The function given is $y = 5 \cos(x)$.

1. Amplitude

The amplitude of a cosine function is determined by the absolute value of the coefficient of the cosine term. In this case:

$\text{Amplitude} = |5| = 5$

The amplitude describes how far the peaks and valleys of the graph extend from the midline (in this case, $y = 0$).

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2. Transformations

The base function is $y = \cos(x)$. The given function $y = 5 \cos(x)$ has been transformed as follows:

• Vertical Stretch: The coefficient $5$ scales the graph vertically, stretching it by a factor of $5$. This means that the maximum value of the graph increases to $5$, and the minimum value decreases to $-5$.

The graph of $y = 5 \cos(x)$ is related to $y = \cos(x)$ by this vertical stretch.

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Summary of Changes:

• The amplitude is increased to $5$, causing the graph to stretch vertically.
• There are no horizontal shifts, vertical shifts, or changes to the period or phase.

Would you like details on how these transformations affect the graph or other specific aspects of the cosine function?

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Relative Questions:

1. How does changing the amplitude affect the period of the cosine function?
2. What transformations occur if a negative coefficient is introduced (e.g., $y = -5 \cos(x)$)?
3. How do vertical stretches or compressions affect the overall shape of trigonometric graphs?
4. What is the significance of the period in a cosine function, and how is it calculated?
5. Can you describe the behavior of the function if a horizontal shift is introduced, such as $y = 5 \cos(x - \pi/2)$?

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Tip:

For cosine functions, the amplitude only affects the height of the graph, not its frequency or horizontal positioning.