`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).