1. **State the problem:** We are given a trigonometric function of the form $$g(x) = a \sin(bx + c) + d$$. We know two points: the function crosses its midline at (1.5, 1.5) and has a minimum at (1, 1). We need to find the exact formula for $$g(x)$$ by determining constants $$a$$, $$b$$, $$c$$, and $$d$$. 2. **Identify the midline:** The function crosses the midline at (1.5,1.5), so the midline value $$d = 1.5$$. 3. **Find amplitude $$a$$:** The minimum point is at (1,1), which is below the midline by amplitude $$a$$. Since the minimum value is $$g(1) = 1$$ and midline $$d=1.5$$, amplitude $$a = 1.5 - 1 = 0.5$$. Therefore, $$a = 0.5$$. 4. **Determine phase shift and frequency:** Since $$g(x) = 0.5 \sin(bx + c) + 1.5$$, the sine function's minimum occurs at $$x=1$$. Recall that $$\sin(\theta)$$ has minimum $$-1$$ at $$\theta = \frac{3\pi}{2} + 2k\pi$$. So we must have: $$b \cdot 1 + c = \frac{3\pi}{2}$$ (taking $$k=0$$). 5. **Use midline crossing:** The function crosses the midline at (1.5, 1.5). At midline, $$\sin(bx + c) = 0$$, since $$g(x) = a\sin(bx+c)+d$$ equals $$d$$ when sine argument is 0. Thus: $$b \cdot 1.5 + c = 0$$ or some multiple of $$\pi$$. For the sine function crossing midline going upward, the argument equals $$0$$. So: $$1.5b + c = 0$$. 6. **Solve system:** From step 4: $$b + c = \frac{3\pi}{2}$$. From step 5: $$1.5b + c = 0$$. Subtract the first from the second: $$(1.5b + c) - (b + c) = 0 - \frac{3\pi}{2}$$ $$0.5b = -\frac{3\pi}{2}$$ $$b = -3\pi$$. 7. **Find $$c$$:** Substitute back to $$b + c = \frac{3\pi}{2}$$: $$-3\pi + c = \frac{3\pi}{2}$$ $$c = \frac{3\pi}{2} + 3\pi = \frac{3\pi}{2} + \frac{6\pi}{2} = \frac{9\pi}{2}$$. 8. **Write the final formula:** $$g(x) = 0.5 \sin(-3\pi x + \frac{9\pi}{2}) + 1.5$$. 9. **Simplify if desired:** Sine is odd and periodic, so we can rewrite to positive frequency: $$g(x) = 0.5 \sin\left(-3\pi x + \frac{9\pi}{2}\right) + 1.5 = 0.5 \sin\left(3\pi x - \frac{9\pi}{2}\right) + 1.5$$. This is the exact expression for $$g(x)$$.