To find the equation of a parabola that opens upward, has a minimum value of 3, and has an axis of symmetry at [tex]\( x = 3 \)[/tex], let's analyse these parts step by step.
### Step 1: Understand the Standard Form of a Parabola
A parabolic equation that opens upward can be written in the standard vertex form:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
In this equation:
- [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
- [tex]\( a \)[/tex] determines the direction and width of the parabola. Since the parabola opens upward, [tex]\( a \)[/tex] will be positive.
### Step 2: Identify the Vertex
From the problem statement, we know:
- The minimum value of the parabola is 3. This means the y-coordinate of the vertex, [tex]\( k \)[/tex], is 3.
- The axis of symmetry is at [tex]\( x = 3 \)[/tex]. This means the x-coordinate of the vertex, [tex]\( h \)[/tex], is 3.
So, the vertex [tex]\( (h, k) \)[/tex] is [tex]\( (3, 3) \)[/tex].
### Step 3: Form the Equation
Substituting [tex]\( h = 3 \)[/tex] and [tex]\( k = 3 \)[/tex] into the standard form, we get:
[tex]\[ f(x) = a(x - 3)^2 + 3 \][/tex]
Since it opens upward, and no stretch or compression is specified, we typically assume [tex]\( a = 1 \)[/tex]. Therefore, the equation simplifies to:
[tex]\[ f(x) = (x - 3)^2 + 3 \][/tex]
### Step 4: Match with Given Options
Now, we need to match this equation with the given options:
A) [tex]\( f(x) = (x-3)^2 - 6 \)[/tex]
B) [tex]\( f(x) = (x+3)^2 + 3 \)[/tex]
C) [tex]\( f(x) = (x+3)^2 - 6 \)[/tex]
D) [tex]\( f(x) = (x-3)^2 + 3 \)[/tex]
The correct match is option D: [tex]\( f(x) = (x-3)^2 + 3 \)[/tex].
### Conclusion
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
