The question shown in the image asks, "Which of these parabolas opens downwards and has its vertex at the point (1, -1)?"

Let's analyze the options:

• For a parabola that opens downwards, the coefficient of the quadratic term must be negative, meaning the parabola will have an inverted shape.
• The vertex of the parabola is the point where it reaches its maximum or minimum. In this case, the vertex is given as (1, -1), meaning that at $x = 1$, $y = -1$.

Now, looking at the graphs provided:

• Graph A: Opens downwards, and the vertex seems to be at the point (1, -1). This graph is a match to the conditions.
• Graph B: Opens upwards, so it is not the correct choice.
• Graph C: Opens upwards as well, so it is also not correct.

Therefore, Graph A is the correct answer.

Would you like more details or have any additional questions?

Relative Questions:

1. How do you determine whether a parabola opens upwards or downwards based on its equation?
2. What is the significance of the vertex in a quadratic function?
3. Can you explain the general form of a quadratic function and how to find the vertex?
4. How would you solve for the vertex algebraically if the equation of the parabola is given?
5. How does the axis of symmetry relate to the vertex of the parabola?

Tip:

Remember, for the quadratic equation $y = ax^2 + bx + c$, if $a > 0$, the parabola opens upwards; if $a < 0$, it opens downwards.