Answer:
a) See below
b) See attachment
Step-by-step explanation:
The graph of g(x) = 2cos(x) - 3 differs from the graph of f(x) = cos(x) in three main ways: amplitude, vertical shift, and midline.
The amplitude of g(x) is 2, which is twice that of f(x), meaning the peaks and troughs are twice as high and twice as low. Additionally, g(x) is shifted downward by 3 units, moving the midline from y = 0 to y = -3. The period and phase of the cosine function remain unchanged, so g(x) still completes one full cycle over an interval of 2π.
The vertical shift moves the midline of the graph from y = 0 to y = -3.
The amplitude of g(x) is 2. This means that the maximum value of the function is 2 units above the midline at y = -1, and the minimum value of the function is 2 units below the midline at y = -5.
The period of the parent function cos(x) is 2π, and this does not change with vertical scaling or shifting. Therefore, g(x) also completes one full cycle over an interval of 0 ≤ x ≤ 2π.
To accurately plot the key points of function g(x), divide the period into four equal intervals:
[tex]0, \dfrac{\pi}{2},\pi,\dfrac{3\pi}{2}, 2\pi[/tex]
Since the function g(x) = 2cos(x) - 3 has not undergone any horizontal shifting or reflection compared to the parent cosine function f(x) = cos(x), the maximum point remains at x = 0, where the parent function also reaches its maximum.
Therefore, the key points are:
To sketch the graph of one full cycle of g(x) = 2cos(x) - 3, plot the key points on a coordinate plane and connect them with a smooth curve.
