To find the equation of the quadratic function $f(x)$, we need to use the given points on the graph and recognize that the quadratic function can generally be written as:

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

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

Step 1: Identify the vertex

From the graph, the vertex is $(-3, -1)$, which means $h = -3$ and $k = -1$. So the equation takes the form:

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

Step 2: Use another point to find $a$

The point $(-6, 8)$ lies on the graph. We substitute $x = -6$ and $f(x) = 8$ into the equation to find $a$:

$8 = a(-6 + 3)^2 - 1$ $8 = a(9) - 1$ $8 + 1 = 9a$ $9 = 9a$ $a = 1$

Step 3: Write the final equation

Now that we know $a = 1$, the equation of the quadratic function is:

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

This is the equation of the quadratic function based on the given graph.

Would you like further clarification on this?

Here are 5 related questions you might find useful:

1. How do you convert a quadratic function in vertex form to standard form?
2. How would the equation change if the parabola opened downward instead?
3. What if the vertex was at the origin, how would the equation look?
4. How can you find the equation if the points provided do not include the vertex?
5. What role does the coefficient $a$ play in the shape and direction of the parabola?

Tip: Always start by identifying the vertex when given a graph of a parabola—it simplifies finding the equation.