The image shows a question asking to "Write an equation (in any form) for the quadratic graphed below." The quadratic graph appears to have a vertex at approximately $(-3, -4)$ and passes through points like $(-4, 0)$ and $(-2, 0)$, suggesting a standard quadratic form.

Approach:

The vertex form of a quadratic equation is:

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

Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ is the coefficient determining the "width" and direction of the parabola.

From the graph:

• The vertex is at $(-3, -4)$, so $h = -3$ and $k = -4$.
• The parabola opens upwards.

We can use one of the intercepts, $(-4, 0)$, to find $a$:

• At $x = -4$, $y = 0$.

Substituting into the vertex form equation: $0 = a(-4 + 3)^2 - 4$ $0 = a(1)^2 - 4$ $a = 4$

Thus, the equation in vertex form is: $y = 4(x + 3)^2 - 4$

Would you like further explanation on any part of the process or conversion to other forms (like standard form)?

Here are 5 related questions to explore:

1. What is the standard form of a quadratic equation?
2. How does the value of $a$ affect the shape of the parabola?
3. What are the advantages of using vertex form over standard form?
4. How can you find the roots of the equation from this form?
5. How would the graph change if the value of $a$ was negative?

Tip: Always double-check the vertex and intercepts from a graph before writing the equation.