Answer:
A) The y-intercept is 3.
Step-by-step explanation:
The general form of the cosine function is:
[tex]y=A\cos (B(x+C))+D[/tex]
where:
- |A| is the amplitude (distance between the midline and the peak).
- 2π/B is the period (horizontal distance between consecutive peaks).
- C is the phase shift (horizontal shift where negative is to the right).
- D is the vertical shift (y = D is the midline).
In the case of y = 3 cos (2x):
Therefore, the amplitude is 3 and the midline is y = 0.
The maximum value of the function is the midline plus the amplitude (0 + 3 = 3), while the minimum value of the function is the midline minus the amplitude (0 - 3 = -3). Therefore, the range of the function is [-3, 3].
To calculate the period, substitute B = 2 into the period formula:
[tex]\textsf{Period}=\dfrac{2\pi}{B}=\dfrac{2\pi}{2}=\pi[/tex]
Therefore, the period is π.
The y-intercept is the point at which the curve intersects the y-axis, so when x = 0. A peak of the parent cosine function y = cos(x) occurs at x = 0. Since the given function has no phase shift, its peak will also occur at x = 0. As the maximum value of the function is y = 3, the y-intercept is 3.
The y-intercept is the point where the curve intersects the y-axis, which occurs when x = 0. A peak of the parent cosine function y = cos(x) occurs at x = 0. Since the given function has no phase shift, its peak will also occur at x = 0. As the maximum value of the function is y = 3, the y-intercept is 3.
In summary, the true statement about the graph of y = 3 cos (2x) is:
[tex]\Large\boxed{\textsf{The $y$-intercept is 3.}}[/tex]
