Parabola Domain Analysis: Finding x-Values Where f(x) > 0

Q: Why doesn't this parabola intersect the x-axis?

When a parabola has no real roots, its discriminant \( b^2 - 4ac . This means the parabola never touches or crosses the x-axis - it stays entirely on one side.

Q: How do I know if the parabola is above or below the x-axis?

Look at two things: the vertex position and the opening direction. If the vertex (the highest or lowest point) is below the x-axis and the parabola opens downward, then \( f(x) everywhere.

Q: What does it mean when f(x) > 0 has no solution?

It means there are no x-values that make the function positive. Since this parabola stays entirely below the x-axis, $ f(x) $ is always negative.

Q: Could the parabola be always positive instead?

Only if it opened upward and had its vertex above the x-axis. But this graph shows a downward-opening parabola with vertex below the x-axis, so it's always negative.

Q: How is this different from a parabola that does intersect the x-axis?

When a parabola intersects the x-axis, it has positive and negative regions separated by the x-intercepts. Here, with no intersections, the function has the same sign everywhere.

Parabola Sign Analysis with Non-Intersecting Graph

The graph of the function below does not intersect the $x$ -axis.

The parabola's vertex is marked A.

Find all values of $x$ where
$f\left(x\right) > 0$ .

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The graph of the function below does not intersect the $x$ -axis.

The parabola's vertex is marked A.

Find all values of $x$ where
$f\left(x\right) > 0$ .

Step-by-step solution

To solve this problem, let's analyze the key characteristics of the parabola:

Since the parabola does not intersect the $x$ -axis, it indicates that it is entirely either above or below the $x$ -axis.
The graph of a parabola $ax^2 + bx + c$ does not intersect the $x$ -axis when its discriminant $b^2 - 4ac$ is negative. Thus, it does not have any real roots.
If the parabola opens upwards, then the function is entirely above the $x$ -axis if $a > 0$ and below if $a < 0$ .
Given the problem indicates the parabola never reaches or crosses the $x$ -axis and the absence of real roots, a positive opening parabola cannot reach positive territory in when not intersecting the x-axis.

Since the parabola's graph neither touches nor crosses the $x$ -axis and isn't stated to be always positive or negative, we conclude:

The function does not have a positive domain.

Final Answer

The function does not have a positive domain.

Key Points to Remember

Essential concepts to master this topic

Non-intersecting Rule: Parabola entirely above or below x-axis when discriminant negative
Sign Determination: Opens downward + no x-intercepts = always negative values
Verification: Check vertex position below x-axis confirms $f(x) < 0$ for all x ✓

Common Mistakes

Avoid these frequent errors

Assuming non-intersecting means always positive
Don't assume a parabola that doesn't touch the x-axis must be positive = wrong conclusion! A downward-opening parabola with no x-intercepts stays entirely below the x-axis. Always check the vertex position and opening direction together.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

Why doesn't this parabola intersect the x-axis?

When a parabola has no real roots, its discriminant $b^2 - 4ac < 0$ . This means the parabola never touches or crosses the x-axis - it stays entirely on one side.

How do I know if the parabola is above or below the x-axis?

Look at two things: the vertex position and the opening direction. If the vertex (the highest or lowest point) is below the x-axis and the parabola opens downward, then $f(x) < 0$ everywhere.

What does it mean when f(x) > 0 has no solution?

It means there are no x-values that make the function positive. Since this parabola stays entirely below the x-axis, $f(x)$ is always negative.

Could the parabola be always positive instead?

Only if it opened upward and had its vertex above the x-axis. But this graph shows a downward-opening parabola with vertex below the x-axis, so it's always negative.

How is this different from a parabola that does intersect the x-axis?

When a parabola intersects the x-axis, it has positive and negative regions separated by the x-intercepts. Here, with no intersections, the function has the same sign everywhere.

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