To determine the correct statements about the given equation of the parabola [tex]\( (y-1)^2 = 6(x-2) \)[/tex], let's analyze the equation step by step.
### Step 1: Understanding the Equation Structure
The given equation is [tex]\( (y-1)^2 = 6(x-2) \)[/tex]. This can be compared to the canonical form of a parabola that opens horizontally:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\( p \)[/tex] determines the direction and width of the parabola.
### Step 2: Identifying the Vertex
Comparing the given equation [tex]\((y-1)^2 = 6(x-2)\)[/tex] with the canonical form [tex]\((y-k)^2 = 4p(x-h)\)[/tex]:
- [tex]\( h = 2 \)[/tex]
- [tex]\( k = 1 \)[/tex]
Thus, the vertex of the parabola is at [tex]\((2, 1)\)[/tex].
### Step 3: Finding the Axis of Symmetry
The axis of symmetry for a parabola in the form [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is the vertical line passing through the vertex, which is:
[tex]\[
x = h
\][/tex]
Therefore, the axis of symmetry is:
[tex]\[
x = 2
\][/tex]
### Step 4: Determining the Direction of the Parabola
In the general form [tex]\((y - k)^2 = 4p(x - h)\)[/tex]:
- If [tex]\( p > 0 \)[/tex], the parabola opens to the right.
- If [tex]\( p < 0 \)[/tex], the parabola opens to the left.
From the given equation [tex]\((y-1)^2 = 6(x-2)\)[/tex]:
- We have [tex]\( 4p = 6 \)[/tex], hence [tex]\( p = \frac{6}{4} = \frac{3}{2} \)[/tex]
Since [tex]\( p > 0 \)[/tex], the parabola opens to the right.
### Conclusion
Given these analyses, we can confirm that the correct statement about the given parabola is:
- The axis of symmetry is the line [tex]\( x = 2 \)[/tex] and the parabola opens to the right.
