To solve part (a) for the function $f(x) = \frac{1}{2}(x - 3)^2 - 2$, we need to:

1. Identify the vertex of the parabola.
2. Choose additional points on either side of the vertex to plot on the graph.

The function is in vertex form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex and $a$ determines the direction and width of the parabola.

Step 1: Identify the Vertex

In this case:

• $h = 3$
• $k = -2$

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

Step 2: Choose Additional Points

To find four other points, substitute different values of $x$ into the function and calculate the corresponding $f(x)$.

1. Let $x = 1$: $f(1) = \frac{1}{2}(1 - 3)^2 - 2 = \frac{1}{2} \cdot 4 - 2 = 2 - 2 = 0$ Point: $(1, 0)$

2. Let $x = 2$: $f(2) = \frac{1}{2}(2 - 3)^2 - 2 = \frac{1}{2} \cdot 1 - 2 = 0.5 - 2 = -1.5$ Point: $(2, -1.5)$

3. Let $x = 4$: $f(4) = \frac{1}{2}(4 - 3)^2 - 2 = \frac{1}{2} \cdot 1 - 2 = 0.5 - 2 = -1.5$ Point: $(4, -1.5)$

4. Let $x = 5$: $f(5) = \frac{1}{2}(5 - 3)^2 - 2 = \frac{1}{2} \cdot 4 - 2 = 2 - 2 = 0$ Point: $(5, 0)$

Answer for Part (a)

The five unique points on the graph of the function are: $(3, -2), (1, 0), (2, -1.5), (4, -1.5), (5, 0)$

These points include the vertex and four other points around it.

Would you like further clarification on any of these steps or details on how to graph it?

Here are five related questions to deepen understanding:

1. How would the graph change if the coefficient $\frac{1}{2}$ were increased or decreased?
2. What does the vertex represent in a real-world context, if this function modeled a real-world scenario?
3. How can we determine if this function has a maximum or minimum value?
4. What is the axis of symmetry for this parabola, and why is it important?
5. How would the graph shift if the vertex were located at a different point, such as $(4, -2)$?

Tip: When dealing with a function in vertex form, the values of $h$ and $k$ provide an easy way to identify the vertex, which is essential for understanding the graph's shape and position.