Answer: 1. x = -2(y - 4)² + 1
2. x = -y² + 5
3. y = -5(x + 1)² + 2
Step-by-step explanation:
Notes: The vertex formula of a parabola is x = a(y - k)² + h or y = a(x - h)² + k
- (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}=(1,4)\qquad \text{Focus}:\bigg(\dfrac{7}{8},4\bigg)\\\\\text{Given}: (h,k)=(1,4)\\\\\\p=focus-vertex=\dfrac{7}{8}-\dfrac{8}{8}=\dfrac{-1}{8}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{8})}=\dfrac{1}{-\frac{1}{2}}=-2[/tex]
Now input a = -2 and (h, k) = (1, 4) into the equation x = a(y - k)² + h
x = -2(y - 4)² + 1
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2)
[tex]\text{Vertex}=(5,0)\qquad \text{Directrix}:x=\dfrac{21}{4}\\\\\text{Given}: (h,k)=(5,0)\\\\\\p=vertex-directrix=\dfrac{20}{4}-\dfrac{21}{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) = (5, 0) into the equation x = a(y - k)² + h
x = -1(y - 0)² + 5 → x = -y² + 5
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3)
[tex]\text{Vertex}=(-1,2)\qquad \text{Directrix}:y=\dfrac{41}{20}\\\\\text{Given}: (h,k)=(-1,2)\\\\\\p=vertex-directrix=\dfrac{40}{20}-\dfrac{41}{20}=\dfrac{-1}{20}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{20})}=\dfrac{1}{-\frac{1}{5}}=-5[/tex]
Now input a = -5 and (h, k) = (-1, 2) into the equation y = a(x - h)² + k
y = -5(x + 1)² + 2
