To find the axis of symmetry and the vertex of the quadratic function \( y = x^2 - 6x - 7 \), we can use the formulas related to the vertex form of a quadratic function.
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Axis of Symmetry: The axis of symmetry for a parabola given by the equation \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
For the function \( y = x^2 - 6x - 7 \):
- \( a = 1 \)
- \( b = -6 \)
Plugging in the values: \[ x = -\frac{-6}{2 \cdot 1} = \frac{6}{2} = 3 \] Thus, the equation of the axis of symmetry is: \[ x = 3 \]
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Vertex: The x-coordinate of the vertex is the same as the axis of symmetry, \( x = 3 \). To find the y-coordinate of the vertex, substitute \( x = 3 \) back into the original equation: \[ y = (3)^2 - 6(3) - 7 \] \[ y = 9 - 18 - 7 \] \[ y = 9 - 25 = -16 \] Therefore, the coordinates of the vertex are \( (3, -16) \).
Summary:
- The equation of the axis of symmetry is: \( x = 3 \)
- The coordinates of the vertex are: \( (3, -16) \)