Problem Analysis:

The question is about identifying the graph of a quadratic function $f(x) = ax^2 + bx + c$, where:

1. $a$, $b$, and $c$ are positive real numbers.
2. The discriminant $\Delta = b^2 - 4ac$ is less than 0 ($\Delta < 0$).

Key Insights:

1. Discriminant Condition ($\Delta < 0$):

   • If $\Delta < 0$, the quadratic function has no real roots. Therefore, the graph does not intersect the $x$-axis.
   • The parabola will either open upwards or downwards depending on the sign of $a$.

2. Sign of $a$:

   • Since $a > 0$, the parabola opens upwards.

3. Behavior of the Function:

   • $c > 0$: The $y$-intercept is positive (the graph crosses the $y$-axis above 0).
   • The graph is a parabola that opens upwards and does not intersect the $x$-axis.

Elimination of Graphs:

• Graph A: Opens upwards and does not intersect the $x$-axis. This is a valid possibility.
• Graph B: Opens upwards but intersects the $x$-axis, which violates the $\Delta < 0$ condition. Not valid.
• Graph C: Opens downwards, violating the condition $a > 0$. Not valid.
• Graph D: Opens upwards but intersects the $x$-axis, which violates the $\Delta < 0$ condition. Not valid.

Correct Answer:

Graph A is the only one that satisfies all the conditions.

Let me know if you'd like further clarification or have any questions!

Related Questions:

1. How is the discriminant related to the number of roots of a quadratic equation?
2. How does the sign of $a$ affect the orientation of a parabola?
3. Can a quadratic equation with $\Delta = 0$ have two distinct roots?
4. What happens if $c < 0$ in $f(x) = ax^2 + bx + c$?
5. How can we algebraically verify whether a graph satisfies $\Delta < 0$?

Tip:

Always check both the discriminant and the sign of $a$ when analyzing quadratic equations, as these determine the parabola's shape and intersection with the axes.