Question

1. $y=-3cos x$, $-2pileq xleq2pi$

Explanation:

Step1: Find the amplitude

The general form of a cosine function is \(y = A\cos(x)\), where \(|A|\) is the amplitude. For the function \(y=-3\cos(x)\), \(A = - 3\). The amplitude is \(|A|=\vert-3\vert = 3\).

Step2: Determine key points

• When \(x = 0\), \(y=-3\cos(0)=-3\times1=-3\)
• When \(x=\frac{\pi}{2}\), \(y =-3\cos(\frac{\pi}{2})=-3\times0 = 0\)
• When \(x=\pi\), \(y=-3\cos(\pi)=-3\times(- 1)=3\)
• When \(x=\frac{3\pi}{2}\), \(y=-3\cos(\frac{3\pi}{2})=-3\times0 = 0\)
• When \(x = 2\pi\), \(y=-3\cos(2\pi)=-3\times1=-3\)
• Due to the even - function property of \(y = \cos(x)\) (\(\cos(-x)=\cos(x)\)), when \(x=-\frac{\pi}{2}\), \(y = 0\); when \(x =-\pi\), \(y = 3\); when \(x=-2\pi\), \(y=-3\)

Step3: Plot the points and draw the graph

Plot the points \((-2\pi,-3),(-\pi,3),(-\frac{\pi}{2},0),(0, - 3),(\frac{\pi}{2},0),(\pi,3),(2\pi,-3)\) on the coordinate plane. Then, connect these points with a smooth curve. The graph of \(y =-3\cos(x)\) is a cosine - type curve with amplitude \(3\), reflected about the \(x\) - axis (because of the negative sign) over the interval \([-2\pi,2\pi]\)

Answer:

The graph of \(y=-3\cos(x)\) for \(-2\pi\leq x\leq2\pi\) is a cosine - like curve with amplitude \(3\), passing through the points \((-2\pi,-3),(-\pi,3),(-\frac{\pi}{2},0),(0, - 3),(\frac{\pi}{2},0),(\pi,3),(2\pi,-3)\) and is symmetric about the \(y\) - axis.