Let's analyze the function \( y = 5x^2 + 50x + 125 \) and assess the provided statements.
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Identify the nature of the roots: The function can be rewritten in the vertex form to find its vertex and determine the nature of its roots. However, we can also use the fact that it is a quadratic equation.
The formula for the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 5 \), \( b = 50 \), and \( c = 125 \). We can calculate the discriminant: \[ b^2 - 4ac = 50^2 - 4(5)(125) = 2500 - 2500 = 0 \] Since the discriminant is 0, there is exactly one real root (a repeated root) at \( x = -5 \).
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Graph characteristics:
- Opening direction: Since the leading coefficient \( a = 5 \) is positive, the graph opens upwards.
- Vertex: To find the vertex, use the formula for the x-coordinate of the vertex, given by: \[ x = \frac{-b}{2a} = \frac{-50}{2 \times 5} = \frac{-50}{10} = -5 \] Therefore, the vertex is at \( (-5, y(-5)) \). We can calculate \( y(-5) \): \[ y(-5) = 5(-5)^2 + 50(-5) + 125 = 5(25) - 250 + 125 = 125 - 250 + 125 = 0 \] This confirms that the vertex is at \( (-5, 0) \).
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Examine each statement:
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Statement #1: The graph opens downward.
- False: The graph opens upward since the coefficient of \( x^2 \) is positive.
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Statement #2: The graph has a vertex to the right of the x-axis.
- False: The vertex \( (-5, 0) \) is exactly on the x-axis, not to the right.
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Statement #3: The graph touches the x-axis once.
- True: The graph touches the x-axis exactly at the repeated root \( x = -5 \).
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Thus, the only accurate statement about the graph of \( y = 5x^2 + 50x + 125 \) is Statement #3. The other statements are false.