Graphing Quadratic Functions In Vertex Form - Free Printable

It looks like you've uploaded an image showing a quadratic function and its graph, but there's a mistake in the information provided. Let’s carefully analyze and correct it.

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📌 Given Function:

$$
y = (x - 2)^2 - 3
$$

This is a quadratic function in vertex form:
$$
y = (x - h)^2 + k
$$
where $(h, k)$ is the vertex of the parabola.

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✔ Step 1: Identify the Vertex

From the equation:
- $ h = 2 $
- $ k = -3 $

So, the vertex is:
$$
V(2, -3)
$$

⚠️ But in your image, it says $ V(2, 3) $ — this is incorrect. The vertex should be at $ (2, -3) $, not $ (2, 3) $. That’s a key error.

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✔ Step 2: Direction of the Parabola

Since the coefficient of the squared term is positive (it's 1), the parabola opens upward.

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✔ Step 3: Domain and Range

- Domain: All real numbers, because any $ x $ can be plugged into the function.
$$
D = (-\infty, \infty)
$$
✔️ This is correct.

- Range: Since the parabola opens upward and the vertex is the minimum point at $ y = -3 $, the smallest value of $ y $ is $ -3 $, and it goes up to infinity.
$$
R = [-3, \infty)
$$
✔️ This is also correct.

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✔ Step 4: Graph Analysis

The graph shown is a parabola opening upward, with:
- Vertex at $ (2, -3) $
- Axis of symmetry: $ x = 2 $ (shown as the blue dashed line)
- As $ x \to \pm\infty $, $ y \to +\infty $ (correctly indicated)

But again, the vertex label is wrong — it shows $ (2, 3) $, which would be above the axis, but the actual vertex is below at $ y = -3 $.

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✔ Final Corrections

| Item | Correct Value | Your Image |
|------|----------------|------------|
| Vertex | $ (2, -3) $ | ✘ $ (2, 3) $ |
| Domain | $ (-\infty, \infty) $ | ✔️ Correct |
| Range | $ [-3, \infty) $ | ✔️ Correct |

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🔍 Summary

✔ Corrected Information:
- Function: $ y = (x - 2)^2 - 3 $
- Vertex: $ \boxed{(2, -3)} $
- Domain: $ (-\infty, \infty) $
- Range: $ [-3, \infty) $
- Opens upward → minimum at vertex

✘ Error: The vertex was incorrectly labeled as $ (2, 3) $ instead of $ (2, -3) $

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✔ How to Fix It:

Update the vertex box from:
> $ V(2, 3) $
to:
> $ \boxed{V(2, -3)} $

Then everything else matches perfectly.

Let me know if you want help plotting more points or transforming the function!