To find an equation for the quadratic function graphed, let's examine key points and characteristics of the parabola:

1. Vertex: The vertex of the parabola is at $(-2, 2)$, indicating the form $y = a(x + 2)^2 + 2$.
2. Direction: The parabola opens downward, meaning $a$ will be negative.
3. Another Point: Observing the point $(0, 0)$, we can use this to solve for $a$.

Using the vertex form:

$y = a(x + 2)^2 + 2$

Substitute $(0, 0)$ to Solve for $a$:

$0 = a(0 + 2)^2 + 2$ $0 = 4a + 2$ $-2 = 4a$ $a = -\frac{1}{2}$

Equation

Now we substitute $a = -\frac{1}{2}$ into the vertex form:

$y = -\frac{1}{2}(x + 2)^2 + 2$

This is the equation for the quadratic function shown in the graph.

Would you like more details on this solution, or do you have any questions?

Here are some related questions you might consider:

1. How would the equation change if the vertex was moved to a different point?
2. What is the significance of the sign of $a$ in a quadratic function?
3. How would the equation look in standard form, $y = ax^2 + bx + c$?
4. Can you determine the axis of symmetry from the vertex form?
5. How would you find the roots of this equation?

Tip: Always check if additional points on the graph can be used to confirm the value of $a$.