x-axis.

For a sinusoidal function of the form $y = A \sin (ω x)$ the $y$ axis is scaled by a factor of $A$ and the $x$ axis is scaled by a factor of $\frac{1}{ω}$ .

Step by step solution

Step 1. Concept Used

If $ω > 0$ , then the amplitude and period of $y = A \sin (ω x)$ and $y = A \cos (ω x)$ are given by

Amplitude $= |A|$

Period $= \frac{2 π}{ω}$

Step 2. Find what amplitude and period represents

The amplitude of a sinusoidal function represents the range of the sinusoidal function.

So for the sinusoidal curve $y = A \sin (ω x)$ and $y = A \cos (ω x)$ the amplitude $|A|$ means that the graph of the function lies between the values $- A$ and $A$ .

The period of the function is given as $T = \frac{2 π}{ω}$ represents the length of the one cycle and after $\frac{2 π}{ω}$ the graph starts repeating.

One cycle is represented by the interval localid="1646473406723" $[0, \frac{2 π}{ω}]$ .

Step 3. Find the scale in each axis

As the yvalue lies between $- A$ and $A$ , so the y-axis is scaled by a factor of $A$ .

As one interval of the cycle is given as $[0, \frac{2 π}{ω}]$ so the x-axis is scaled by a factor of $\frac{1}{ω}$ .

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Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis.

Short Answer

Step by step solution

Step 1. Concept Used

Step 2. Find what amplitude and period represents

Step 3. Find the scale in each axis

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