To determine the quadratic function $f(x) = a(x - h)^2 + k$ for the given graph, we need to identify the vertex and use additional points to solve for $a$.

Step 1: Identify the Vertex $(h, k)$

From the graph, we can see the vertex is at $(2, -3)$. Therefore:

• $h = 2$
• $k = -3$

So far, the function looks like: $f(x) = a(x - 2)^2 - 3$

Step 2: Use Another Point to Find $a$

We’ll use the point $(-1, 2)$ to find $a$. Substitute $x = -1$ and $f(x) = 2$ into the equation: $2 = a(-1 - 2)^2 - 3$ $2 = a(3)^2 - 3$ $2 = 9a - 3$ Add 3 to both sides: $5 = 9a$ Divide by 9: $a = \frac{5}{9}$

Final Equation

Now, substitute $a = \frac{5}{9}$, $h = 2$, and $k = -3$ back into the equation: $f(x) = \frac{5}{9}(x - 2)^2 - 3$

So, the quadratic function is: $f(x) = \frac{5}{9}(x - 2)^2 - 3$

Would you like further details or have any questions?

Here are some related questions to consider:

1. How can we determine the axis of symmetry from the vertex form of a quadratic function?
2. What are other methods to convert a quadratic function to standard form?
3. How does changing the value of $a$ affect the graph of the function?
4. How can we find the x-intercepts of this quadratic function?
5. How does the vertex form make it easier to graph parabolas?

Tip: Remember that the vertex form of a quadratic function makes it easy to read the vertex $(h, k)$ directly from the equation!