Parabola Vertex on X-axis: Analyzing Positive Value Behavior

Q: Why can't we determine if the parabola is always positive?

Because we need to know the sign of coefficient a! If $ a > 0 $, the parabola opens upward and is always positive except at the vertex. If \( a , it opens downward and is always negative except at the vertex.

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

It means the vertex has coordinates $ (h, 0) $ where the y-coordinate is zero. This gives us the vertex form $ y = a(x-h)^2 $ since k = 0.

Q: How do I know which way a parabola opens?

Look at the coefficient a in $ y = a(x-h)^2 + k $:If $ a > 0 $: opens upward (U-shape)If \( a : opens downward (∩-shape)

Q: Can a parabola with vertex on x-axis ever be negative?

Yes! If the coefficient a is negative, the parabola opens downward. The vertex at $ (h, 0) $ is the highest point, so all other y-values are negative.

Q: What should I do when information is missing from a problem?

Carefully identify what's given and what's needed. If key information like the sign of a coefficient is missing, the answer is often "Cannot be determined" rather than making assumptions.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

If the vertex of a parabola is on the x-axis, then the function is always positive except for the vertex point.

2

Step-by-step solution

To determine if a parabola with its vertex on the x-axis is always positive except for the vertex point, consider the vertex form of a quadratic function: $y = a(x-h)^2 + k$

Since the vertex is on the x-axis, we have $k = 0$ , so the equation simplifies to $y = a(x-h)^2$ .
The vertex of the parabola is the point $(h, 0)$ .
If $a > 0$ , the parabola opens upwards, meaning $y$ values are always non-negative but equal zero at the vertex.
If $a < 0$ , the parabola opens downwards, indicating the vertex is at a maximum point, and all other $y$ values are negative.
Therefore, the behavior of the parabola (whether it is always positive except at the vertex) is dependent on the sign of $a$ , which is not specified in the problem.

Without knowing the sign of $a$ , we cannot definitively determine if the parabola is always positive except at the vertex. Thus, the answer is that the outcome "cannot be determined" from the given information.

Therefore, the correct answer choice is Cannot be determined.

3

Final Answer

Cannot be determined

Key Points to Remember

Essential concepts to master this topic

Vertex Form: When vertex is on x-axis, equation becomes $y = a(x-h)^2$
Parameter Analysis: Sign of coefficient a determines upward (a > 0) or downward (a < 0) opening
Check Dependencies: Verify all required parameters are given before making conclusions ✓

Common Mistakes

Avoid these frequent errors

Assuming parabola direction without knowing coefficient sign
Don't assume the parabola opens upward just because the vertex is on the x-axis = wrong conclusion about positive values! The direction depends entirely on whether coefficient a is positive or negative. Always check what information is actually provided before determining parabola behavior.

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

Why can't we determine if the parabola is always positive?

+

Because we need to know the sign of coefficient a! If $a > 0$ , the parabola opens upward and is always positive except at the vertex. If $a < 0$ , it opens downward and is always negative except at the vertex.

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

+

It means the vertex has coordinates $(h, 0)$ where the y-coordinate is zero. This gives us the vertex form $y = a(x-h)^2$ since k = 0.

How do I know which way a parabola opens?

+

Look at the coefficient a in $y = a(x-h)^2 + k$ :

If $a > 0$ : opens upward (U-shape)
If $a < 0$ : opens downward (∩-shape)

Can a parabola with vertex on x-axis ever be negative?

+

Yes! If the coefficient a is negative, the parabola opens downward. The vertex at $(h, 0)$ is the highest point, so all other y-values are negative.

What should I do when information is missing from a problem?

+

Carefully identify what's given and what's needed. If key information like the sign of a coefficient is missing, the answer is often "Cannot be determined" rather than making assumptions.

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Parabola Vertex on X-axis: Analyzing Positive Value Behavior

Parabola Analysis with Missing Parameter Information

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Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't we determine if the parabola is always positive?

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

How do I know which way a parabola opens?

Can a parabola with vertex on x-axis ever be negative?

What should I do when information is missing from a problem?

🌟 Unlock Your Math Potential

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