To analyze the graph of the quadratic function \( y = 5x^2 + 50x + 125 \) and its repeated root of \( x = -5 \), we can examine the properties of the quadratic function:
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Repeated Root: The fact that \( x = -5 \) is a repeated root means that the graph touches the x-axis at this point but does not cross it. Therefore, statement 1 ("The graph touches the x-axis once") is true.
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Vertex Location: The vertex of a quadratic function in standard form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). For this function:
- \( a = 5 \)
- \( b = 50 \)
Now calculating: \[ x = -\frac{50}{2 \times 5} = -\frac{50}{10} = -5 \] Since the vertex \( x = -5 \) is exactly at the repeated root (the point where the graph touches the x-axis), it indicates that the vertex is at the x-axis. Since \( -5 \) is to the left of the x-axis, statement 2 ("The graph has a vertex to the right of the x-axis") is false.
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Direction of Opening: The graph of a quadratic function opens upward if \( a > 0 \) and downward if \( a < 0 \). Here, \( a = 5 \), which is positive, so the graph opens upward. Therefore, statement 3 ("The graph opens downward") is false.
In summary, the only accurate statement about the graph is:
- The graph touches the x-axis once.
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