x²+6x-y+3= 0.

Solve for the values of h, k, and p. Find the equation. Write the given parts. Sketch the possible graph.

The given equation is:

\[
x^2 + 6x - y + 3 = 0
\]

This equation represents a parabola, and we need to rewrite it in a standard form to find the values of \(h\), \(k\), and \(p\), which are typically the parameters of the vertex form of a parabola. Let's go step by step.

### Step 1: Rearrange the equation
First, let's isolate \(y\) on one side:

\[
y = x^2 + 6x + 3
\]

### Step 2: Complete the square
Now, to put this equation in a form where we can easily identify the vertex, we complete the square on the \(x\)-terms.

Take the coefficient of \(x\) (which is 6), divide by 2, and square it:

\[
\left(\frac{6}{2}\right)^2 = 9
\]

Now, add and subtract 9 within the equation:

\[
y = (x^2 + 6x + 9) - 9 + 3
\]

\[
y = (x + 3)^2 - 6
\]

### Step 3: Vertex form
We have now rewritten the equation in vertex form:

\[
y = (x + 3)^2 - 6
\]

### Step 4: Identify the vertex and focal length
From the equation \(y = (x + 3)^2 - 6\), we can identify the following:

- The vertex \((h, k)\) is \((-3, -6)\).
- Since this is a vertical parabola, we also know the value of \(p\) is the distance from the vertex to the focus. In this case, the equation is already in the standard form \(y = (x - h)^2 + k\), and there is no extra coefficient, so \(p = 1\).

### Step 5: Write the given parts
- **Vertex (h, k):** \((-3, -6)\)
- **Focal distance (p):** \(1\)

### Step 6: Sketch the possible graph
The graph will be a parabola opening upwards with its vertex at \((-3, -6)\) and a focal distance of \(1\).

Would you like me to create a sketch of the parabola?

### Follow-up Questions:
1. Can you identify the axis of symmetry from the equation?
2. What is the directrix of this parabola?
3. How would you describe the shape and direction of the graph?
4. Can you find the coordinates of the focus?
5. How would the equation change if the parabola opened downward?

#### Tip:
When completing the square, always remember to balance the equation by adding and subtracting the same value.

The query submitted by the user: x²+6x-y+3= 0. Solve for the values of h, k, and p. Find the equation. Write the given parts. Sketch the possible graph.

Learn how to solve the parabola equation x²+6x-y+3=0 by finding the vertex (h, k) and the focal distance (p) using completing the square. This problem covers essential algebraic concepts such as the vertex form of a parabola and the properties of parabolas. Suitable for Grades 9-12.

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