# What are the vertex, focus and directrix of # y=3x^2+8x+17 #?

##### 1 Answer

**Vertex**

**Focus**

**Directrix**

**Labelled Graph** is also available

#### Explanation:

We are given the **quadratic**

Coefficient of the **greater than Zero**

Hence, our **Parabola Opens Up** and we will also have a **Vertical Axis of Symmetry**

We need bring our quadratic function to the form given below:

Consider

Note that, we need to keep both the **constant term** on the other side.

To find the **Vertex**, we will **Complete the Square on x**

Divide every single term by

What value goes into the

Divide the coefficient of the **x.term** by **Square**.

The answer goes into the

Factor **Left-hand Side (LHS)** to get

We can rewrite to bring it to the required form given below:

whered

Hence, our **Vertex** will be

**Vertex**

Using

Hence,

**Focus** is always on the **Axis of Symmetry**

**Focus** is also **inside the Parabola**

**Focus** will have the same **x.Value as the Vertex** because it lies on the **Axis of Symmetry**

The **Axis of Symmetry** is at

The **Directrix** is always **Perpendicular** to the **Axis of Symmetry**

The **Value of P** tells us **how far** the **Focus is** from the **Vertex**

The **Value of P** also tells us **how far** the **Directrix is** from the **Vertex**

Since we know that **Focus** is **Vertex**

Our **Focus** is also **Vertex** and lies on the **Axis of Symmetry**

Also, **Focus** is **inside our parabola**.

So, the **Location of the Focus** is given by

**Focus**

**Directrix** is always **Perpendicular to the Axis of Symmetry**

**required equation of the Directrix** and also **lies on the Axis of Symmetry**

Please refer to the graph below:

A **labelled graph** given below with a few intermediate calculations shows on it might also be useful