To express the given parabola in standard form [tex]\((y - k)^2 = 4p(x - h)\)[/tex], we need to determine the values for [tex]\(p\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] based on the given vertex, directrix, and focus.
Given:
- Vertex [tex]\((h, k) = (2, 2)\)[/tex]
- Directrix [tex]\(x = 2 - \sqrt{2}\)[/tex]
- Focus [tex]\((2 + \sqrt{2}, 2)\)[/tex]
### Step-by-Step Solution:
1. Identify the coordinates for vertex [tex]\((h, k)\)[/tex]:
The vertex is provided as [tex]\((2, 2)\)[/tex]. Therefore, the values for [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are:
[tex]\[
h = 2
\][/tex]
[tex]\[
k = 2
\][/tex]
2. Determine the value for [tex]\(p\)[/tex]:
The value for [tex]\(p\)[/tex] is the distance from the vertex to the directrix or the focus. Since the focus is at [tex]\((2 + \sqrt{2}, 2)\)[/tex], we calculate the horizontal distance from the vertex to the focus:
[tex]\[
p = (2 + \sqrt{2}) - 2 = \sqrt{2} \approx 1.4142135623730951 \quad (\text{rounded to the nearest hundredth: } 1.41)
\][/tex]
### Conclusion:
- The value for [tex]\(p\)[/tex] is:
[tex]\[
p = 1.41
\][/tex]
- The value for [tex]\(h\)[/tex] is:
[tex]\[
h = 2
\][/tex]
- The value for [tex]\(k\)[/tex] is:
[tex]\[
k = 2
\][/tex]
Now we can write the equation of the parabola in standard form:
[tex]\[
(y - 2)^2 = 4 \cdot 1.41 \cdot (x - 2)
\][/tex]
Thus, the final equation in standard form is:
[tex]\[
(y - 2)^2 = 5.64 \cdot (x - 2)
\][/tex]
