The graph of the function below intersects the X-axis at one point A (the vertex of the parabola).
Find all values of
where .
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The graph of the function below intersects the X-axis at one point A (the vertex of the parabola).
Find all values of
where .
To identify the conditions where , we need to analyze the nature of the quadratic function as represented on the provided graph.
Based on the problem, the graph intersects the x-axis exactly at one point, recognized as point A, the vertex. In a quadratic function , if the vertex intersects at the x-axis and nowhere else, it means the graph is tangent to the x-axis at that vertex.
To determine if the function is positive, examine the orientation: - If , the parabola opens upwards, making it have a minimum at the vertex. - If , the parabola opens downwards, making it have a maximum at the vertex. Given that the problem states the parabola intersects the x-axis only at the vertex, the parabola opens downward. This is inferred from the phrased graph where no areas reach above the x-axis.
Therefore, the function never reaches a value greater than zero, as the parabola is concave down, and the vertex sits on the x-axis.
Conclusively, the range where is nonexistent given the parameters of the problem.
Therefore, the solution is that there are no such values.
No such values
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 \( f\left(x\right) > 0 \).
Look at the direction of the curve! If the parabola looks like a smile (∪), it opens upward. If it looks like a frown (∩), it opens downward. In this problem, the parabola opens downward.
When a parabola touches the x-axis at exactly one point, that point is the vertex. For a downward-opening parabola, the vertex is the highest point. If the highest point is on the x-axis (where y = 0), then all other points are below it!
f(x) > 0 means strictly greater than zero (positive values only). f(x) ≥ 0 includes zero itself. In this problem, f(x) = 0 at point A, but f(x) is never greater than zero.
No! The problem states the parabola intersects the x-axis at one point only. This mathematical fact tells us the complete behavior: it's a downward parabola with vertex on the x-axis, so no positive values exist anywhere.
The key clue is that it intersects the x-axis at exactly one point. If it were upward-opening, the vertex would be the minimum, and the parabola would have positive values everywhere else. Since we see it touches only once, it must open downward.
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