Question

calculator
which function is shown on the graph?
$f(x) = -\frac{1}{2}sin x$
$f(x) = \frac{1}{2}sin x$
$f(x) = \frac{1}{2}cos x$
$f(x) = -\frac{1}{2}cos x$

Explanation:

Step1: Check x=0 value

At $x=0$, the graph has a positive y-value.

• For $f(x)=-\frac{1}{2}\sin x$: $f(0)=-\frac{1}{2}\sin0=0$ (does not match)
• For $f(x)=\frac{1}{2}\sin x$: $f(0)=\frac{1}{2}\sin0=0$ (does not match)
• For $f(x)=\frac{1}{2}\cos x$: $f(0)=\frac{1}{2}\cos0=\frac{1}{2}$ (positive, matches)
• For $f(x)=-\frac{1}{2}\cos x$: $f(0)=-\frac{1}{2}\cos0=-\frac{1}{2}$ (negative, does not match)

Step2: Verify midline/shape

The graph has the shape of a cosine curve (starts at a peak, oscillates) with amplitude $\frac{1}{2}$, consistent with $f(x)=\frac{1}{2}\cos x$.

Answer:

$f(x)=\frac{1}{2}\cos x$