Yes! Let's determine the quadratic function $f(x)$ that matches the given graph.

Step 1: Identify Key Points

From the image, we see the graph has two labeled points:

• Vertex: $(1,4)$
• Another point: $(0,2)$

Since the vertex is given, we use the vertex form of a quadratic function: $f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex.

Step 2: Substitute the Vertex

Since the vertex is $(1,4)$, we substitute $h = 1$ and $k = 4$: $f(x) = a(x - 1)^2 + 4$

Step 3: Solve for $a$

Using the second point $(0,2)$, substitute $x = 0$ and $f(0) = 2$:

$2 = a(0 - 1)^2 + 4$

$2 = a(1) + 4$

$2 - 4 = a$

$a = -2$

Step 4: Write the Final Equation

Now that we have $a = -2$, the quadratic function is:

$f(x) = -2(x - 1)^2 + 4$

This is the equation of the quadratic function that fits the given graph.

Would you like a verification or a different form (like standard form)? Let me know if you have any questions!

Related Questions:

1. How do you convert a quadratic equation from vertex form to standard form?
2. What does the value of $a$ represent in a quadratic equation?
3. How can you determine the axis of symmetry from a quadratic equation?
4. What is the process for finding the x-intercepts of a quadratic function?
5. How do transformations affect the graph of a quadratic function?

Tip: The vertex form is useful when you know the vertex and another point, as it simplifies solving for $a$.