The image shows a graph of the quadratic function $y = ax^2 + bx + c$ and asks which of the following statements are correct based on the graph:

1. $a > 0$
2. $b > 0$
3. $c > 0$
4. $b^2 < 4ac$

Analysis:

• Nature of the Parabola: The graph is a parabola that opens downwards, which means the coefficient of $x^2$ (i.e., $a$) must be negative, so $a < 0$. Therefore, statement (i) is incorrect.

• Sign of $b$: The graph shows a parabola that intersects the x-axis at two distinct points, implying the quadratic equation has two real roots. This suggests that the discriminant $b^2 - 4ac$ must be greater than or equal to zero. If the parabola is symmetrical about the y-axis or shifted, it can provide information about $b$. However, without clear asymmetry, determining $b$ based solely on the graph is challenging.

• Sign of $c$: The y-intercept of the graph is at a positive value on the y-axis, indicating that $c > 0$. Therefore, statement (iii) is correct.

• Discriminant Condition $b^2 < 4ac$: Since the parabola intersects the x-axis at two points, the discriminant must satisfy $b^2 - 4ac \geq 0$. This means statement (iv), which states $b^2 < 4ac$, is incorrect.

Conclusion:

The correct answer is (iii) only.

Would you like more details or have any further questions?

Here are some related questions to expand your understanding:

1. How do you derive the formula for the discriminant of a quadratic equation?
2. How does the sign of $b$ affect the position of the vertex of a parabola?
3. What are the geometric interpretations of the coefficients $a$, $b$, and $c$ in a quadratic function?
4. How does the vertex form of a quadratic equation relate to its standard form?
5. What conditions on the coefficients ensure that a parabola has no real roots?

Tip: Always analyze the graph's shape and key points (like intercepts and vertex) to determine the signs of the coefficients in a quadratic function.