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We are given the quadratic function \( f(x) = (x + 45)^2 + 60 \), and the task is to find which set of data corresponds to this function. Let's break it down step-by-step.

### General form of the function:

This is a quadratic equation written in vertex form:

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

where:
- \( (h, k) \) is the **vertex** of the parabola.
- The parabola opens **upward** if the coefficient of \( (x - h)^2 \) is positive, and it opens **downward** if it's negative.
- The **axis of symmetry** is the line \( x = h \).

### 1. Direction of the parabola:
- In this case, the equation is \( f(x) = (x + 45)^2 + 60 \), which can be rewritten as \( f(x) = (x - (-45))^2 + 60 \).
- Since the coefficient of \( (x + 45)^2 \) is positive, the parabola opens **upward**.

### 2. Vertex of the parabola:
- The vertex is \( (-45, 60) \).

### 3. Axis of symmetry:
- The axis of symmetry is the vertical line passing through the x-coordinate of the vertex, so it's \( x = -45 \).

### Now, let's compare this to the options provided:

- **Set A**: Vertex \( (-60, -45) \), Axis of Symmetry \( x = -60 \), Parabola opens upward — Incorrect.
- **Set B**: Vertex \( (60, 45) \), Axis of Symmetry \( x = 60 \), Parabola opens downward — Incorrect.
- **Set C**: Vertex \( (-45, 60) \), Axis of Symmetry \( x = -45 \), Parabola opens upward — **Correct**.
- **Set D**: Vertex \( (45, 60) \), Axis of Symmetry \( x = 45 \), Parabola opens downward — Incorrect.

### Conclusion:
The correct set is **Set C**.

Would you like further details or have any questions?

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Here are 5 questions that expand on this explanation:
1. What is the significance of the vertex form of a quadratic equation?
2. How do you determine the direction in which a parabola opens?
3. What is the relationship between the vertex and the axis of symmetry?
4. How can you convert a quadratic equation from standard form to vertex form?
5. How does the value of the vertex affect the graph of a parabola?

**Tip**: The vertex form of a quadratic equation is particularly useful for easily identifying the vertex and the direction of the parabola without completing the square.

Which set of data is correct for the quadratic relation f(x) = (x + 45)^2 + 60?

Explore how to determine the correct vertex, axis of symmetry, and parabola opening direction for the quadratic equation f(x) = (x + 45)^2 + 60. Suitable for Grades 9-12 students, this example highlights key concepts like the vertex form of a parabola and how to interpret data sets.

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