Find the amplitude, period, and displacement of the function. Graph the function. Show at least one period

Type the amplitude, period, and displacement of the function.
Amplitude Period Displacement
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers

Ask by Grant Hodgson. in the United States

Jan 19,2025

Question

Find the amplitude, period, and displacement of the function. Graph the function. Show at least one period

Type the amplitude, period, and displacement of the function.
Amplitude Period Displacement
(Simplify your answer. Type an exact answer, using as needed. Use integers or fractions for any numbers

Ask by Grant Hodgson. in the United States

Jan 19,2025

UpStudy AI · Accepted Answer

It seems that I am unable to generate the graph at this moment. However, I can guide you on how to graph the function $$ y = 9 \sin(4 \pi x + 7) $$ over one period.

### Steps to Graph the Function

1. **Set Up the Axes**:
   - Draw the x-axis and y-axis.
   - Label the x-axis from $$ 0 $$ to $$ \frac{1}{2} $$.
   - Label the y-axis to accommodate values from $$ -9 $$ to $$ 9 $$ (since the amplitude is $$ 9 $$).

2. **Identify Key Points**:
   - The sine function starts at $$ 0 $$ when $$ x = 0 $$.
   - Calculate the key points within one period:
     - At $$ x = 0 $$: $$ y = 9 \sin(7) $$
     - At $$ x = \frac{1}{8} $$: $$ y = 9 \sin(4\pi \cdot \frac{1}{8} + 7) = 9 \sin(\frac{\pi}{2} + 7) $$ (which will be maximum)
     - At $$ x = \frac{1}{4} $$: $$ y = 9 \sin(4\pi \cdot \frac{1}{4} + 7) = 9 \sin(2\pi + 7) $$ (which will return to 0)
     - At $$ x = \frac{3}{8} $$: $$ y = 9 \sin(4\pi \cdot \frac{3}{8} + 7) $$ (which will be minimum)
     - At $$ x = \frac{1}{2} $$: $$ y = 9 \sin(4\pi \cdot \frac{1}{2} + 7) = 9 \sin(2\pi + 7) $$ (which will return to 0)

3. **Plot the Points**:
   - Plot the calculated points on the graph.
   - Connect the points smoothly to form the sine wave.

4. **Label the Graph**:
   - Indicate the amplitude and period on the graph.

### Final Summary
- **Amplitude**: $$ 9 $$
- **Period**: $$ \frac{1}{2} $$
- **Displacement**: $$ -\frac{7}{4\pi} $$

You can use graphing software or a graphing calculator to visualize the function accurately. If you have any further questions or need additional assistance, feel free to ask!
Amplitude = 9 Period = ½ Displacement = –7/(4π)

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