To determine the graph that correctly represents the function [tex]\( f(x) = \frac{1}{2}(x-3)^2 - 6 \)[/tex], let's analyze its key characteristics:
1. Vertex of the Parabola:
The function is in the vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
- Here, [tex]\( h = 3 \)[/tex] and [tex]\( k = -6 \)[/tex].
- Thus, the vertex of this parabola is at [tex]\( (3, -6) \)[/tex].
2. Direction of Opening:
- The coefficient of [tex]\( (x-3)^2 \)[/tex] is [tex]\(\frac{1}{2}\)[/tex], which is positive, indicating that the parabola opens upwards.
3. Plotting Key Points:
To better understand the shape of the parabola, we can evaluate the function at several points. Here are the values for specific [tex]\( x \)[/tex]-values:
- [tex]\( f(0) = \frac{1}{2}(0-3)^2 - 6 = -1.5 \)[/tex]
- [tex]\( f(1) = \frac{1}{2}(1-3)^2 - 6 = -4.0 \)[/tex]
- [tex]\( f(2) = \frac{1}{2}(2-3)^2 - 6 = -5.5 \)[/tex]
- [tex]\( f(3) = \frac{1}{2}(3-3)^2 - 6 = -6.0 \)[/tex]
- [tex]\( f(4) = \frac{1}{2}(4-3)^2 - 6 = -5.5 \)[/tex]
- [tex]\( f(5) = \frac{1}{2}(5-3)^2 - 6 = -4.0 \)[/tex]
- [tex]\( f(6) = \frac{1}{2}(6-3)^2 - 6 = -1.5 \)[/tex]
Given these points and the characteristics of the parabola, you should look for a graph that:
- Has a vertex at [tex]\( (3, -6) \)[/tex].
- Opens upwards.
- Passes through the points mentioned above.
When you compare these criteria to the given options, you will be able to identify the correct graph!
