Parabola Analysis: Finding f(x) < 0 with Non-Intersecting X-Axis

Q: How can a parabola never cross the x-axis?

When a parabola opens upward but its vertex is above the x-axis, or opens downward with vertex below the x-axis, it never touches or crosses the x-axis. This happens when the discriminant \( b^2 - 4ac .

Q: If the parabola doesn't intersect the x-axis, how do I know if it's always positive or negative?

Look at the vertex position and opening direction! If it opens upward and vertex is above x-axis, it's always positive. If it opens upward and vertex is below x-axis, it's always negative.

Q: What does 'the domain is always negative' mean?

This means \( f(x) for all possible x-values. The function output is negative no matter what x you choose. It's the entire real number line where the function is negative.

Q: Can I use the vertex to determine the sign of the function?

Absolutely! The vertex is the extreme point of the parabola. If the vertex is below the x-axis and the parabola opens upward, then every point on the parabola is below the x-axis, making \( f(x) everywhere.

Q: Why isn't the answer 'no solution' if we want f(x) < 0?

Because the parabola is entirely below the x-axis! This means \( f(x) for every x-value. The solution is all real numbers, not no solution.

Parabola Analysis with Non-Intersecting Graphs

The graph of the function below the 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

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Understand the problem

The graph of the function below the 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 decide where $f(x) < 0$ for the given parabola, observe the following:

The parabola does not intersect the x-axis, indicating it is either entirely above or below the x-axis.
If the parabola were entirely above the x-axis for $f(x) > 0$ , it would contradict the question by not giving a valid interval for $f(x) < 0$ .
Therefore, the correct conclusion is that the parabola is entirely below the x-axis, meaning $f(x) < 0$ for all $x$ .

Based on the understanding of quadratic functions and their graph behavior, the function does not intersect the x-axis implies it is always negative.

Hence, the domain where $f(x) < 0$ is for all $x$ . This leads us to choose:

The domain is always negative.

Final Answer

The domain is always negative.

Key Points to Remember

Essential concepts to master this topic

Graph Reading: Non-intersecting parabolas are entirely above or below x-axis
Technique: If parabola opens upward and never crosses x-axis, always positive
Check: Pick any x-value and verify y-coordinate sign matches conclusion ✓

Common Mistakes

Avoid these frequent errors

Assuming parabolas always have x-intercepts
Don't assume every parabola crosses the x-axis = missing that some stay entirely above or below! When discriminant is negative, parabolas never intersect the x-axis. Always examine the graph's position relative to the x-axis first.

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

How can a parabola never cross the x-axis?

When a parabola opens upward but its vertex is above the x-axis, or opens downward with vertex below the x-axis, it never touches or crosses the x-axis. This happens when the discriminant $b^2 - 4ac < 0$ .

If the parabola doesn't intersect the x-axis, how do I know if it's always positive or negative?

Look at the vertex position and opening direction! If it opens upward and vertex is above x-axis, it's always positive. If it opens upward and vertex is below x-axis, it's always negative.

What does 'the domain is always negative' mean?

This means $f(x) < 0$ for all possible x-values. The function output is negative no matter what x you choose. It's the entire real number line where the function is negative.

Can I use the vertex to determine the sign of the function?

Absolutely! The vertex is the extreme point of the parabola. If the vertex is below the x-axis and the parabola opens upward, then every point on the parabola is below the x-axis, making $f(x) < 0$ everywhere.

Why isn't the answer 'no solution' if we want f(x) < 0?

Because the parabola is entirely below the x-axis! This means $f(x) < 0$ for every x-value. The solution is all real numbers, not no solution.

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