Question

Part 1: Use the information provided to write the vertex form equation of each parabola.

Answer 1

Answer:   $\bold{1.\quad y=-\dfrac{1}{2}(x-9)^2+10}$

                2.   x = (y - 1)² + 4

                $\bold{3.\quad x=\dfrac{1}{2}(y-2)^2+7}$

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

     $\bullet\quad a=\dfrac{1}{4p}$

1)

$\text{Vertex}=(9, 10)\qquad \text{Focus}=\bigg(9,\dfrac{19}{2}\bigg)\\\\\text{Given}:(h, k)=(9, 10)\\\\\\p=focus-vertex=\dfrac{19}{2}-\dfrac{20}{2}=\dfrac{-1}{2}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{2})}=\dfrac{1}{-2}=-\dfrac{1}{2}$

Now input a = -1/2 and (h, k) = (9, 10) into the equation y = a(x - h)² + k

$\bold{y=-\dfrac{1}{2}(x-9)^2+10}$

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2)

$\text{Vertex}=(4, 1)\qquad \text{Focus}=\bigg(\dfrac{17}{4},1\bigg)\\\\\text{Given}:(h, k)=(4, 1)\\\\\\p=focus-vertex=\dfrac{17}{4}-\dfrac{16}{4}=\dfrac{1}{4}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{1}{4})}=\dfrac{1}{1}=1$

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)

$\text{Vertex}=(7, 2)\qquad \text{Focus}=\bigg(\dfrac{15}{2},2\bigg)\\\\\text{Given}:(h, k)=(7, 2)\\\\\\p=focus-vertex=\dfrac{15}{2}-\dfrac{14}{2}=\dfrac{1}{2}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{1}{2})}=\dfrac{1}{2}$

Now input a = 1/2 and (h, k) = (7, 2) into the equation x = a(y - k)² + h

$\bold{x=\dfrac{1}{2}(y-2)^2+7}$