Func Transformations Q2 Select the function $$ f(x)=\cos (x) $$ and check the Plot Function box to graph $$ f(x)=\cos (x) $$. Use the $$ a $$-slider and b-slider to graph the function $$ f(x)=3 \cos (4 x) $$. The value of $$ b $$ affects the points Use the interactive figure to find your answer. Use the left and right arrow keys to move along a slider as needed. Click here to launch the interactive figure.

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It seems like you're working with an interactive figure to explore transformations of the cosine function. While I can't directly interact with the figure, I can help you understand how the transformations work.

The function $$ f(x) = \cos(x) $$ is the basic cosine function, which has a period of $$ 2\pi $$ and an amplitude of 1.

When you transform this function to $$ f(x) = 3 \cos(4x) $$, the transformations are as follows:

1. **Amplitude**: The coefficient in front of the cosine function (in this case, 3) affects the amplitude. The amplitude of $$ f(x) = 3 \cos(4x) $$ is 3, meaning the graph will stretch vertically by a factor of 3. The maximum value will be 3 and the minimum value will be -3.

2. **Period**: The coefficient of $$ x $$ (in this case, 4) affects the period of the function. The period of a cosine function is given by the formula $$ \frac{2\pi}{|b|} $$, where $$ b $$ is the coefficient of $$ x $$. For $$ f(x) = 3 \cos(4x) $$, the period is $$ \frac{2\pi}{4} = \frac{\pi}{2} $$. This means the function will complete one full cycle in $$ \frac{\pi}{2} $$ units along the x-axis.

3. **Phase Shift**: If there were a horizontal shift (for example, $$ f(x) = 3 \cos(4(x - c)) $$), it would shift the graph to the right by $$ c $$ units if $$ c $$ is positive, or to the left if $$ c $$ is negative. However, in your case, there is no horizontal shift.

4. **Vertical Shift**: If there were a constant added or subtracted from the function (for example, $$ f(x) = 3 \cos(4x) + d $$), it would shift the graph up or down by $$ d $$ units. Again, in your case, there is no vertical shift.

To visualize this, you can use the sliders to adjust the amplitude and period in the interactive figure. As you move the sliders, you should see how the graph of $$ f(x) = 3 \cos(4x) $$ changes compared to the original $$ f(x) = \cos(x) $$.

If you have specific questions about the transformations or need help with a particular aspect of the interactive figure, feel free to ask!
The transformation of $$ f(x) = \cos(x) $$ to $$ f(x) = 3 \cos(4x) $$ increases the amplitude to 3 and changes the period to $$ \frac{\pi}{2} $$. There's no phase or vertical shift. Use the interactive figure to see these changes.

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