To determine which equation represents a parabola that opens upward, with a minimum value of 3 and an axis of symmetry at [tex]\( x = 3 \)[/tex], let's first recall the general form of a parabola that opens upwards:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex of the parabola, where [tex]\( h \)[/tex] is the x-coordinate and [tex]\( k \)[/tex] is the y-coordinate. For a parabola that opens upward, [tex]\( a \)[/tex] is a positive constant.
Given the following:
- The minimum value of the parabola is 3. This means the y-coordinate of the vertex [tex]\( k = 3 \)[/tex].
- The axis of symmetry is at [tex]\( x = 3 \)[/tex]. This indicates the x-coordinate of the vertex [tex]\( h = 3 \)[/tex].
Thus, the vertex of the parabola is located at [tex]\( (3, 3) \)[/tex].
Substituting [tex]\( h = 3 \)[/tex] and [tex]\( k = 3 \)[/tex] into the general form of the parabola, we get:
[tex]\[ f(x) = a(x - 3)^2 + 3 \][/tex]
To find the correct function from the given options, let's examine each one:
A. [tex]\( f(x) = (x + 3)^2 - 6 \)[/tex]
- The term [tex]\( (x + 3)^2 \)[/tex] suggests a vertex form of [tex]\( f(x) = (x - (-3))^2 \)[/tex], so the axis of symmetry is [tex]\( x = -3 \)[/tex], which is incorrect.
B. [tex]\( f(x) = (x + 3)^2 + 3 \)[/tex]
- Here the term [tex]\( (x + 3)^2 \)[/tex] gives the axis of symmetry at [tex]\( x = -3 \)[/tex], which is incorrect.
C. [tex]\( f(x) = (x - 3)^2 - 6 \)[/tex]
- The term [tex]\( (x - 3)^2 \)[/tex] gives the axis of symmetry at [tex]\( x = 3 \)[/tex], but the minimum value here would be [tex]\( -6 \)[/tex] (since [tex]\( k = -6 \)[/tex]), which does not match our given minimum value of 3.
D. [tex]\( f(x) = (x - 3)^2 + 3 \)[/tex]
- The term [tex]\( (x - 3)^2 \)[/tex] gives the axis of symmetry at [tex]\( x = 3 \)[/tex], and the minimum value is at [tex]\( y = 3 \)[/tex], which matches our given conditions.
Therefore, the correct answer is:
[tex]\[ \boxed{ \text{D. } f(x) = (x - 3)^2 + 3 } \][/tex]
