Question: Which is the correct equation for the parabola shown in the graph?
y = 2(x - 1)^2 + 4
y = -2(x - 1)^2 + 4
y = 2(x + 1)^2 + 4
y = -2(x + 1)^2 + 4
Graph: downward-opening parabola with vertex at (1, 4), so position_hint = center
1. **State the problem:** We need to find the correct equation of a parabola given its graph.
2. **Identify key features:** The parabola opens downward and has its vertex at the point $$(1, 4)$$.
3. **Recall the vertex form of a parabola:**
$$y = a(x - h)^2 + k$$
where $$(h, k)$$ is the vertex and $$a$$ determines the direction and width.
4. **Apply the vertex:** Since the vertex is $$(1, 4)$$, the equation must be of the form:
$$y = a(x - 1)^2 + 4$$
5. **Determine the sign of $$a$$:** The parabola opens downward, so $$a < 0$$.
6. **Check the options:**
- $$y = 2(x - 1)^2 + 4$$ (opens upward, incorrect)
- $$y = -2(x - 1)^2 + 4$$ (opens downward, vertex correct, plausible)
- $$y = 2(x + 1)^2 + 4$$ (vertex at $$(-1,4)$$, incorrect)
- $$y = -2(x + 1)^2 + 4$$ (vertex at $$(-1,4)$$, incorrect)
7. **Conclusion:** The correct equation is:
$$y = -2(x - 1)^2 + 4$$