Consider the function f(x) = 4 sin((x − 3)) + 6. State the amplitude A, period P, a - midline. State the phase shift and vertical translation. In the full period [0, P], state the maximum a minimum y-values and their corresponding x-values. Enter the exact answers. Amplitude: A = 4 Period: P = Midline: y = Number The phase shift is Click for List The vertical translation is Click for List
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
![### Consider the Function
\[ f(x) = 4 \sin\left(\frac{\pi}{2} (x - 3)\right) + 6 \]
1. **State the amplitude \( A \), period \( P \), and midline.**
2. **State the phase shift and vertical translation.**
3. **In the full period \([0, P]\), state the maximum and minimum \( y \)-values and their corresponding \( x \)-values.**
**Enter the exact answers:**
- **Amplitude: \( A \) =** `4`
- **Period: \( P \) =** `_____`
- **Midline: \( y \) =** `_____`
- **The phase shift is** `[Click for List]`.
- **The vertical translation is** `[Click for List]`.
---
### Hints for the maximum and minimum values of \( f(x) \):
- **The maximum value** of \( y = \sin(x) \) is \( 1 \) and the corresponding \( x \) values are \( x = \frac{\pi}{2} \) and multiples of \( 2\pi \) less than and more than this \( x \) value. You may want to solve \(\frac{\pi}{2} (x - 3) = \frac{\pi}{2}\) and...
- **The minimum value** of \( y = \sin(x) \) is \( -1 \) and the corresponding \( x \) values are \( x = \frac{3\pi}{2} \) and...
_(Note: The hints section appears to be truncated and missing parts which would normally provide more detailed guidance on finding the specific x-values for the maximum and minimum y-values within the period)_
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