To analyze the given quadratic function [tex]\( f(x)=3(x-2)^2+4 \)[/tex], we can use the standard form for a quadratic equation:
[tex]\[ f(x) = a(x-h)^2 + k \][/tex]
Here, the function is already in this standard form, allowing us to directly identify the parameters.
1. Understanding the parameters:
- [tex]\( a \)[/tex] determines the "width" and direction of the parabola (whether it opens upwards or downwards). In this case, [tex]\( a = 3 \)[/tex].
- [tex]\( h \)[/tex] is the x-coordinate of the vertex of the parabola.
- [tex]\( k \)[/tex] is the y-coordinate of the vertex of the parabola.
2. Identifying the axis of symmetry:
The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is given by:
[tex]\[
x = h
\][/tex]
In this equation, [tex]\( h = 2 \)[/tex]. Therefore, the axis of symmetry is:
[tex]\[
x = 2
\][/tex]
3. Identifying the vertex:
The vertex of the parabola is the point [tex]\( (h, k) \)[/tex]. From the function [tex]\( f(x)=3(x-2)^2+4 \)[/tex], we see that [tex]\( h = 2 \)[/tex] and [tex]\( k = 4 \)[/tex]. Therefore, the vertex is:
[tex]\[
(2, 4)
\][/tex]
So, summarizing the result:
- Axis of symmetry: [tex]\( x = 2 \)[/tex]
- Vertex: [tex]\( (2, 4) \)[/tex]
Thus, the answers are:
[tex]\[
x = 2
\][/tex]
Vertex: [tex]\( 2, 4 \)[/tex]
