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Let's analyze the given quadratic function \( f(x) = \frac{1}{2}(x + 3)^2 + 4 \).

### Step 1: Identify the Vertex
The function is written in vertex form \( f(x) = a(x - h)^2 + k \), where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) determines the stretch or compression of the parabola.

In your case:
- \( h = -3 \) (since \( x + 3 \) means \( h = -3 \)),
- \( k = 4 \),
So, the vertex is \( (-3, 4) \).

### Step 2: Identify the Stretch or Compression
The coefficient \( a = \frac{1}{2} \) indicates how the graph is stretched or compressed:
- Since \( a = \frac{1}{2} \) is positive but less than 1, the parabola is **wider** (compressed vertically) by a factor of \( \frac{1}{2} \).

### Final Answer:
- The correct option is: "The vertex is \((-3, 4)\). The parabola is compressed shorter by a factor of \( \frac{1}{2} \)."

Let me know if you would like further clarification!

### Related Questions:
1. How would the graph look if the coefficient \( a \) was negative?
2. What effect does changing \( h \) have on the graph?
3. How does changing \( k \) affect the position of the vertex?
4. What is the equation of the parent function for this quadratic?
5. How would the graph change if \( a = 2 \) instead of \( \frac{1}{2} \)?

**Tip:** Always observe the sign and value of \( a \) to determine the stretch/compression and the direction of the parabola.

Describe the graph of the quadratic function f(x) = 1/2 (x + 3)^2 + 4 by giving its vertex and stating how it is stretched or compressed when compared to the parent quadratic function.

Learn how to describe the graph of the quadratic function f(x) = 1/2 (x + 3)^2 + 4. Identify the vertex at (-3, 4) and understand how the parabola is compressed vertically by a factor of 1/2. Explore key concepts in algebra and graph transformations. Suitable for high school students in Grades 9-12.

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