Answer :
To find the axis of symmetry and the vertex for the function [tex]\( f(x) = 3(x - 2)^2 + 4 \)[/tex], we will use the vertex form of a parabolic function, which is generally written as:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
In this form:
- [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
- The axis of symmetry is the vertical line [tex]\( x = h \)[/tex].
For the given function [tex]\( f(x) = 3(x - 2)^2 + 4 \)[/tex]:
- The value of [tex]\( h \)[/tex], which is the [tex]\( x \)[/tex]-coordinate of the vertex, is [tex]\( 2 \)[/tex].
- The value of [tex]\( k \)[/tex], which is the [tex]\( y \)[/tex]-coordinate of the vertex, is [tex]\( 4 \)[/tex].
Thus, the vertex of the function is [tex]\( (2, 4) \)[/tex].
The axis of symmetry, being the vertical line that passes through the vertex, is:
[tex]\[ x = 2 \][/tex]
So, the step-by-step solution is:
1. Identify the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] from the function [tex]\( f(x) = 3(x - 2)^2 + 4 \)[/tex]. Here, [tex]\( h = 2 \)[/tex] and [tex]\( k = 4 \)[/tex].
2. The vertex of the parabola is the point [tex]\( (h, k) \)[/tex], which is [tex]\( (2, 4) \)[/tex].
3. The axis of symmetry is the vertical line [tex]\( x = h \)[/tex], which is [tex]\( x = 2 \)[/tex].
Therefore, the final answers are:
[tex]\[ \text{Axis of symmetry: } x = 2 \][/tex]
[tex]\[ \text{Vertex: } (2, 4) \][/tex]
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
In this form:
- [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
- The axis of symmetry is the vertical line [tex]\( x = h \)[/tex].
For the given function [tex]\( f(x) = 3(x - 2)^2 + 4 \)[/tex]:
- The value of [tex]\( h \)[/tex], which is the [tex]\( x \)[/tex]-coordinate of the vertex, is [tex]\( 2 \)[/tex].
- The value of [tex]\( k \)[/tex], which is the [tex]\( y \)[/tex]-coordinate of the vertex, is [tex]\( 4 \)[/tex].
Thus, the vertex of the function is [tex]\( (2, 4) \)[/tex].
The axis of symmetry, being the vertical line that passes through the vertex, is:
[tex]\[ x = 2 \][/tex]
So, the step-by-step solution is:
1. Identify the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] from the function [tex]\( f(x) = 3(x - 2)^2 + 4 \)[/tex]. Here, [tex]\( h = 2 \)[/tex] and [tex]\( k = 4 \)[/tex].
2. The vertex of the parabola is the point [tex]\( (h, k) \)[/tex], which is [tex]\( (2, 4) \)[/tex].
3. The axis of symmetry is the vertical line [tex]\( x = h \)[/tex], which is [tex]\( x = 2 \)[/tex].
Therefore, the final answers are:
[tex]\[ \text{Axis of symmetry: } x = 2 \][/tex]
[tex]\[ \text{Vertex: } (2, 4) \][/tex]
