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### Part a: Determining the equation in vertex form

The quadratic function given has a clear vertex at \((2, -4)\), based on the graph. The equation of a parabola in vertex form is:

\[
y = a(x - h)^2 + k
\]

Where \((h, k)\) is the vertex of the parabola. In this case, \((h, k) = (2, -4)\), so the equation becomes:

\[
y = a(x - 2)^2 - 4
\]

To find the value of \(a\), we can use another point on the graph. From observation, one of the points on the graph is the y-intercept at \((0, 0)\).

Substitute \((x, y) = (0, 0)\) into the equation:

\[
0 = a(0 - 2)^2 - 4
\]
\[
0 = a(4) - 4
\]
\[
4a = 4 \implies a = 1
\]

Thus, the equation of the quadratic function is:

\[
y = (x - 2)^2 - 4
\]

### Part b: Domain and Range

- **Domain**: Since this is a quadratic function, the domain is all real numbers, expressed in interval notation as:
  \[
  \text{Domain} = (-\infty, \infty)
  \]
- **Range**: The minimum value of the parabola occurs at the vertex \((2, -4)\), and since it opens upwards, the range is:
  \[
  \text{Range} = [-4, \infty)
  \]

Would you like more details on any part of this problem or clarification?

Here are five related questions to explore:
1. How do you convert a quadratic function from vertex form to standard form?
2. What is the significance of the coefficient \(a\) in a quadratic equation?
3. How would you find the axis of symmetry for this parabola?
4. Can the range of the parabola change if the vertex is shifted horizontally?
5. How does the graph of the function change if the value of \(a\) is negative?

**Tip**: In vertex form, the vertex \((h, k)\) provides important information about the function's graph, including its minimum or maximum point.

The graph of the quadratic function on the right has integer coordinates for the vertex and y-intercept. Determine an equation for the function in vertex form and state the domain and range in proper notation.

Learn how to find the equation of a quadratic function in vertex form and determine the domain and range. This problem demonstrates how to use the vertex and y-intercept of a parabola. Suitable for students in Grades 9-11.

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