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Let's analyze the function \( f(x) = -\frac{1}{4}(x - 1)^2 + 4 \) to determine its vertex and whether the parabola opens up or down.

### Step 1: Vertex form of a quadratic function
The function is in **vertex form**, which is generally written as:
\[
f(x) = a(x - h)^2 + k
\]
Here, \( (h, k) \) represents the **vertex**, and \( a \) determines whether the parabola opens up or down:
- If \( a > 0 \), the parabola opens **up**.
- If \( a < 0 \), the parabola opens **down**.

### Step 2: Identify the vertex
From the given equation \( f(x) = -\frac{1}{4}(x - 1)^2 + 4 \), we can directly compare with the general form:
- \( h = 1 \)
- \( k = 4 \)

So, the vertex is \( (1, 4) \).

### Step 3: Determine if the graph opens up or down
Here, \( a = -\frac{1}{4} \), which is **negative**. Therefore, the parabola opens **down**.

### Conclusion:
The vertex is \( (1, 4) \), and the parabola opens **down**.

The correct option from the choices is: **(1, 4); opens down**.

Do you have any questions or want further clarification?

Here are five related questions:
1. How can you determine the axis of symmetry from a quadratic equation in vertex form?
2. What effect does changing the value of \( a \) have on the graph of a quadratic function?
3. How do you convert a quadratic equation from standard form to vertex form?
4. Can you explain the significance of the vertex in a quadratic graph?
5. How would the graph change if the \( a \) value was positive instead?

**Tip:** The sign of \( a \) in a quadratic equation always controls whether the parabola opens up or down.

Identify the vertex and whether the graph opens up or down for the function f(x) = -1/4(x - 1)^2 + 4.

Learn how to determine the vertex and whether the graph opens up or down for the quadratic function f(x) = -1/4(x - 1)^2 + 4. This example explains the vertex form of a quadratic equation and the role of the 'a' coefficient in determining the direction of the parabola.

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