which of the following is the graph of $y = -3 \\cdot \\cos(3x)$?

Explanation:

Step1: Analyze the function ( y = -3\cos(3x) )

The general form of a cosine function is ( y = A\cos(Bx) ), where ( |A| ) is the amplitude and the period is ( \frac{2\pi}{|B|} ). For ( y = -3\cos(3x) ), the amplitude ( |A| = 3 ), so the graph oscillates between ( -3 ) and ( 3 ). The period is ( \frac{2\pi}{3} ), which is much smaller than ( 2\pi ) (so it's a compressed cosine curve). The negative sign reflects the graph over the x - axis.

Step2: Check the key points

• At ( x = 0 ), ( y=-3\cos(0)=-3\times1 = - 3 ). So the graph should pass through ( (0, - 3) ).
• Let's check the first graph: It has a different shape (more like a sine - wave with multiple oscillations, but our function is a cosine - based function with period ( \frac{2\pi}{3} ), but the first graph's key point at ( x = 0 ) doesn't match ( y=-3 )).
• Second graph: At ( x = 0 ), the y - value is 0, which doesn't match ( y=-3 ) at ( x = 0 ).
• Third graph: At ( x = 0 ), the y - value is 3, which is the opposite of what we need (( y=-3 ) at ( x = 0 )).
• Fourth graph: At ( x = 0 ), the y - value is - 3, which matches our calculation. Also, the period and the reflection (due to the negative sign) are consistent with this graph.

Answer:

The fourth graph (the bottom - most graph among the given options)