Final answer:
The function g(x) = 3cos(x) represents a vertical scaling of the original cosine function f(x) = cos(x) by a factor of 3, which increases the amplitude of the waveform without changing its frequency or period.
Step-by-step explanation:
When we compare the function f(x) = cos(x) with the transformed function g(x) = 3cos(x), the transformation applied is a vertical scaling of the original function by a factor of 3. This means that the amplitude of the cosine function is multiplied by 3, resulting in a graph where all of the y-values are three times higher than in the original function. The amplitude represents the maximum value of the wave above or below the midline of the graph, and this transformation does not cause any horizontal shifts (phase shifts) or reflections; it solely affects the height of the peaks and depth of the troughs in the graph of the cosine function.
It's important to note that the frequency and period of the function remain unchanged. Frequency, which is indicative of how many cycles the function completes in a certain interval, is not affected by vertical scaling. Similarly, the period, the horizontal length of one complete cycle, remains the same. This means that the shape of the graph is the same as the original cosine function, just vertically stretched.
In summary, the transformation from f(x) = cos(x) to g(x) = 3cos(x) results in a function where each point on the graph of f(x) moves straight up or down to a new position that is three times further from the x-axis, while still lying exactly above or beneath its original position.