To determine the vertex and axis of symmetry of the quadratic function \( y = x^2 - 10x + 3 \), we can use the formula for the vertex of a parabola in the form \( y = ax^2 + bx + c \).
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Identifying coefficients:
- Here, \( a = 1 \), \( b = -10 \), and \( c = 3 \).
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Finding the x-coordinate of the vertex: The x-coordinate of the vertex can be found using the formula: \[ x = -\frac{b}{2a} \] Plugging in the values of \( a \) and \( b \): \[ x = -\frac{-10}{2 \cdot 1} = \frac{10}{2} = 5 \]
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Finding the y-coordinate of the vertex: To find the y-coordinate, substitute \( x = 5 \) back into the original equation: \[ y = (5)^2 - 10(5) + 3 = 25 - 50 + 3 = -22 \]
Thus, the vertex is: \[ (5, -22) \]
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Finding the axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex and can be found using the x-coordinate of the vertex. Therefore, the axis of symmetry is: \[ x = 5 \]
Summary:
- The vertex of the parabola is \( (5, -22) \).
- The axis of symmetry is \( x = 5 \).