Question

The standard form of the equation of a parabola is y = 2x2 + 16x + 17. What is the vertex form of the equation?

Answer 1

To convert the equation y = 2x2 + 16x + 17 to its vertex form, complete the square to obtain y = 2(x + 4)^2 - 15, which is the desired vertex form.

Step-by-step explanation:

Converting the standard of the equation of a parabola y = 2x2 + 16x + 17 to the vertex form requires completing the square. First, factor out the coefficient of the x2 term from the x terms:

y = 2(x2 + 8x) + 17

Next, add and subtract (8/2)2 = 16 inside the parenthesis to complete the square:

y = 2(x2 + 8x + 16 - 16) + 17

Which simplifies to:

y = 2((x + 4)2 - 16) + 17

Then distribute the 2 and combine like terms:

y = 2(x + 4)2 - 32 + 17

Finally, the vertex form of the equation is:

y = 2(x + 4)2 - 15

Answer 2

The vertex form looks like y-k = a(x-h)^2. We must re-write y = 2x^2 + 16x + 17 in this vertex form.

Note that y = 2x^2 + 16 x + 17 can be re-written as y = 2(x^2 + 8x) + 17

Let's complete the square: to x^2 + 8x add 4^2, and then subtract 4^2:

y = 2(x^2 + 8x + 16 - 16) + 17. This become y = 2(x^2 + 8x + 16) + 1, or

y = 2(x+4)^2 + 1 (answer), or y - 1 = 2(x-[-4])^2 + 1. The vertex is
at (-4, 1).