The given function is:

$f(x) = \frac{1}{2}(x + 2)^2 - 3$

We are comparing this function to the parent function, $f(x) = x^2$.

Analyzing the Transformation:

1. Vertex Form: The function is in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex and $a$ determines the direction and stretch/compression.

   Here, $a = \frac{1}{2}$, $h = -2$, and $k = -3$.

   So, the vertex is $(-2, -3)$.

2. Effect of $a = \frac{1}{2}$:

   • Since $0 < a < 1$, the graph is compressed vertically compared to the parent function, which means it is wider.
   • The graph opens upward because $a > 0$, but not as steeply as the parent function.

3. Vertex:

   • The vertex has shifted to $(-2, -3)$, meaning 2 units to the left and 3 units down from the origin.

Answer:

The correct answer is:

• Compressed since $a$ is less than 1 with a vertex at (-2, -3).

Do you want further details or have any questions?

Here are 5 related questions for practice:

1. How would the graph change if $a = 2$ instead of $\frac{1}{2}$?
2. What happens to the graph of $f(x)$ if the vertex is moved to $(0, 0)$?
3. How does changing $k$ affect the position of the vertex in $f(x) = a(x - h)^2 + k$?
4. If $a$ were negative, how would the graph of the function be affected?
5. What is the significance of the value $h$ in the vertex form of a quadratic function?

Tip: The value of $a$ controls both the direction (upward or downward) and the width of the parabola.