The given image shows a quadratic function $y = ax^2 + bx + c$ and a graph of the parabola. The options involve conditions on $ab$ and $a + b + c$. Let’s analyze the graph and the choices step-by-step:

Observations from the graph:

1. The parabola opens upwards, which means $a > 0$.
2. The parabola crosses the x-axis twice, indicating that the quadratic equation has two real roots. Therefore, the discriminant ($\Delta = b^2 - 4ac$) must be positive.
3. The product of the roots is $\frac{c}{a}$, and the sum of the roots is $-\frac{b}{a}$. While this might not be directly needed, it helps analyze the graph.

Considering the statements:

• Sign of $ab$: Since $a > 0$, the sign of $ab$ depends on $b$. From the graph, the parabola's axis is on the left of the origin (negative $x$-coordinate), which indicates $b > 0$. Thus, $ab > 0$.
• Sign of $a + b + c$: The y-intercept of the parabola is $c$. Since the graph starts below the x-axis but eventually rises, $c$ and $a + b + c$ must be greater than 0.

Correct answer:

The correct statement is A: $ab > 0$ and $a + b + c > 0$.

Would you like a detailed explanation of the conditions for each statement?

Here are 5 questions that expand on this topic:

1. What is the role of the discriminant ($\Delta$) in determining the number of roots for a quadratic equation?
2. How does the sign of $a$ affect the direction of a parabola?
3. Can $a + b + c$ provide insights into the behavior of a quadratic function for $x = 1$?
4. What happens to the product $ab$ when $b < 0$ and $a > 0$?
5. How can the vertex form of a quadratic equation ($y = a(x-h)^2 + k$) help analyze the parabola’s properties?

Tip:

For quadratic functions, always verify the sign of $a$ first, as it determines the overall shape of the graph (upwards or downwards).