Question

nalyzing a cosine graph
here is the graph of a cosine function in the form $y = a\cos(bx)$.
which statements are true? choose three correct answers.
the period is $\pi$
the graph is stretched vertically by a factor of 2
the frequency is $\frac{1}{\pi}$
the graph is horizontally compressed by a factor of $\frac{1}{2}$
the period is $2\pi$
this question requires at least 3 answers.

Explanation:

Step1: Identify amplitude

The standard cosine function $y=\cos(x)$ has amplitude 1. This graph has a maximum of 2 and minimum of -2, so amplitude $a=2$. This means vertical stretch by factor 2.

Step2: Calculate period

Count the distance between two consecutive peaks: from $0$ to $\pi$, so period $T=\pi$.

Step3: Find horizontal compression factor

Standard period is $2\pi$. The new period is $\pi = \frac{2\pi}{2}$, so $b=2$. A horizontal compression by factor $\frac{1}{2}$ gives $y=\cos(2x)$.

Step4: Calculate frequency

Frequency $f=\frac{1}{T} = \frac{1}{\pi}$.

Answer:

• The period is $\pi$
• The graph is stretched vertically by a factor of 2
• The graph is horizontally compressed by a factor of $\frac{1}{2}$