This diagram illustrates how the coefficients a, h, and k in the vertex form equation f(x) = a(x-h)^2 + k affect the graph of a quadratic function compared to the parent function.

Let’s solve this step by step.

We are given the vertex form of a quadratic function:
f(x) = a(x - h)² + k

This form tells us important things about the graph (a parabola):

- The vertex is at the point (h, k).
- The axis of symmetry is the vertical line x = h.
- The value of a tells us if the parabola opens up or down, and how wide or narrow it is. But for finding the vertex and axis of symmetry, we only need h and k.

Now let’s look at each graph labeled a, b, and k. We’ll find the vertex from the graph, then write the equation in vertex form.

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Graph a:

Looking at graph a:
- The vertex is at the bottom of the U-shape.
- It’s located at (0, 1) — that means x=0, y=1.
→ So, h = 0, k = 1

Plug into the formula:
f(x) = a(x - 0)² + 1 → f(x) = a(x)² + 1

But wait — we also see that the shape looks just like the parent function f(x)=x², which has a=1. Since there’s no stretching or flipping shown, we assume a = 1.

So for graph a:
f(x) = 1(x - 0)² + 1 → or simply f(x) = x² + 1

Vertex: (0, 1)
Axis of symmetry: x = 0

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Graph b:

Looking at graph b:
- Vertex is at (2, 3) — move right 2, up 3 from origin.
→ So, h = 2, k = 3

Again, assuming same shape as parent function (no stretch/compress), so a = 1

Equation:
f(x) = 1(x - 2)² + 3 → f(x) = (x - 2)² + 3

Vertex: (2, 3)
Axis of symmetry: x = 2

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Graph k:

Looking at graph k:
- Vertex is at (-1, -2) — left 1, down 2.
→ So, h = -1, k = -2

Assume a = 1 again.

Equation:
f(x) = 1(x - (-1))² + (-2) → simplify:
f(x) = (x + 1)² - 2

Vertex: (-1, -2)
Axis of symmetry: x = -1

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Now, fill in the blanks in the center box:

> Vertex: _______
> Axis of Symmetry: _______

Wait — actually, looking back at the original image layout, those blanks are probably meant to be filled for ONE specific graph? But since the problem shows three graphs (a, b, k) and asks to use the vertex form, I think the intention is to match each graph with its equation and identify vertex and axis for each.

However, the central box says “Vertex: ___” and “Axis of Symmetry: ___” without specifying which graph. That might be confusing.

But notice — the arrows go FROM the big formula TO each graph. So perhaps the task is: For each graph (a, b, k), write the correct equation using f(x)=a(x-h)²+k, and also state the vertex and axis of symmetry for that graph.

Since the user didn’t specify which one to focus on, but the central box is empty, maybe they want the general idea? Or perhaps it's a template to fill per graph.

Actually, re-examining — the central box is likely meant to be filled out AFTER choosing one graph? But the instruction isn't clear.

Wait — perhaps the student is supposed to pick one graph and complete the vertex and axis for it? But all three are shown.

Alternatively, maybe the central box is for the GENERAL rule? Like:

In f(x) = a(x - h)² + k,

→ Vertex is (h, k)
→ Axis of symmetry is x = h

That makes sense! Because the title is “Transformations of Quadratic Functions — Vertex Form”, and the central box is between the formula and the graphs — probably meant to remind you what h and k represent.

Yes — that must be it.

So regardless of the graphs, the key takeaway is:

In the equation f(x) = a(x - h)² + k:

- The vertex is the point (h, k)
- The axis of symmetry is the line x = h

You don’t plug in numbers yet — this is the definition.

Then, for each graph, you find h and k from the vertex point, and write the equation.

So to answer the central box:

> Vertex: (h, k)
> Axis of Symmetry: x = h

And then for each graph:

Graph a: vertex (0,1) → equation: f(x) = (x - 0)² + 1 = x² + 1
Graph b: vertex (2,3) → equation: f(x) = (x - 2)² + 3
Graph k: vertex (-1,-2) → equation: f(x) = (x + 1)² - 2

All with a=1 assumed unless otherwise stated.

Final Answer:

For the central box:
Vertex: (h, k)
Axis of Symmetry: x = h

For graph a: f(x) = x² + 1, vertex (0,1), axis x=0
For graph b: f(x) = (x - 2)² + 3, vertex (2,3), axis x=2
For graph k: f(x) = (x + 1)² - 2, vertex (-1,-2), axis x=-1

But since the question seems to ask primarily for filling the central box based on the formula, here’s the core answer:

Final Answer:
Vertex: (h, k)
Axis of Symmetry: x = h