9. Atlikite funkcijos $$ f(x)=\sin x $$ transformacijas ir nubraižykite funkcijos grafiko eskizą: $$ \begin{array}{ll} ext { a) } y=f\left(x-\frac{\pi}{3} ight) ; & ext { b) } y=f\left(x+\frac{\pi}{4} ight) ext {; } \ ext { c) } y=f(x)+3 ; & ext { d) } y=f(x)-4\end{array} $$

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9. Atlikite funkcijos $$ f(x)=\sin x $$ transformacijas ir nubraižykite funkcijos grafiko eskizą: $$ \begin{array}{ll}	ext { a) } y=f\left(x-\frac{\pi}{3}ight) ; & 	ext { b) } y=f\left(x+\frac{\pi}{4}ight) 	ext {; } \ 	ext { c) } y=f(x)+3 ; & 	ext { d) } y=f(x)-4\end{array} $$

UpStudy AI · Accepted Answer

It seems that I am currently unable to create plots directly. However, I can guide you on how to sketch the graphs of the transformed functions based on the transformations we discussed.

### Sketching the Graphs

1. **For $$ y = \sin\left(x - \frac{\pi}{3}\right) $$**:
   - Start with the basic sine wave.
   - Shift the entire graph to the right by $$ \frac{\pi}{3} $$ units.
   - The key points (like $$ (0, 0) $$, $$ \left(\frac{\pi}{2}, 1\right) $$, etc.) will now be at $$ \left(\frac{\pi}{3}, 0\right) $$, $$ \left(\frac{\pi}{2} + \frac{\pi}{3}, 1\right) $$, etc.

2. **For $$ y = \sin\left(x + \frac{\pi}{4}\right) $$**:
   - Again, start with the basic sine wave.
   - Shift the graph to the left by $$ \frac{\pi}{4} $$ units.
   - The key points will now be at $$ \left(-\frac{\pi}{4}, 0\right) $$, $$ \left(\frac{\pi}{2} - \frac{\pi}{4}, 1\right) $$, etc.

3. **For $$ y = \sin x + 3 $$**:
   - Start with the basic sine wave.
   - Shift the entire graph upwards by 3 units.
   - The midline of the sine wave will now be at $$ y = 3 $$ instead of $$ y = 0 $$.

4. **For $$ y = \sin x - 4 $$**:
   - Start with the basic sine wave.
   - Shift the entire graph downwards by 4 units.
   - The midline of the sine wave will now be at $$ y = -4 $$.

### Summary of Key Points
- The amplitude of the sine function remains 1 for all transformations.
- The period remains $$ 2\pi $$.
- The horizontal shifts affect the x-coordinates of the key points, while the vertical shifts affect the y-coordinates.

You can use graphing software or a graphing calculator to visualize these transformations accurately. If you have any specific questions about the transformations or need further assistance, feel free to ask!
Here are the transformed functions and their descriptions: a) $$ y = \sin\left(x - \frac{\pi}{3}\right) $$: Shifts the sine graph to the right by $$ \frac{\pi}{3} $$ units. b) $$ y = \sin\left(x + \frac{\pi}{4}\right) $$: Shifts the sine graph to the left by $$ \frac{\pi}{4} $$ units. c) $$ y = \sin x + 3 $$: Shifts the sine graph upwards by 3 units. d) $$ y = \sin x - 4 $$: Shifts the sine graph downwards by 4 units. These transformations move the sine wave horizontally and vertically without changing its shape or period.

Atlikite funkcijos transformacijas ir
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