Find the amplitude and perlod of the function

amplitüde
period

Sketch its graph. (Select the correct graph.)

Ask by Ball Newton. in the United States

Mar 24,2025

Question

Find the amplitude and perlod of the function

amplitüde
period

Sketch its graph. (Select the correct graph.)

Ask by Ball Newton. in the United States

Mar 24,2025

UpStudy AI · Accepted Answer

1. The given function is 
   $$
   y=-4 \sin (2\pi x).
   $$
   
2. The amplitude of a sine function $$ y = a \sin(bx) $$ is given by $$ |a| $$. Here, $$ a = -4 $$, so
   $$
   \text{Amplitude} = |-4| = 4.
   $$

3. The period of a sine function $$ y = \sin(bx) $$ is given by 
   $$
   \text{Period} = \frac{2\pi}{|b|}.
   $$
   In this case, $$ b = 2\pi $$, so
   $$
   \text{Period} = \frac{2\pi}{2\pi} = 1.
   $$

4. Hence, the amplitude is $$ \boxed{4} $$ and the period is $$ \boxed{1} $$.

5. To sketch the graph of $$ y = -4 \sin(2\pi x) $$:
   - Start at the origin $$ (0, 0) $$.
   - The basic sine curve $$ \sin(2\pi x) $$ has been reflected vertically because of the negative sign.
   - The maximum and minimum values of the reflected sine are $$ -4 $$ and $$ 4 $$ respectively.
   - One period extends from $$ x = 0 $$ to $$ x = 1 $$.
   
   A rough sketch over one period is as follows:
   
   - At $$ x = 0 $$: $$ y = -4 \sin(0) = 0 $$.
   - At $$ x = \frac{1}{4} $$: $$ y = -4 \sin\left(2\pi \cdot \frac{1}{4}\right) = -4 \sin\left(\frac{\pi}{2}\right) = -4 \times 1 = -4 $$.
   - At $$ x = \frac{1}{2} $$: $$ y = -4 \sin\left(2\pi \cdot \frac{1}{2}\right) = -4 \sin(\pi) = 0 $$.
   - At $$ x = \frac{3}{4} $$: $$ y = -4 \sin\left(2\pi \cdot \frac{3}{4}\right) = -4 \sin\left(\frac{3\pi}{2}\right) = -4 \times (-1) = 4 $$.
   - At $$ x = 1 $$: $$ y = -4 \sin(2\pi) = 0 $$.

6. The graph is a sine wave starting at the origin, reaching a minimum of $$ -4 $$ at $$ x=\frac{1}{4} $$, returning to $$ 0 $$ at $$ x=\frac{1}{2} $$, reaching a maximum of $$ 4 $$ at $$ x=\frac{3}{4} $$, and then returning to $$ 0 $$ at $$ x=1 $$. This pattern repeats with period $$ 1 $$.

Thus, the answers are:

- Amplitude: $$ 4 $$
- Period: $$ 1 $$.
Amplitude: 4 Period: 1

Find the amplitude and perlod of the function

amplitüde
period

Sketch its graph. (Select the correct graph.)

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Find the amplitude and perlod of the function amplitüde period Sketch its graph. (Select the correct graph.)

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Find the amplitude and perlod of the function

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Sketch its graph. (Select the correct graph.)