Parabola Property: Vertex on X-axis and Positive Value Analysis

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

When the vertex is on the x-axis, it means the y-coordinate of the vertex is zero. So the vertex point is $ (h, 0) $ instead of $ (h, k) $.

Q: Why does the parabola equal zero at the vertex?

Because the vertex form is $ y = a(x - h)^2 + k $. When the vertex is on the x-axis, k = 0. At the vertex point $ x = h $, we get $ y = a(h - h)^2 + 0 = 0 $.

Q: Can an upward-opening parabola have negative values?

It depends on where the vertex is! If the vertex is above the x-axis, all values are positive. If it's on the x-axis, only the vertex equals zero. If it's below the x-axis, some values are negative.

Q: How do I know if the statement is correct or incorrect?

Read carefully! The statement says 'always positive except at the vertex'. Since the function equals zero (not positive) at the vertex, this phrasing is technically incorrect. It should say 'non-negative' or 'zero at vertex, positive elsewhere.'

Q: What's the difference between 'positive' and 'non-negative'?

Positive: Greater than zero (y > 0)Non-negative: Greater than or equal to zero (y ≥ 0)Zero is non-negative but not positive!

Parabola Analysis with Vertex Positioning

If the vertex of the parabola is on the x-axis and the parabola is bending upwards, then the function is always positive except at the vertex point.

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

If the vertex of the parabola is on the x-axis and the parabola is bending upwards, then the function is always positive except at the vertex point.

Step-by-step solution

To solve this problem, let's analyze the function:

Step 1: Given the parabola is in vertex form $y = a(x - h)^2 + k$ where $a > 0$ for an upward opening.
Step 2: Since the vertex is on the x-axis, $k = 0$ . Thus, the equation simplifies to $y = a(x - h)^2$ .
Step 3: Analyze values:
- At $x = h$ , the vertex, $y = a(h - h)^2 = 0$ .
- For $x \neq h$ , $y = a(x - h)^2 > 0$ since $a > 0$ and any square is non-negative, but only zero at $h$ .
Step 4: Therefore, the function $y$ is zero only at the vertex $(h, 0)$ and positive for all other $x$ .

The statement in the problem says the function is always positive except at the vertex. As we see, the function is indeed zero only at the vertex and positive elsewhere, meaning the statement provided is incorrect in its description if one understands it as implying it should never reach zero, which technically it does only at the vertex.

Therefore, the correct answer is Incorrect.

Final Answer

Incorrect

Key Points to Remember

Essential concepts to master this topic

Vertex Form: When vertex is on x-axis, k=0 so y=a(x-h)²
Value Analysis: At vertex x=h, y=0; elsewhere y>0 when a>0
Check: Function equals zero at vertex, positive everywhere else ✓

Common Mistakes

Avoid these frequent errors

Confusing 'always positive' with 'never zero'
Don't say the function is always positive when it equals zero at the vertex = incorrect statement! The function value is exactly zero at one point. Always distinguish between 'positive everywhere except one point' and 'always positive.'

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

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

When the vertex is on the x-axis, it means the y-coordinate of the vertex is zero. So the vertex point is $(h, 0)$ instead of $(h, k)$ .

Why does the parabola equal zero at the vertex?

Because the vertex form is $y = a(x - h)^2 + k$ . When the vertex is on the x-axis, k = 0. At the vertex point $x = h$ , we get $y = a(h - h)^2 + 0 = 0$ .

Can an upward-opening parabola have negative values?

It depends on where the vertex is! If the vertex is above the x-axis, all values are positive. If it's on the x-axis, only the vertex equals zero. If it's below the x-axis, some values are negative.

How do I know if the statement is correct or incorrect?

Read carefully! The statement says 'always positive except at the vertex'. Since the function equals zero (not positive) at the vertex, this phrasing is technically incorrect. It should say 'non-negative' or 'zero at vertex, positive elsewhere.'

What's the difference between 'positive' and 'non-negative'?

Positive: Greater than zero (y > 0)
Non-negative: Greater than or equal to zero (y ≥ 0)

Zero is non-negative but not positive!

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