Answer :
(Part 1) Writing another function with the same graph as y = 2 cos(x) - 1:
To write another function that has the same graph as y = 2 cos(x) - 1, we need to apply transformations that don't change the shape of the graph. We can apply a horizontal shift, which means adding or subtracting a constant inside the argument of the cosine function.
Let's create a new function by applying a horizontal shift of π units to the right.
The new function will be:
\[ y = 2 \cos(x - \pi) - 1 \]
This horizontal shift moves the graph of the cosine function π units to the right on the x-axis. However, because the cosine function is periodic with a period of 2π, this shift doesn't change the overall shape of the graph; it looks exactly the same as the original function y = 2 cos(x) - 1.
(Part 2) Comparing the graphs y = 2 cos(x) - 1 and y = 2 cos(2x) - 1:
First, let's discuss how the graphs of these two functions are alike:
1. Amplitude: The amplitude of both graphs is 2. The amplitude of a cosine function is the coefficient in front of the cos() term, which is 2 in both cases. The amplitude represents the maximum distance from the center line (y value without the cosine function) of the graph to its peak or trough.
2. Vertical Shift: Both graphs are shifted downward by 1 unit, which is indicated by the "-1" at the end of each function. This means that the center line of both graphs is y = -1, and the entire graph oscillates around this horizontal line.
Now, let's look at how they are different:
1. Period: The period of a cosine function is the length of one complete cycle of the curve. For a standard cosine function y = cos(x), the period is 2Ï€. However, when there is a coefficient of x inside the cosine, the period becomes \( \frac{2\pi}{|B|} \) where B is the coefficient of x.
- For y = 2 cos(x) - 1, the period is \( \frac{2\pi}{1} = 2\pi \). This is because there is no coefficient multiplying x, so it is just 1.
- For y = 2 cos(2x) - 1, the period is \( \frac{2\pi}{2} = \pi \). This is because the coefficient of x is 2 (B = 2), which compresses the graph horizontally, making it complete its cycle in half the length.
2. Frequency: Frequency is the number of cycles the function completes in a given interval. The graph of y = 2 cos(2x) - 1 has a higher frequency than the graph of y = 2 cos(x) - 1. It completes twice as many cycles over the same length on the x-axis because of the coefficient 2 in front of the x, which makes the function oscillate twice as fast.
To summarize, both graphs share the same amplitude and vertical shift, but they have different periods and frequencies, with y = 2 cos(2x) - 1 having half the period and twice the frequency of y = 2 cos(x) - 1.
