1. **State the problem:** We have a function defined as $g(x) = a\cos(2x) + d$, where $a$ and $d$ are constants. We are given some values of $g(x)$ at specific $x$ values and need to find the missing values in the table. 2. **Given values:** $$ \begin{array}{c|c} x & g(x) \\\hline 0 & 4 \\ \frac{\pi}{2} & ? \\ \frac{3\pi}{4} & 1 \\ \pi & ? \\ \frac{3\pi}{2} & ? \end{array} $$ 3. **Use the function formula:** $$g(x) = a\cos(2x) + d$$ 4. **Write equations for known points:** - At $x=0$: $$g(0) = a\cos(0) + d = a(1) + d = a + d = 4$$ - At $x=\frac{3\pi}{4}$: $$g\left(\frac{3\pi}{4}\right) = a\cos\left(2 \times \frac{3\pi}{4}\right) + d = a\cos\left(\frac{3\pi}{2}\right) + d = a(0) + d = d = 1$$ 5. **From the second equation, find $d$:** $$d = 1$$ 6. **Substitute $d=1$ into the first equation to find $a$:** $$a + 1 = 4 \implies a = 3$$ 7. **Now we have $a=3$ and $d=1$, so the function is:** $$g(x) = 3\cos(2x) + 1$$ 8. **Calculate missing values:** - At $x=\frac{\pi}{2}$: $$g\left(\frac{\pi}{2}\right) = 3\cos(\pi) + 1 = 3(-1) + 1 = -3 + 1 = -2$$ - At $x=\pi$: $$g(\pi) = 3\cos(2\pi) + 1 = 3(1) + 1 = 3 + 1 = 4$$ - At $x=\frac{3\pi}{2}$: $$g\left(\frac{3\pi}{2}\right) = 3\cos(3\pi) + 1 = 3(-1) + 1 = -3 + 1 = -2$$ 9. **Final completed table:** $$ \begin{array}{c|c} x & g(x) \\\hline 0 & 4 \\ \frac{\pi}{2} & -2 \\ \frac{3\pi}{4} & 1 \\ \pi & 4 \\ \frac{3\pi}{2} & -2 \end{array} $$