Question: Write an equation for the graph shown.
$(x - 1)$
$(2, 1)$
$(1, 0)$
Graph shape: an increasing S-shaped cubic curve with inflection near $(2, 1)$ and passing through the labeled points $(1, 0)$ and $(2, 1)$, position_hint = center
1. **State the problem:** We need to find an equation for a cubic curve that passes through the points $(1,0)$ and $(2,1)$ and has an S-shape with an inflection near $(2,1)$.
2. **General form of a cubic function:** A cubic function can be written as $$y = a(x - h)^3 + k$$ where $(h,k)$ is the inflection point.
3. **Use the inflection point:** Since the inflection is near $(2,1)$, we set $h=2$ and $k=1$, so the function is $$y = a(x - 2)^3 + 1$$
4. **Use the point $(1,0)$ to find $a$:** Substitute $x=1$, $y=0$:
$$0 = a(1 - 2)^3 + 1$$
$$0 = a(-1)^3 + 1$$
$$0 = -a + 1$$
$$a = 1$$
5. **Final equation:** Substitute $a=1$ back:
$$y = (x - 2)^3 + 1$$
6. **Check the point $(2,1)$:**
$$y = (2 - 2)^3 + 1 = 0 + 1 = 1$$ which matches the point.
**Answer:** The equation of the cubic curve is $$y = (x - 2)^3 + 1$$.