Analyzing Parabola Properties: Vertex on X-axis with Upward Orientation

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

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Step-by-step written solution

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1

Understand the problem

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

2

Step-by-step solution

To evaluate the statement about the parabola, we need to understand its properties precisely:

Step 1: Given the parabola's vertex is on the x-axis, we can write its equation in the form y=a(xh)2 y = a(x-h)^2 , where a>0 a > 0 indicates that it opens upwards.

Step 2: The vertex form of a quadratic function has the vertex (h,0)(h, 0) with the vertex lying directly on the x-axis. Since a>0a > 0, the parabola opens upwards, implying (h,0)(h, 0) is the minimum point.

Step 3: For points other than the vertex, (xh)2(x-h)^2 is always non-negative. Since aa is positive, y=a(xh)20 y = a(x-h)^2 \geq 0. Therefore, the function value at the vertex is zero, and for all other xx, the function value is positive.

Step 4: Analyze if the function can be negative: With a>0a > 0, the value a(xh)2 a(x-h)^2 cannot be negative for any value of xx.

Conclusion: The given statement that the function is "always negative except for the vertex point" is incorrect since outside the vertex, the function is always non-negative and can be positive. Hence, the statement is incorrect.

Therefore, the correct answer is Incorrect.

3

Final Answer

Incorrect

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

AAABBBCCCX

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