What is the vertex of the function f (x) = 1/2x2 + 3x + 3/2?

The coordinates for the vertex are ( - 32, 574) or (-1.5,14.25).

Explanation:

The first thing to do is replicate the function in vertex form, y=a (x-h) 2 + k. The way I personally do this is to find what h value would make the x2 and x values, but in this case, it would be wise to separate the negative coefficient.

This would make the problem y=-1 (x2 + 3x-12). Expanding the first part of the vertex form, we find that the vertex form can be rewritten as y=a (x2 - 2xh + h2) + k.

That means we already have x2 in both forms and know that - 2xh=3x. Now we need to solve for h. We can divide both sides by - 2x, leaving the left side with h, and making the left side simplify to - 32. This is the x value of the vertex, using knowledge that in vertex form, (h, k) are the coordinates of the vertex.

Then we can substitute the value for x at the vertex with any x in the original equation. We get y = - ( - 32) 2 + 3 ( - 32) + 12. ( - 32) 2 is 94 and 3 ( - 32) is - 184 while 12 is 484. Using common denominators, the equation becomes y = - 9+18+48 4.

This simplifies to y = 574.

The coordinates for the vertex are ( - 32, 574) or (-1.5,14.25)