1. **State the problem:** Find the exact formula for the function $g(x) = a \sin(bx + c) + d$ given that it has a minimum point at $(1, 1)$ and crosses its midline at $(1.5, 1.5)$. 2. **Identify key points and parameters:** - Midline is the horizontal centerline, so $d = 1.5$. - Minimum point at $(1, 1)$ means the lowest value is $1$. 3. **Calculate amplitude $a$:** Amplitude = distance from midline to minimum or maximum. $$a = |d - \text{minimum}| = |1.5 - 1| = 0.5$$ Since it is a sine function with minimum lower than midline, $a = -0.5$ (to reflect a downward shift). 4. **Use the property of sine function at minimum:** Minimum of sine is at $\sin(\theta) = -1$. So, $$a \sin(b \cdot 1 + c) + d = 1 \implies -0.5 \sin(b + c) + 1.5 = 1$$ Simplify: $$-0.5 \sin(b + c) = -0.5 \implies \sin(b + c) = 1$$ 5. **Use intercept at midline:** At $x=1.5$, function crosses the midline $y=1.5$, so $$g(1.5) = a \sin(b \cdot 1.5 + c) + d = 1.5$$ Substitute known values: $$-0.5 \sin(1.5b + c) + 1.5 = 1.5 \implies \sin(1.5b + c) = 0$$ 6. **Solve for $b$ and $c$ using sine properties:** Since $\sin(b + c) = 1$, then $$b + c = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}$$ Since $\sin(1.5b + c) = 0$, then $$1.5b + c = k\pi, \quad k \in \mathbb{Z}$$ 7. **Set $k=0$ for intercept point:** $$1.5b + c = 0$$ From minimum point (choose $k=0$): $$b + c = \frac{\pi}{2}$$ 8. **Solve system:** From $1.5b + c = 0$, $$c = -1.5b$$ Substitute into $b + c = \frac{\pi}{2}$: $$b - 1.5b = \frac{\pi}{2} \implies -0.5b = \frac{\pi}{2} \implies b = -\pi$$ Then, $$c = -1.5(-\pi) = 1.5\pi$$ 9. **Write final formula:** $$g(x) = -0.5 \sin(-\pi x + 1.5 \pi) + 1.5$$ Alternatively, use sine identity $\sin(-\theta) = -\sin(\theta)$: $$g(x) = -0.5 (-\sin(\pi x - 1.5 \pi)) + 1.5 = 0.5 \sin(\pi x - 1.5 \pi) + 1.5$$ Either form is exact. The initial form matches minimum at $x=1$.