# What are the vertex, focus and directrix of  y=(x-1)^2-1 ?

Nov 9, 2017

Vertex $\left(1 , - 1\right)$

Focus $\left(1 , - 0.75\right)$
Directrix $y = - 1.25$

#### Explanation:

Given -

$y = {\left(x - 1\right)}^{2} - 1$

We have to find the vertex, focus and directrix.

The given equation is in the form.

$y = a {\left(x - h\right)}^{2} + k$

It is in the vertex form. Then its vertex is $\left(h , k\right)$

Obtain the values of $h$ and $k$ from the given equation.

$h = 1$ x-coordinate of the vertex
$k = - 1$ y coordinate of the vertex

Then the vertex of the given equation is $\left(1 , - 1\right)$

If the term $a$ in the equation $y = a {\left(x - h\right)}^{2} + k$ is positive, it means the curve opens up.

In the given equation $a = 1$. one is a positive value, hence the parabola of the given equation opens up.

The standard form of a parabola of this type is as follows.
[for a parabola whose vertex is away from origin]

${\left(x - h\right)}^{2} = 4 a \left(y - k\right)$

Where

4 is a constant

$a$ is the distance between vertex and focus.

We shall rewrite the given equation like this

$y = {\left(x - 1\right)}^{2} - 1$

${\left(x - 1\right)}^{2} - 1 = y$

${\left(x - 1\right)}^{2} = y + 1$

It can be written as

${\left(x - 1\right)}^{2} = \left(y + 1\right)$

It appears, there is no $4 a$ term.We shall bring it like this.

${\left(x - 1\right)}^{2} = \left(y + 1\right)$ this equation can be written like this

${\left(x - 1\right)}^{2} = 1 \times \left(y + 1\right)$

One is equal to $4 \times \frac{1}{4}$

So we can replace one with $4 \times \frac{1}{4}$ and rewrite

${\left(x - 1\right)}^{2} = 4 \times \frac{1}{4} \times \left(y + 1\right)$

Now we have the constant term 4. $\frac{1}{4}$ is $a$

The distance between vertex and focus or vertex and directrix is $a = \frac{1}{4}$

We can find the focus and directrix.

The point which lies vertically at a distance of 0.25 above vertex is focus.

Focus is $\left(1 , \left(- 1 + 0.25\right)\right)$; $\left(1 , - 0.75\right)$

Find the point the lies vertically at a distance of 0.25 below vertex.
Use its y- coordinate to find the equation of directrix.

$\left(1 , \left(- 1 + \left(- 25\right)\right)\right)$
$\left(1 , \left(- 1 - 25\right)\right)$
$\left(1 , - 1.25\right)$

Directrix $y = - 1.25$