Parabola Property Analysis: Upward-Facing Curve with Vertex Below X-axis

Q: If the vertex is below the x-axis, doesn't that mean the whole parabola is negative?

No! The vertex being negative only means the minimum value is negative. Since the parabola opens upward, it will eventually cross the x-axis and become positive for large values of |x|.

Q: How can I tell where the parabola becomes positive?

The parabola crosses the x-axis (becomes zero) at $ x = h \pm \sqrt{\frac{-k}{a}} $ where (h,k) is the vertex. For x-values outside this range, the parabola is positive.

Q: How do I visualize this concept?

Imagine a U-shaped curve with its bottom point below the x-axis. As you move left or right from the bottom, the curve eventually rises above the x-axis, showing positive y-values.

Q: Does the direction the parabola opens matter?

Absolutely! This problem specifically mentions an upward-opening parabola. If it opened downward with vertex below the x-axis, it would indeed always be negative.

Q: Can you give me a specific example?

Consider $ y = x^2 - 4 $. The vertex is at (0, -4), below the x-axis. But when x = 3, y = 9 - 4 = 5, which is positive!

Parabola Sign Analysis with Vertex Positioning

If a parabola is bending upwards and its vertex is below the x-axis, then it is always negative.

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Step-by-step written solution

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Understand the problem

If a parabola is bending upwards and its vertex is below the x-axis, then it is always negative.

Step-by-step solution

The solution to this problem involves understanding the behavior of parabolas. Given that the parabola opens upwards, indicated by $a > 0$ , and the vertex $(h,k)$ is below the x-axis, $k < 0$ , here's the detailed explanation:
1. The vertex of the parabola, $(h, k)$ , is at the lowest point because the parabola opens upwards.
2. With $k < 0$ , the value of $f(h) = k$ is negative. However, as $x$ moves away from the vertex, the function increases since it opens upwards.

Therefore, for large $|x|$ , $f(x)$ becomes positive. For instance, at $x = h \pm \sqrt{\frac{-k}{a}}$ , the parabola can cross the x-axis and become positive.

Given that the parabola will eventually have positive $y$ -values for either very large or very small $x$ , the function is not always negative. Hence, the statement that the parabola is always negative is Incorrect.

Final Answer

Incorrect

Key Points to Remember

Essential concepts to master this topic

Vertex Rule: Upward parabola vertex is the minimum point
Sign Change: Function crosses x-axis at $x = h \pm \sqrt{\frac{-k}{a}}$
Check Domain: Test values far from vertex show positive y-values ✓

Common Mistakes

Avoid these frequent errors

Assuming parabola sign stays constant throughout domain
Don't think that just because the vertex is negative, the entire parabola stays negative = ignores x-axis crossings! This misses how upward parabolas eventually become positive as x moves away from the vertex. Always check the parabola's behavior at multiple points across its domain.

Practice Quiz

Test your knowledge with interactive questions

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
$f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

If the vertex is below the x-axis, doesn't that mean the whole parabola is negative?

No! The vertex being negative only means the minimum value is negative. Since the parabola opens upward, it will eventually cross the x-axis and become positive for large values of |x|.

How can I tell where the parabola becomes positive?

The parabola crosses the x-axis (becomes zero) at $x = h \pm \sqrt{\frac{-k}{a}}$ where (h,k) is the vertex. For x-values outside this range, the parabola is positive.

What if the parabola never crosses the x-axis?

If the vertex is below the x-axis but the parabola doesn't cross it, then the parabola would be always negative. This happens when the discriminant is negative. But the question asks about the general case!

How do I visualize this concept?

Imagine a U-shaped curve with its bottom point below the x-axis. As you move left or right from the bottom, the curve eventually rises above the x-axis, showing positive y-values.

Does the direction the parabola opens matter?

Absolutely! This problem specifically mentions an upward-opening parabola. If it opened downward with vertex below the x-axis, it would indeed always be negative.

Can you give me a specific example?

Consider $y = x^2 - 4$ . The vertex is at (0, -4), below the x-axis. But when x = 3, y = 9 - 4 = 5, which is positive!

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