Rewrite the quadratic function $$ y=3 x^{2}+9 x-1 $$ in vertex form.

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To rewrite the quadratic function $$ y = 3x^2 + 9x - 1 $$ in vertex form, we will complete the square. The vertex form of a quadratic function is given by:

$$
y = a(x - h)^2 + k
$$

where $$(h, k)$$ is the vertex of the parabola.

### Step 1: Factor out the coefficient of $$x^2$$

First, we factor out the coefficient of $$x^2$$ from the first two terms:

$$
y = 3(x^2 + 3x) - 1
$$

### Step 2: Complete the square

Next, we need to complete the square for the expression inside the parentheses. To do this, we take the coefficient of $$x$$ (which is 3), divide it by 2, and square it:

$$
\left(\frac{3}{2}\right)^2 = \frac{9}{4}
$$

Now, we add and subtract this value inside the parentheses:

$$
y = 3\left(x^2 + 3x + \frac{9}{4} - \frac{9}{4}\right) - 1
$$

This simplifies to:

$$
y = 3\left((x + \frac{3}{2})^2 - \frac{9}{4}\right) - 1
$$

### Step 3: Distribute and simplify

Now, we distribute the 3:

$$
y = 3(x + \frac{3}{2})^2 - 3 \cdot \frac{9}{4} - 1
$$

Calculating $$3 \cdot \frac{9}{4}$$:

$$
3 \cdot \frac{9}{4} = \frac{27}{4}
$$

So we have:

$$
y = 3(x + \frac{3}{2})^2 - \frac{27}{4} - 1
$$

Now, convert -1 to a fraction with a denominator of 4:

$$
-1 = -\frac{4}{4}
$$

Thus, we can combine the constants:

$$
y = 3(x + \frac{3}{2})^2 - \frac{27}{4} - \frac{4}{4} = 3(x + \frac{3}{2})^2 - \frac{31}{4}
$$

### Final Result

The vertex form of the quadratic function is:

$$
y = 3\left(x + \frac{3}{2}\right)^2 - \frac{31}{4}
$$
The quadratic function $$ y = 3x^2 + 9x - 1 $$ in vertex form is $$ y = 3\left(x + \frac{3}{2}\right)^2 - \frac{31}{4} $$.

Rewrite the quadratic function \( y=3 x^{2}+9 x-1 \) in vertex form.

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