Question

analyzing a sine graph
which of the following statements are true? choose three correct answers.
the amplitude is $\frac{1}{2}$.
the equation of the graph is $y = \frac{1}{2}\sin(2x)$
the graph is stretched vertically by a factor of 2.
the midline has the equation $x=0$.
the graph is compressed horizontally by a factor of $\frac{1}{2}$.
this question requires at least 3 answers.

Explanation:

Step1: Identify the amplitude

The maximum distance from the midline (x-axis) to the peak is $\frac{1}{2}$, so amplitude = $\frac{1}{2}$.

Step2: Identify horizontal compression

The standard sine wave $y=\sin(x)$ has period $2\pi$. This graph completes 4 cycles in $2\pi$, so period is $\frac{2\pi}{4}=\frac{\pi}{2}$. For $y=\sin(Bx)$, period = $\frac{2\pi}{B}$. Solving $\frac{2\pi}{B}=\frac{\pi}{2}$ gives $B=2$, meaning horizontal compression by $\frac{1}{2}$.

Step3: Verify the equation

Combining amplitude $\frac{1}{2}$ and horizontal compression factor 2, the equation is $y=\frac{1}{2}\sin(2x)$.

Step4: Evaluate other statements

Vertical stretch would mean amplitude >1, which is false. Midline is $y=0$, not $x=0$, so that is false.

Answer:

The amplitude is $\frac{1}{2}$.
The equation of the graph is $y = \frac{1}{2}\sin(2x)$
The graph is compressed horizontally by a factor of $\frac{1}{2}$