To determine the quadratic function from the given graph, we can use the vertex form of a quadratic equation:

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

Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ is a constant that determines the shape and direction of the parabola.

Step 1: Identify the vertex from the graph

From the graph, the vertex appears to be at the point $(-2, 9)$. Therefore, we have: $h = -2, \quad k = 9$

Step 2: Use another point on the parabola

Another point on the graph is $(0, 5)$. This point will help us find the value of $a$.

Step 3: Plug the point into the equation

Using the vertex form and substituting $f(0) = 5$, we get: $5 = a(0 - (-2))^2 + 9$ Simplify: $5 = a(2)^2 + 9$ $5 = 4a + 9$ Solve for $a$: $5 - 9 = 4a$ $-4 = 4a$ $a = -1$

Step 4: Write the quadratic equation

Now that we know $a = -1$, $h = -2$, and $k = 9$, we can write the equation as: $f(x) = -(x + 2)^2 + 9$

Thus, the quadratic function is: $f(x) = -(x + 2)^2 + 9$

Would you like more details, or have any further questions on this? Here are some related questions for further exploration:

1. How do we confirm that the vertex is the maximum point of the parabola?
2. Can we rewrite the equation in standard form $ax^2 + bx + c$?
3. What is the axis of symmetry for this quadratic function?
4. How does changing the value of $a$ affect the graph of a parabola?
5. What are the x-intercepts of the quadratic function?

Tip: The sign of $a$ in the vertex form tells you whether the parabola opens upward or downward. A negative $a$ means it opens downward.