To determine which graph represents a parabola whose corresponding quadratic equation has imaginary roots, we need to consider the following:

A quadratic equation has imaginary roots when the discriminant $\Delta = b^2 - 4ac$ is negative. This implies that the graph of the quadratic function $y = ax^2 + bx + c$:

• Does not intersect the $x$-axis (since real roots correspond to the $x$-intercepts of the graph).

Analyzing the provided graphs:

1. Graph 1 (Top): This parabola opens downwards and crosses the $x$-axis, meaning it has real roots.
2. Graph 2 (Second from Top): This parabola opens upwards and does not touch the $x$-axis, meaning its roots are imaginary.
3. Graph 3 (Third from Top): This parabola opens upwards and crosses the $x$-axis, meaning it has real roots.
4. Graph 4 (Bottom): This parabola opens downwards and crosses the $x$-axis, meaning it has real roots.

Correct Answer: Graph 2

Would you like a detailed explanation of the discriminant and its relationship to the graph?

Relative Questions:

1. How can you calculate the discriminant of a quadratic equation?
2. What does it mean geometrically when a parabola has one real root?
3. How do you determine the vertex of a parabola using its equation?
4. How does the sign of the coefficient $a$ in $y = ax^2 + bx + c$ affect the direction of the parabola?
5. Can a parabola ever have exactly two imaginary roots?

Tip:

When analyzing the roots of a quadratic function, always link the discriminant to the graph's intersection with the $x$-axis for a clearer understanding.