Determine the amplitude, period, and phase
shift (if any) of the given function. Graph the
function.

What is the amplitude of ?
(Simplify your answer. Type an
exact answer, using as needed.)

Ask by Dawson Pollard. in the United States

Dec 08,2024

Question

Determine the amplitude, period, and phase
shift (if any) of the given function. Graph the
function.

What is the amplitude of ?
(Simplify your answer. Type an
exact answer, using as needed.)

Ask by Dawson Pollard. in the United States

Dec 08,2024

UpStudy AI · Accepted Answer

To analyze the function $$ y = \sin x + 7 $$, we can identify the amplitude, period, and phase shift.

1. **Amplitude**: The amplitude of a sine function $$ y = A \sin(Bx + C) + D $$ is given by the absolute value of $$ A $$. In this case, the function can be rewritten as:
   $$
   y = 1 \cdot \sin(x) + 7
   $$
   Here, $$ A = 1 $$. Therefore, the amplitude is:
   $$
   \text{Amplitude} = |A| = |1| = 1
   $$

2. **Period**: The period of a sine function is given by the formula $$ \frac{2\pi}{|B|} $$. In this case, $$ B = 1 $$ (since there is no coefficient in front of $$ x $$). Thus, the period is:
   $$
   \text{Period} = \frac{2\pi}{|1|} = 2\pi
   $$

3. **Phase Shift**: The phase shift is determined by the term $$ C $$ in the function $$ y = A \sin(Bx + C) + D $$. In this case, there is no horizontal shift since there is no $$ C $$ term. Therefore, the phase shift is:
   $$
   \text{Phase Shift} = 0
   $$

4. **Vertical Shift**: The function $$ y = \sin x + 7 $$ has a vertical shift of 7 units upward.

### Summary:
- **Amplitude**: $$ 1 $$
- **Period**: $$ 2\pi $$
- **Phase Shift**: $$ 0 $$

### Graphing the Function:
To graph $$ y = \sin x + 7 $$:
- The sine function oscillates between -1 and 1. Therefore, $$ \sin x $$ oscillates between -1 and 1, and when we add 7, the function $$ y = \sin x + 7 $$ oscillates between $$ 6 $$ and $$ 8 $$.
- The midline of the graph is at $$ y = 7 $$.
- The graph will have peaks at $$ y = 8 $$ and troughs at $$ y = 6 $$.

You can sketch the graph by plotting points for key angles (like $$ 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi $$) and noting the vertical shift. The graph will repeat every $$ 2\pi $$ units.
Amplitude: 1, Period: $$ 2\pi $$, Phase Shift: 0

Determine the amplitude, period, and phase
shift (if any) of the given function. Graph the
function.

What is the amplitude of ?
(Simplify your answer. Type an
exact answer, using as needed.)

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Solución

Summary:

Graphing the Function:

Beyond the Answer

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Latest Trigonometry Questions

Determine the amplitude, period, and phase shift (if any) of the given function. Graph the function. What is the amplitude of ? (Simplify your answer. Type an exact answer, using as needed.)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Summary:

Graphing the Function:

Beyond the Answer

preguntas relacionadas

Latest Trigonometry Questions

Determine the amplitude, period, and phase
shift (if any) of the given function. Graph the
function.

What is the amplitude of ?
(Simplify your answer. Type an
exact answer, using as needed.)