Question

6 multiple choice 1 point what is the equation of the graph? $y = \frac{3}{4}cos(2x)$ $y = \frac{3}{4}sin(2x)$ $y = \frac{3}{2}cosx$ $y = \frac{3}{2}sinx$

Explanation:

Step1: Analyze the general form of trigonometric functions

The general form of a cosine function is \(y = A\cos(Bx)\), and for a sine function is \(y = A\sin(Bx)\). The amplitude is \(|A|\) and the period is \(T=\frac{2\pi}{|B|}\).

Step2: Check the value at \(x = 0\)

When \(x = 0\), for \(y=\frac{3}{4}\cos(2x)\), \(y=\frac{3}{4}\cos(0)=\frac{3}{4}\). For \(y=\frac{3}{4}\sin(2x)\), \(y=\frac{3}{4}\sin(0) = 0\). For \(y=\frac{3}{2}\cos x\), \(y=\frac{3}{2}\cos(0)=\frac{3}{2}\). For \(y=\frac{3}{2}\sin x\), \(y=\frac{3}{2}\sin(0)=0\). The graph has a non - zero value at \(x = 0\), so we can eliminate the sine functions (\(y=\frac{3}{4}\sin(2x)\) and \(y=\frac{3}{2}\sin x\)).

Step3: Analyze the amplitude

The amplitude of the graph is less than \(1\). The amplitude of \(y=\frac{3}{2}\cos x\) is \(\frac{3}{2}=1.5>1\), and the amplitude of \(y = \frac{3}{4}\cos(2x)\) is \(\frac{3}{4}=0.75<1\).

Answer:

\(y=\frac{3}{4}\cos(2x)\)