Question: Which equation is for a parabola that opens down?
y = 2(x - 4)(x + 3)
y = -3x² + 6x -7
y = x² - 3x - 5
y = 3(x - 4)² - 2
1. **State the problem:** We need to determine which of the given equations represents a parabola that opens down.
2. **Recall the rule for parabolas:**
A parabola given by the quadratic function $$y = ax^2 + bx + c$$ opens:
- **Upward** if $$a > 0$$
- **Downward** if $$a < 0$$
3. **Analyze each equation:**
- $$y = 2(x - 4)(x + 3)$$
Expanding:
$$y = 2(x^2 - x - 12) = 2x^2 - 2x - 24$$
Here, $$a = 2 > 0$$, so it opens **up**.
- $$y = -3x^2 + 6x - 7$$
Here, $$a = -3 < 0$$, so it opens **down**.
- $$y = x^2 - 3x - 5$$
Here, $$a = 1 > 0$$, so it opens **up**.
- $$y = 3(x - 4)^2 - 2$$
Expanding the square:
$$y = 3(x^2 - 8x + 16) - 2 = 3x^2 - 24x + 48 - 2 = 3x^2 - 24x + 46$$
Here, $$a = 3 > 0$$, so it opens **up**.
4. **Conclusion:**
Only the equation $$y = -3x^2 + 6x - 7$$ has a negative leading coefficient $$a$$, so it represents a parabola that opens down.
**Final answer:** $$y = -3x^2 + 6x - 7$$