To find the axis of symmetry and the vertex for the function [tex]\( f(x) = 3(x - 2)^2 + 4 \)[/tex], let's analyze the function step-by-step.
1. Identify the form of the quadratic function:
The given function is in the vertex form of a quadratic equation, which is
[tex]\[
f(x) = a(x - h)^2 + k
\][/tex]
In this form, [tex]\((h, k)\)[/tex] represents the vertex of the parabola and [tex]\(x = h\)[/tex] is the axis of symmetry.
2. Extract the values of [tex]\(h\)[/tex] and [tex]\(k\)[/tex]:
By comparing the given function [tex]\( f(x) = 3(x - 2)^2 + 4 \)[/tex] with the standard vertex form [tex]\( f(x) = a(x - h)^2 + k \)[/tex], we can identify the values:
[tex]\[
h = 2
\][/tex]
[tex]\[
k = 4
\][/tex]
3. Determine the axis of symmetry:
The axis of symmetry for the parabola is a vertical line that passes through the vertex. Since [tex]\( h = 2 \)[/tex], the axis of symmetry is:
[tex]\[
x = 2
\][/tex]
4. Determine the vertex:
The vertex of the parabola is the point [tex]\((h, k)\)[/tex]. Using the identified values:
[tex]\[
h = 2
\][/tex]
[tex]\[
k = 4
\][/tex]
Thus, the vertex is:
[tex]\[
(2, 4)
\][/tex]
Therefore, the axis of symmetry is [tex]\(x = 2\)[/tex] and the vertex is [tex]\((2, 4)\)[/tex].
[tex]\[
\text{Axis of Symmetry: } x = 2
\][/tex]
[tex]\[
\text{Vertex: } (2, 4)
\][/tex]
