To complete the square for the expression \( y = -x^2 + 8x + 7 \) and determine the vertex and axis of symmetry, you need to follow a few steps in words:
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Identify the quadratic term: First, notice that the leading coefficient of the quadratic term is negative. To make completing the square easier, factor out the negative sign from the quadratic and linear terms.
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Rearranging the expression: Write the equation in a form that isolates the quadratic and linear terms. This means you express the equation as \( y = -(x^2 - 8x) + 7 \).
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Completing the square: Focus on the expression in the parentheses. To complete the square, you take half of the coefficient of the linear term (which is eight), square it, and add and subtract this value inside the parentheses. This will create a perfect square trinomial.
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Rewriting the expression: After adding and subtracting the squared value, rewrite the equation to express it as a perfect square. This step will yield an expression in the form of a squared term minus a constant.
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Identifying the vertex: The vertex of the quadratic expression is found in the completed square form, which shows the highest or lowest point of the parabola. The coordinates of the vertex will be determined from the perfect square's structure.
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Determining the axis of symmetry: The axis of symmetry can be found from the vertex. Since the axis of symmetry for a parabola given by this form is a vertical line that passes through the x-coordinate of the vertex.
In summary, after completing the square, the vertex will represent the point where the maximum or minimum value of the quadratic occurs, and the axis of symmetry will be the vertical line that passes through this vertex.