This graph shows a quadratic function with a vertex at the point (4, -4), and it opens upwards, meaning the parabola has a minimum at this vertex. The vertex form of a quadratic function is generally expressed as:

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

Where:

• $(h, k)$ is the vertex of the parabola
• $a$ is a constant that determines the width and direction (upward or downward) of the parabola.

From the graph:

• The vertex is at $(h, k) = (4, -4)$.
• The graph opens upwards, so $a$ is positive.

To determine $a$, we can use another point on the graph. The point $(2, 4)$ appears to lie on the parabola. Substituting this point into the vertex form equation will allow us to solve for $a$:

$4 = a(2 - 4)^2 - 4$

$4 = a(-2)^2 - 4$

$4 = 4a - 4$

$8 = 4a$

$a = 2$

Thus, the equation of the quadratic function in vertex form is:

$y = 2(x - 4)^2 - 4$

Would you like more details or further explanation on this?

Here are some questions related to this topic:

1. What is the standard form of a quadratic equation, and how does it compare to the vertex form?
2. How can we convert from vertex form to standard form?
3. How would the graph change if the value of $a$ were negative?
4. What does the parameter $a$ represent geometrically in a quadratic function?
5. Can you explain the process for completing the square to convert a quadratic equation to vertex form?

Tip: When working with quadratics, the value of $a$ determines whether the parabola opens upwards ($a > 0$) or downwards ($a < 0$).