Question

What is the focus of the parabola given the equation y=x^2+4x-1

Answer 1

Let's first rewrite it in vertex form.
Start by completing the square.
y = x² + 4x + 4 - 4 - 1
y = (x + 2)² - 5
(x + 2)² = y + 5

Now, 4a = 1; a = $\frac{1}{4}$

So, the vertex form is (x + 2)² = 4($\frac{1}{4}$)(y + 5)
So, we know that the vertex is at (-2, -5).
Since it's a concave up parabola, we just have to change the y-value to find the focus.

∴ F(-2, -5 + $\frac{1}{4}$)
F(-2, $\frac{-19}{4}$)