Question

describe the graph of the function $f(x) = \cos(2x)$ as a transformation of its parent function. (1 point) \bigcirc the graph is a horizontal compression by a factor of $\frac{1}{2}$. \bigcirc the graph is a vertical stretch by a factor of 2. \bigcirc the graph is a horizontal stretch by a factor of 2. \bigcirc the graph is a vertical compression by a factor of $\frac{1}{2}$.

Explanation:

Step1: Recall Transformations of Cosine

For a function \( y = \cos(Bx) \), the horizontal transformation is determined by \( B \). If \( |B|>1 \), it's a horizontal compression; if \( 0<|B|<1 \), it's a horizontal stretch. The factor of compression/stretch is \( \frac{1}{|B|} \).

Step2: Analyze \( f(x)=\cos(2x) \)

Here, \( B = 2 \). Since \( |B| = 2>1 \), it's a horizontal compression. The factor of compression is \( \frac{1}{2} \) (because \( \frac{1}{|B|}=\frac{1}{2} \)). Vertical transformations involve coefficients outside the cosine function, so options about vertical stretch/compression are incorrect. A horizontal stretch would occur for \( B<1 \), so that's also incorrect.

Answer:

The graph is a horizontal compression by a factor of \(\frac{1}{2}\) (the first option).