Rewrite the quadratic equation:  y = 1/2x^2 - 4x - 5 into Standard Form to determine the Vertex, whether it opens up or down, and is taller or shorter.

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baxthum8 | High School Teacher | (Level 3) Associate Educator

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`y = 1/2x^2 - 4x - 5` is written in standard form: `ax^2 + bx + c = 0.`

To find the vertex, whether it opens up or down, and is taller or shorter, you will want to rewrite in vertex form: `y = a(x - h)^2 + k`

(h, k) represents the vertex of the quadratic equation. If a < 0, then it opens down.  If a > 0 then it opens up.  By taller or shorter, I'll assume you mean stretched or compressed.  If 0 < a < 1, then the equation will be compressed.  If a > 1 then it will be stretched.

To rewrite the equation, we will need to complete the square.

`y = 1/2x^2 - 4x - 5`

`y = (1/2x^2 - 4x) - 5`

`y = 1/2(x^2 - 8x) - 5`

`y = 1/2(x^2 - 8x + 16) - 5 - 8`

`y = 1/2(x - 4)^2 - 13`

Therefore, the vertex is (4, -13).  Since a > 0, the quadratic opens upward.  Since 0<a<1 then the graph is compressed (shorter).

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