The graph of the function below intersects the -axis at point A (the vertex of the parabola).
Find all values of where.
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The graph of the function below intersects the -axis at point A (the vertex of the parabola).
Find all values of where.
To solve this problem, we need to determine when is negative by analyzing the graph provided.
The graph shows a quadratic function shaped as a parabola. Importantly, the parabola intersects the x-axis precisely at point A, which is its vertex. From this, we can deduce two possible scenarios:
1. If the parabola opens upwards (convex), the vertex represents the minimum point. Thus, the y-value at the vertex is greater than any other point on the function, implying there is no region where since the lowest point is zero.
2. If it were to open downwards, point A would be the maximum, and could be negative elsewhere, but this contradicts the given information that point A is a vertex on the x-axis, suggesting the opening is upwards.
Since the graph passes through the x-axis only at vertex A and that is the minimum point, the parabola opens upwards. Therefore, the function never takes negative values as it only touches the x-axis without crossing it.
Thus, the conclusion is that there are no values of for which .
Hence, the function has no negative domain.
The function has no negative domain.
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
\( f\left(x\right) > 0 \).
Look at the shape direction! If it looks like a 'U', it opens upward. If it looks like an upside-down 'U', it opens downward. The graph shows a U-shape.
When the vertex touches the x-axis, that's the turning point of the parabola. For upward-opening parabolas, this is the lowest point, so the function never goes below zero.
Yes! When a parabola has its vertex on the x-axis, it touches the axis at exactly one point but doesn't cross over to the other side.
Since the parabola opens upward and its lowest point (vertex) is at zero on the x-axis, all other points must be above the x-axis, making everywhere.
If it opened downward with vertex on the x-axis, then the vertex would be the highest point, and the function would be negative everywhere else except at the vertex.
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