To graph the focus and directrix for the given equation of the parabola [tex]\( x = -\frac{1}{8}(y-3)^2 + 1 \)[/tex], we need to decode the necessary components step by step:
### 1. Recognize the Structure:
The given equation is already in a form similar to the conic standard form. We identify the given equation as:
[tex]\[ x = -\frac{1}{8}(y - 3)^2 + 1 \][/tex]
### 2. Identify the Vertex:
The vertex form of a parabola [tex]\( x = a(y - k)^2 + h \)[/tex] allows us to easily identify the vertex [tex]\((h, k)\)[/tex]:
- [tex]\( h = 1 \)[/tex]
- [tex]\( k = 3 \)[/tex]
So, the vertex of the parabola is at:
[tex]\[ (1, 3) \][/tex]
### 3. Determine the Parameter [tex]\( p \)[/tex]:
In the standard form [tex]\((y - k)^2 = 4p(x - h)\)[/tex], we need to identify [tex]\( p \)[/tex], which controls the focal width and direction (the distance from the vertex to the focus and directrix).
### 4. Map it to Standard Form:
Rewriting [tex]\( x = -\frac{1}{8}(y - 3)^2 + 1 \)[/tex], we see:
[tex]\[ -\frac{1}{8}(y - 3)^2 = x - 1 \][/tex]
This implies:
[tex]\[ (y - 3)^2 = -8(x - 1) \][/tex]
Thus, the coefficient [tex]\( -8 \)[/tex] in front of [tex]\((x - 1)\)[/tex] corresponds to [tex]\( 4p \)[/tex]:
[tex]\[ 4p = -8 \Rightarrow p = -2 \][/tex]
### 5. Locate the Focus:
The focus of a parabola that opens horizontally is at a distance [tex]\( p \)[/tex] from the vertex along the axis of symmetry. For our parabola, since [tex]\( p \)[/tex] is negative, it opens to the left.
Given the vertex is at [tex]\( (1, 3) \)[/tex]:
[tex]\[ \text{Focus} = (h + p, k) = (1 - 2, 3) = (-1, 3) \][/tex]
### 6. Identify the Directrix:
The directrix is a vertical line that is [tex]\( p \)[/tex] units away from the vertex on the opposite side of the focus:
[tex]\[ \text{Directrix}: x = h - p \][/tex]
[tex]\[ x = 1 - (-2) = 1 + 2 = 3 \][/tex]
So, the directrix is the vertical line:
[tex]\[ x = 3 \][/tex]
### Summary:
- Focus: The coordinates are [tex]\((-1, 3)\)[/tex].
- Directrix: The equation of the vertical line is [tex]\( x = 3 \)[/tex].
Finally, you can graph these components as follows:
1. Select the "Point" tool to plot the focus at the coordinates [tex]\((-1, 3)\)[/tex].
2. Select the "Line" tool to draw the directrix, which is a vertical line at [tex]\( x = 3 \)[/tex].
Ensure your graph is correctly labeled by marking the vertex at [tex]\((1, 3)\)[/tex] for a complete and clear representation.
