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.