The question is asking about the properties of a quadratic function's graph, given its visual representation.

1. Understanding the Graph:

   • The graph is a parabola opening upwards, indicating that the coefficient $a$ (from the general form $ax^2 + bx + c$) is positive ($a > 0$).
   • The parabola intersects the x-axis at two distinct points, which means it has two real roots. This implies that the discriminant ($D$) of the quadratic equation is greater than zero ($D > 0$).

2. Conclusion:

   • The properties of the function are: $a > 0$ and $D > 0$.
   • Therefore, the correct answer is option a. $a > 0$, $D > 0$.

Would you like more details on the quadratic discriminant or any other aspect of the problem?

Related Questions:

1. What is the general form of a quadratic function, and how do its coefficients affect the graph?
2. How do you calculate the discriminant of a quadratic equation, and what does it signify?
3. What are the vertex and axis of symmetry of a quadratic function?
4. How do you determine whether a parabola opens upwards or downwards?
5. What are the roots of a quadratic function, and how can they be found graphically and algebraically?

Tip: The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by $D = b^2 - 4ac$. It determines the nature of the roots:

• $D > 0$: Two distinct real roots.
• $D = 0$: One real root (repeated).
• $D < 0$: No real roots (complex roots).