https://file.aiquickdraw.com/mathbot/file/1725788152932c7sv00wv.jpg

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.

The adjoining figure shows the graph of y = ax^2 + bx + c. Which of the following is correct? (i) a > 0 (ii) b > 0 (iii) c > 0 (iv) b^2 < 4ac

This problem involves analyzing the graph of a quadratic function to determine the correctness of statements about the coefficients a, b, and c. Key concepts include the discriminant and the opening direction of the parabola.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation