To find the equation of the quadratic function, we use the vertex form of a quadratic equation:

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

where:

• $(h, k)$ is the vertex of the parabola.
• $a$ is a constant that determines the shape and direction of the parabola.

Step 1: Identify $h$ and $k$

Given that the vertex is $(-3, -1)$:

• $h = -3$
• $k = -1$

So the equation becomes:

$P(x) = a(x + 3)^2 - 1$

Step 2: Use the Point $(6, 323)$ to Solve for $a$

We substitute $x = 6$ and $P(x) = 323$ into the equation:

$323 = a(6 + 3)^2 - 1$

$323 = a(9)^2 - 1$

$323 = 81a - 1$

$324 = 81a$

$a = 4$

Step 3: Write the Equation

Substituting $a = 4$ back into the vertex form:

$P(x) = 4(x + 3)^2 - 1$

Step 4: Expand to Standard Form

Now, let's expand this equation to put it in standard form $P(x) = ax^2 + bx + c$:

$P(x) = 4(x^2 + 6x + 9) - 1$

$P(x) = 4x^2 + 24x + 36 - 1$

$P(x) = 4x^2 + 24x + 35$

Final Answer

The equation of the quadratic function is:

$\boxed{P(x) = 4x^2 + 24x + 35}$

Would you like a detailed breakdown of any specific step or have any questions?

Further Exploration Questions

1. How would the equation change if the vertex were different?
2. What is the significance of the coefficient $a$ in the quadratic equation?
3. How can you verify if the function is concave up or concave down based on the value of $a$?
4. What other methods could you use to find the equation of a quadratic function?
5. Can you find the x-intercepts (roots) of the function $P(x) = 4x^2 + 24x + 35$?

Tip

Always double-check your vertex and additional point when using the vertex form. It's crucial for solving for the correct value of $a$.