Answer:
Equation: [tex](x-2)^2=16(y-2)[/tex]
Vertex: [tex](2,2)[/tex]
Directrix: [tex]y=-2[/tex]
Focus: [tex](2,6)[/tex]
Step-by-step explanation:
Standard Form of Vertical Parabola:
- Equation -> [tex](x-h)^2=4p(y-k)[/tex] where [tex]p\neq 0[/tex]
- Vertex -> [tex](h,k)[/tex]
- Directrix -> [tex]y=k-p[/tex]
- Focus -> [tex](h,k+p)[/tex]
We know that our vertex is [tex](h,k)[/tex] -> [tex](2,2)[/tex], therefore, we can determine the value of [tex]p[/tex] by selecting a point from the parabola as [tex](x,y)[/tex]:
[tex](x-h)^2=4p(y-k)[/tex]
[tex](x-2)^2=4p(y-2)[/tex]
[tex](6-2)^2=4p(3-2)[/tex]
[tex](4)^2=4p(1)[/tex]
[tex]16=4p[/tex]
[tex]4=p[/tex]
Therefore:
- Directrix -> [tex]y=k-p[/tex] -> [tex]y=2-4[/tex] -> [tex]y=-2[/tex]
- Focus -> [tex](h,k+p)[/tex] -> [tex](2,2+4)[/tex] -> [tex](2,6)[/tex]
Conclusion:
- Equation -> [tex](x-2)^2=16(y-2)[/tex]
- Vertex -> [tex](2,2)[/tex]
- Directrix -> [tex]y=-2[/tex]
- Focus -> [tex](2,6)[/tex]
Review the graph for a visual
