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

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

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1

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.

2

Step-by-step solution

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

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

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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