Answer: The graph passes through \((0,4)\), \(\left(\frac{\pi}{4},0\right)\), \(\left(\frac{\pi}{2},4\right)\), \(\left(\frac{3\pi}{4},0\right)\), and \((\pi,4)\).
Answer: The graph passes through \((0,4)\), \(\left(\frac{\pi}{4},0\right)\), \(\left(\frac{\pi}{2},4\right)\), \(\left(\frac{3\pi}{4},0\right)\), and \((\pi,4)\).
The graph of \(y=4\cos2x\) has amplitude \(4\) and period \(\pi\).
The absolute value reflects any part below the \(x\)-axis above the \(x\)-axis, so \(y=\left|4\cos2x\right|\) is always non-negative.
The graph meets the \(x\)-axis when
\(\cos2x=0.\)
For \(0\leq x\leq\pi\), this gives
\(2x=\frac{\pi}{2},\frac{3\pi}{2},\)
so
\(x=\frac{\pi}{4},\frac{3\pi}{4}.\)
At \(x=0,\frac{\pi}{2},\pi\), the value of \(\left|4\cos2x\right|\) is \(4\).
Therefore the key points are
\((0,4),\quad \left(\frac{\pi}{4},0\right),\quad \left(\frac{\pi}{2},4\right),\quad \left(\frac{3\pi}{4},0\right),\quad (\pi,4).\)
