Using the equation f(x)=1/4(x+5)^2+2, find the vertex.
 a
(5, -2)
 b
(5, 2)
 c
(-5, -2)
 d
(-5, 2)

The given equation is \( f(x) = \frac{1}{4}(x + 5)^2 + 2 \).

This equation is in the vertex form of a parabola, which is generally written as:

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

In this form, \((h, k)\) represents the vertex of the parabola.

### Comparing the given equation with the vertex form:

- \( a = \frac{1}{4} \)
- \( h = -5 \) (since \( x + 5 \) can be rewritten as \( x - (-5) \))
- \( k = 2 \)

So, the vertex of the parabola is \((-5, 2)\).

### The correct answer is:
**d) \((-5, 2)\)**

Would you like further details or have any questions about this? Here are some related questions to deepen your understanding:

1. How does changing the value of \( a \) affect the graph of the parabola?
2. What happens to the vertex if we change the value of \( h \) in the equation?
3. How would you rewrite the equation if the vertex were \((3, -4)\)?
4. What is the axis of symmetry for the parabola given by the equation?
5. How would you determine if the parabola opens upwards or downwards?
6. If the equation was \( f(x) = -\frac{1}{4}(x + 5)^2 + 2 \), where would the vertex be?
7. What is the minimum or maximum value of the function, and how is it related to the vertex?
8. How can you find the y-intercept of the given equation?

**Tip:** The vertex form of a quadratic function is especially useful for easily identifying the vertex and the direction in which the parabola opens.

Learn how to find the vertex of the quadratic equation f(x) = (1/4)(x + 5)^2 + 2 using the vertex form of a parabola. Understand the concepts of vertex coordinates (h, k) and how to determine them from the given equation. Suitable for high school students studying quadratic equations.

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