Parabola Property Analysis: Vertex Position Below X-axis and Positive Values

Q: If a parabola opens upward, doesn't that mean it's always positive?

Not necessarily! An upward-opening parabola has a minimum value at its vertex. If that minimum is below the x-axis, the parabola will have negative values near the vertex.

Q: What does it mean for the vertex to be below the x-axis?

When the vertex is below the x-axis, the y-coordinate of the vertex is negative. Since this is the lowest point of an upward parabola, the function has negative values at least at that point.

Q: Can an upward parabola ever cross the x-axis if its vertex is below?

Yes! If the vertex is below the x-axis but not too far down, the parabola will cross the x-axis at two points, creating a U-shape that dips below zero between those crossing points.

Q: How can I tell if a parabola is always positive?

For an upward-opening parabola to be always positive, its vertex must be above the x-axis (k > 0 in vertex form). If k ≤ 0, the parabola will have non-positive values.

Q: What's the difference between vertex form and standard form?

Vertex form: $ y = a(x-h)^2 + k $ clearly shows vertex at (h, k)Standard form: $ y = ax^2 + bx + c $ requires completing the square to find vertex

Quadratic Function Analysis with Vertex Properties

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

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

Step-by-step solution

To analyze this problem, we'll follow these steps:

Step 1: Understand the structure of a quadratic function.
Step 2: Explore the implications of the vertex location below the x-axis.
Step 3: Analyze specific conditions where the function might not be always positive.

Step 1:
A parabola $y = ax^2 + bx + c$ opens upwards if $a > 0$ .

Step 2:
The vertex form of a quadratic is $y = a(x-h)^2 + k$ , where $(h, k)$ is the vertex. If $k < 0$ , the vertex is below the x-axis, making $y$ negative at $x = h$ (the minimum point for an upward-opening parabola).

Step 3:
Since $y$ at the vertex is $k < 0$ , it implies the function is negative at least at this point. Thus, the function cannot always be positive, as there exists at least one point where it is non-positive (negative).

Therefore, the assertion that the parabola is always positive is incorrect.

The correct answer is: Incorrect.

Final Answer

Incorrect

Key Points to Remember

Essential concepts to master this topic

Vertex Rule: Upward parabola minimum occurs at vertex position
Technique: Use vertex form $y = a(x-h)^2 + k$ where k is y-coordinate
Check: If vertex below x-axis, then k < 0 means negative values exist ✓

Common Mistakes

Avoid these frequent errors

Assuming upward parabolas are always positive
Don't think upward-opening means always positive = wrong conclusion! The vertex position determines the minimum value. Always check if the vertex y-coordinate is above or below the x-axis first.

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 a parabola opens upward, doesn't that mean it's always positive?

Not necessarily! An upward-opening parabola has a minimum value at its vertex. If that minimum is below the x-axis, the parabola will have negative values near the vertex.

What does it mean for the vertex to be below the x-axis?

When the vertex is below the x-axis, the y-coordinate of the vertex is negative. Since this is the lowest point of an upward parabola, the function has negative values at least at that point.

Can an upward parabola ever cross the x-axis if its vertex is below?

Yes! If the vertex is below the x-axis but not too far down, the parabola will cross the x-axis at two points, creating a U-shape that dips below zero between those crossing points.

How can I tell if a parabola is always positive?

For an upward-opening parabola to be always positive, its vertex must be above the x-axis (k > 0 in vertex form). If k ≤ 0, the parabola will have non-positive values.

What's the difference between vertex form and standard form?

Vertex form: $y = a(x-h)^2 + k$ clearly shows vertex at (h, k)
Standard form: $y = ax^2 + bx + c$ requires completing the square to find vertex

Is there a quick way to check my answer?

Yes! Remember: if the vertex is below the x-axis AND the parabola opens upward, then the statement "always positive" is false because the vertex itself gives a negative y-value.

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