To analyze the graph of the quadratic function \(y = 5x^2 + 50x + 125\) with the repeated root at \(x = -5\), we can evaluate each of the statements given.
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Determining if the graph opens downward: The leading coefficient of \(x^2\) in the equation \(y = 5x^2 + 50x + 125\) is \(5\), which is positive. This means that the graph opens upward.
- Statement #1: The graph opens downward. (False)
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Finding the vertex: The vertex of a parabola in the form \(y = ax^2 + bx + c\) can be found using the vertex formula \(x = -\frac{b}{2a}\). Here, \(a = 5\) and \(b = 50\). \[ x = -\frac{50}{2 \cdot 5} = -\frac{50}{10} = -5 \] The vertex is at \(x = -5\), which is to the left of the x-axis (i.e., less than \(0\)).
- Statement #2: The graph has a vertex to the right of the x-axis. (False)
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Understanding the x-axis intersections: A quadratic function with a repeated root means it touches the x-axis at a single point. Since \(x = -5\) is the repeated root, the graph will indeed touch the x-axis at this point.
- Statement #3: The graph touches the x-axis once. (True)
Based on this analysis, the correct statement is:
Statement #3: The graph touches the x-axis once.
So the correct answer is Statement # \(3\).