This graphic organizer breaks down the vertex form equation, helping students visualize how changing a, h, and k shifts and stretches the parent parabola.

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

Now, look at each labeled part in the diagram:

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Part “a” — This shows a parabola that opens upward and is narrower than the parent function.

In the vertex form, a controls the width and direction.

- If |a| > 1 → parabola is narrower
- If 0 < |a| < 1 → parabola is wider
- If a > 0 → opens up
- If a < 0 → opens down

Since this graph opens up and is narrower than the parent function (which has a = 1), then a must be greater than 1. For example, a = 2 or a = 3 would make it narrower.

But we don’t need to pick an exact number unless asked — just understand what “a” represents.

→ So, “a” affects the width and direction of the parabola.

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Part “h” — This graph is shifted left or right compared to the parent function.

In the formula: f(x) = a(x - h)² + k

The h value shifts the graph horizontally.

- If h is positive → shift RIGHT
- If h is negative → shift LEFT

Looking at the graph labeled “h”, it looks like the vertex is moved to the right (say, x = 2). That means h = 2.

So, “h” gives the x-coordinate of the vertex and shifts the graph left/right.

Axis of symmetry is x = h.

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Part “k” — This graph is shifted up or down.

In the formula, k is added at the end.

- If k is positive → shift UP
- If k is negative → shift DOWN

The graph labeled “k” looks like it’s shifted up (maybe y = 3). So k = 3.

→ So, “k” gives the y-coordinate of the vertex and shifts the graph up/down.

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Vertex and Axis of Symmetry

From the formula:
Vertex = (h, k)
Axis of Symmetry = x = h

So for any quadratic in vertex form, you can read these directly from the equation.

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Parent Function

The parent function is f(x) = x². Its vertex is at (0, 0), and it opens upward with a = 1.

All other graphs are transformations of this one.

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Now, let’s fill in the blanks based on understanding:

In the box labeled “Vertex: ___ , Axis of Symmetry: ___”

We’re not given specific numbers, but since this is connected to the general formula, we use variables:

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

That’s the standard answer when working with the general vertex form.

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Final Answer:

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