Answer: [tex]\bold{1.\quad y=-\dfrac{1}{2}(x-9)^2+10}[/tex]
2. x = (y - 1)² + 4
[tex]\bold{3.\quad x=\dfrac{1}{2}(y-2)^2+7}[/tex]
Step-by-step explanation:
Notes: The vertex form of a parabola is y = a(x - h)² + k or x = a(y - k)² + h
- (h, k) is the vertex
- p is the distance from the vertex to the focus
[tex]\bullet\quad a=\dfrac{1}{4p}[/tex]
1)
[tex]\text{Vertex}=(9, 10)\qquad \text{Focus}=\bigg(9,\dfrac{19}{2}\bigg)\\\\\text{Given}:(h, k)=(9, 10)\\\\\\p=focus-vertex=\dfrac{19}{2}-\dfrac{20}{2}=\dfrac{-1}{2}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{2})}=\dfrac{1}{-2}=-\dfrac{1}{2}\\[/tex]
Now input a = -1/2 and (h, k) = (9, 10) into the equation y = a(x - h)² + k
[tex]\bold{y=-\dfrac{1}{2}(x-9)^2+10}[/tex]
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2)
[tex]\text{Vertex}=(4, 1)\qquad \text{Focus}=\bigg(\dfrac{17}{4},1\bigg)\\\\\text{Given}:(h, k)=(4, 1)\\\\\\p=focus-vertex=\dfrac{17}{4}-\dfrac{16}{4}=\dfrac{1}{4}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{1}{4})}=\dfrac{1}{1}=1\\[/tex]
Now input a = 1 and (h, k) = (4, 1) into the equation x = a(y - k)² + h
x = 1(y - 1)² + 4 → x = (y - 1)² + 4
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3)
[tex]\text{Vertex}=(7, 2)\qquad \text{Focus}=\bigg(\dfrac{15}{2},2\bigg)\\\\\text{Given}:(h, k)=(7, 2)\\\\\\p=focus-vertex=\dfrac{15}{2}-\dfrac{14}{2}=\dfrac{1}{2}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{1}{2})}=\dfrac{1}{2}[/tex]
Now input a = 1/2 and (h, k) = (7, 2) into the equation x = a(y - k)² + h
[tex]\bold{x=\dfrac{1}{2}(y-2)^2+7}[/tex]
