Question

a cosine function has been reflected over the $x$-axis and horizontally compressed by a factor of $\frac{1}{3}$. which equation matches this description?
$y = -cosleft(\frac{1}{3}x\
ight)$
$y = cos\left(-\frac{1}{3}x\
ight)$
$y = \cos(-3x)$
$y = -cos(3x)$

Explanation:

Step1: Reflect over x-axis

For a parent function $y = \cos(x)$, reflecting over the x-axis multiplies the function by -1, giving $y = -\cos(x)$.

Step2: Horizontal compression by $\frac{1}{3}$

A horizontal compression by factor $\frac{1}{a}$ replaces $x$ with $ax$. Here, $\frac{1}{a}=\frac{1}{3}$ so $a=3$, replacing $x$ with $3x$ gives $y = -\cos(3x)$.

Step3: Verify cosine even property

$\cos(-x)=\cos(x)$, so options with negative inside cosine do not represent reflection over x-axis.

Answer:

$y = -\cos(3x)$