QUESTION 1 . 1.1 By completing the table below, draw the graphs of the following given functions on $$ t $$ same set of axes provided where $$ x \in\left[0^{\circ} ; 360^{\circ} ight] $$. Label your graphs clearly by showi the turning points, $$ x $$ - intercepts and the $$ y $$ - intercepts. The graphs of a should be sketched on one system of axes and of $$ \mathbf{b} $$ on the same system of axes and of $$ \mathbf{c} $$ on the same system of axes a) $$ f(x)=\sin (x) $$ and $$ g(x)=2 \sin (x) $$ and $$ h(x)=\sin (2 x) $$; b) $$ f(x)=\cos (x) $$ and $$ g(x)=3 \cos (x) $$ and $$ h(x)=\cos (3 x) $$, c) $$ f(x)= an (x) $$ and $$ g(x)=2 an (x) $$ and $$ h(x)= an (2 x) $$.

Question

QUESTION 1 . 1.1 By completing the table below, draw the graphs of the following given functions on $$ t $$ same set of axes provided where $$ x \in\left[0^{\circ} ; 360^{\circ}ight] $$. Label your graphs clearly by showi the turning points, $$ x $$ - intercepts and the $$ y $$ - intercepts. The graphs of a should be sketched on one system of axes and of $$ \mathbf{b} $$ on the same system of axes and of $$ \mathbf{c} $$ on the same system of axes a) $$ f(x)=\sin (x) $$ and $$ g(x)=2 \sin (x) $$ and $$ h(x)=\sin (2 x) $$; b) $$ f(x)=\cos (x) $$ and $$ g(x)=3 \cos (x) $$ and $$ h(x)=\cos (3 x) $$, c) $$ f(x)=	an (x) $$ and $$ g(x)=2 	an (x) $$ and $$ h(x)=	an (2 x) $$.

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It seems that I am unable to generate the graphs directly. However, I can guide you on how to plot these functions using graphing software or a graphing calculator. Here’s how you can do it:

### Instructions for Graphing

1. **Choose a Graphing Tool**: You can use graphing software like Desmos, GeoGebra, or a graphing calculator.

2. **Set the Range**: Make sure to set the $$ x $$-axis to range from $$ 0^\circ $$ to $$ 360^\circ $$.

3. **Plot the Functions**:
   - For **Set a**:
     - Plot $$ f(x) = \sin(x) $$
     - Plot $$ g(x) = 2\sin(x) $$
     - Plot $$ h(x) = \sin(2x) $$
   - For **Set b**:
     - Plot $$ f(x) = \cos(x) $$
     - Plot $$ g(x) = 3\cos(x) $$
     - Plot $$ h(x) = \cos(3x) $$
   - For **Set c**:
     - Plot $$ f(x) = 	an(x) $$
     - Plot $$ g(x) = 2	an(x) $$
     - Plot $$ h(x) = 	an(2x) $$

4. **Label the Graphs**: 
   - Clearly label each function with its name and color code.
   - Mark the turning points, $$ x $$-intercepts, and $$ y $$-intercepts on the graph.

5. **Turning Points and Intercepts**:
   - For each function, you can use the previously calculated turning points and intercepts to annotate the graph.

### Example of Key Points to Plot

- **Set a**:
  - $$ f(x) = \sin(x) $$: Turning points at $$ (90, 1) $$, $$ (270, -1) $$
  - $$ g(x) = 2\sin(x) $$: Same turning points, but $$ y $$-values are scaled.
  - $$ h(x) = \sin(2x) $$: Turning points at $$ (45, 1) $$, $$ (135, -1) $$, etc.

- **Set b**:
  - $$ f(x) = \cos(x) $$: Turning points at $$ (0, 1) $$, $$ (180, -1) $$
  - $$ g(x) = 3\cos(x) $$: Same turning points, but $$ y $$-values are scaled.
  - $$ h(x) = \cos(3x) $$: Turning points at $$ (0, 1) $$, $$ (60, -1) $$, etc.

- **Set c**:
  - $$ f(x) = 	an(x) $$: Vertical asymptotes at $$ x = 90^\circ, 270^\circ $$
  - $$ g(x) = 2	an(x) $$: Same vertical asymptotes, scaled vertically.
  - $$ h(x) = 	an(2x) $$: Vertical asymptotes at $$ x = 45^\circ, 135^\circ, 225^\circ, 315^\circ $$

### Conclusion

By following these steps, you should be able to create the graphs for the functions provided in the problem. If you have any specific questions about the functions or need further assistance, feel free to ask!
To graph the functions $$ f(x) = \sin(x) $$, $$ g(x) = 2\sin(x) $$, and $$ h(x) = \sin(2x) $$ on the same set of axes for $$ x \in [0^\circ, 360^\circ] $$, follow these steps: 1. **Set Up the Axes**: Draw the $$ x $$-axis from $$ 0^\circ $$ to $$ 360^\circ $$ and the $$ y $$-axis to accommodate the range of the functions. 2. **Plot $$ f(x) = \sin(x) $$**: - **Turning Points**: Maximum at $$ 90^\circ $$ and $$ 270^\circ $$, minimum at $$ 180^\circ $$. - **Intercepts**: $$ x $$-intercepts at $$ 0^\circ, 180^\circ, 360^\circ $$, $$ y $$-intercept at $$ 0 $$. 3. **Plot $$ g(x) = 2\sin(x) $$**: - **Scaling**: Vertically scale the $$ y $$-values by a factor of 2. - **Intercepts**: Same as $$ f(x) $$. 4. **Plot $$ h(x) = \sin(2x) $$**: - **Period**: The period is halved, so the function completes two cycles within $$ 360^\circ $$. - **Intercepts**: $$ x $$-intercepts at $$ 0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ $$, $$ y $$-intercept at $$ 0 $$. 5. **Label the Graphs**: - Clearly label each function with its name and color code. - Mark the turning points and intercepts on the graph. 6. **Repeat for Sets b and c**: - **Set b**: $$ f(x) = \cos(x) $$, $$ g(x) = 3\cos(x) $$, $$ h(x) = \cos(3x) $$ - **Intercepts**: $$ x $$-intercepts at $$ 90^\circ, 270^\circ $$ for $$ f(x) $$, scaled accordingly for others. - **Set c**: $$ f(x) = 	an(x) $$, $$ g(x) = 2	an(x) $$, $$ h(x) = 	an(2x) $$ - **Asymptotes**: Vertical asymptotes at $$ 90^\circ, 270^\circ $$ for $$ f(x) $$, scaled for others. By following these steps, you can accurately graph all six functions on the same set of axes, clearly labeling their turning points, $$ x $$-intercepts, and $$ y $$-intercepts.

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