To find the equation of the quadratic function $g(x)$ given the points on its graph, we can use the vertex form of a quadratic equation:

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

where $(h, k)$ is the vertex of the parabola. From the graph, we see that the vertex is $(-2, -3)$, so we can substitute these values into the equation:

$g(x) = a(x + 2)^2 - 3$

Next, we will use another point on the parabola to solve for the value of $a$. The point $(-4, -7)$ is on the graph, so substituting these coordinates into the equation:

$-7 = a(-4 + 2)^2 - 3$ $-7 = a(2)^2 - 3$ $-7 = 4a - 3$ $4a = -7 + 3$ $4a = -4$ $a = -1$

Now that we know $a = -1$, we can write the full equation of the quadratic function:

$g(x) = -(x + 2)^2 - 3$

Expanding this equation: $g(x) = -(x^2 + 4x + 4) - 3$ $g(x) = -x^2 - 4x - 4 - 3$ $g(x) = -x^2 - 4x - 7$

So, the equation of the quadratic function is: $g(x) = -x^2 - 4x - 7$

Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:

1. What other forms can a quadratic equation be written in?
2. How do you find the vertex of a quadratic function if it is not given in the graph?
3. What is the axis of symmetry of the quadratic function $g(x) = -x^2 - 4x - 7$?
4. How does changing the value of $a$ affect the direction of the parabola?
5. Can you derive the standard form of a quadratic equation from its factored form?

Tip: When working with quadratic equations, always identify the vertex and direction of the parabola to understand its properties better.