Let's analyze the function [tex]\( f(x) = 3(x-2)^2 + 4 \)[/tex].
### Step 1: Identify the standard form
The function [tex]\( f(x) = 3(x-2)^2 + 4 \)[/tex] is in the vertex form of a quadratic function, which is given by:
[tex]\[ f(x) = a(x-h)^2 + k \][/tex]
where:
- [tex]\( a \)[/tex] is the coefficient of the squared term,
- [tex]\( (h, k) \)[/tex] is the vertex of the parabola,
- [tex]\( x = h \)[/tex] is the axis of symmetry.
### Step 2: Identify the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex]
In the given function [tex]\( f(x) = 3(x-2)^2 + 4 \)[/tex], we can see that:
- The term inside the parenthesis, [tex]\((x-2)\)[/tex], indicates [tex]\( h = 2 \)[/tex].
- The constant term outside the parenthesis, [tex]\( +4 \)[/tex], indicates [tex]\( k = 4 \)[/tex].
### Step 3: Determine the vertex
The vertex of the quadratic function is given by the point [tex]\((h, k)\)[/tex]. Therefore, for the function [tex]\( f(x) = 3(x-2)^2 + 4 \)[/tex]:
- The vertex is located at [tex]\( (h, k) = (2, 4) \)[/tex].
### Step 4: Determine the axis of symmetry
The axis of symmetry for a parabola in the vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex] is given by the vertical line that passes through [tex]\( h \)[/tex]. Hence, the axis of symmetry is:
- [tex]\( x = h = 2 \)[/tex].
### Final Answer
- Axis of symmetry: [tex]\( x = 2 \)[/tex]
- Vertex: [tex]\( (2, 4) \)[/tex]
So, you would fill in:
[tex]$
x=
$[/tex]
[tex]$2$[/tex]
Vertex: ( [tex]$2$[/tex], [tex]$4$[/tex] )
