The graph provided shows a parabola, and we are tasked with finding the quadratic function in the form:

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

From the graph:

• The vertex is at $(-3, 12)$, so $h = -3$ and $k = 12$.
• The point $(0, 3)$ lies on the parabola, which we can use to find the value of $a$.

Step 1: Write the general form

Using the vertex $(-3, 12)$, we substitute into the vertex form:

$f(x) = a(x + 3)^2 + 12$

Step 2: Use the point $(0, 3)$ to find $a$

Now substitute $x = 0$ and $f(x) = 3$ into the equation to solve for $a$:

$3 = a(0 + 3)^2 + 12$ $3 = a(9) + 12$ $3 - 12 = 9a$ $-9 = 9a$ $a = -1$

Step 3: Final equation

Now substitute $a = -1$ back into the equation:

$f(x) = -(x + 3)^2 + 12$

This is the equation of the quadratic function whose graph is shown.

Would you like any more details on this? Here are 5 related questions to explore further:

1. How do you find the vertex form of a quadratic equation in other cases?
2. What happens to the parabola if the value of $a$ is positive?
3. How do you convert the vertex form to standard form of a quadratic function?
4. How do you determine the direction of opening of a parabola from the vertex form?
5. How can we use other points to verify or adjust the quadratic equation?

Tip: In vertex form, $a$ controls the width and direction of the parabola, while $h$ and $k$ determine its location.