To analyze the graph of the quadratic equation \( y = 5x^2 + 50x + 125 \), let’s break down the properties of the equation step by step.
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Determine the direction of the parabola: The quadratic formula is given by \( ax^2 + bx + c \). In this case, \( a = 5 \) (which is positive), indicating that the parabola opens upward, not downward.
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Vertex of the parabola: The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). Here, \( b = 50 \) and \( a = 5 \): \[ x = -\frac{50}{2 \cdot 5} = -\frac{50}{10} = -5 \] The vertex's x-coordinate is thus \(-5\). Since the vertex is at \(-5\), which is to the left of the y-axis, it is not to the right of the x-axis.
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Roots of the equation: The equation has a repeated root (a double root) at \( x = -5 \). This implies that the parabola touches the x-axis at this point but does not cross it. Therefore, the graph touches the x-axis once.
Summary of the statements:
- A: The graph opens downward. (Incorrect; it opens upward.)
- B: The graph has a vertex to the right of the x-axis. (Incorrect; the vertex is at \( x = -5\), which is to the left.)
- C: The graph touches the x-axis once. (Correct; it has a double root at \( x = -5\).)
Thus, the only correct statement is C: The graph touches the x-axis once.
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