Parabola Property Analysis: Upward-Facing Curve with Vertex Below X-axis

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

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Step-by-step written solution

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1

Understand the problem

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

2

Step-by-step solution

The solution to this problem involves understanding the behavior of parabolas. Given that the parabola opens upwards, indicated by a>0 a > 0 , and the vertex (h,k)(h,k) is below the x-axis, k<0 k < 0 , here's the detailed explanation:
1. The vertex of the parabola, (h,k)(h, k), is at the lowest point because the parabola opens upwards.
2. With k<0k < 0, the value of f(h)=kf(h) = k is negative. However, as xx moves away from the vertex, the function increases since it opens upwards.

Therefore, for large x|x|, f(x)f(x) becomes positive. For instance, at x=h±kax = h \pm \sqrt{\frac{-k}{a}}, the parabola can cross the x-axis and become positive.

Given that the parabola will eventually have positive yy-values for either very large or very small xx, the function is not always negative. Hence, the statement that the parabola is always negative 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 \( f\left(x\right) > 0 \).

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