Question

Part 2: Use the information provided to write the vertex form equation of each parabola.

Answer 1

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

    $\bullet\quad a=\dfrac{1}{4p}$

1)

$\text{Vertex}=(1,4)\qquad \text{Focus}:\bigg(\dfrac{7}{8},4\bigg)\\\\\text{Given}: (h,k)=(1,4)\\\\\\p=focus-vertex=\dfrac{7}{8}-\dfrac{8}{8}=\dfrac{-1}{8}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{8})}=\dfrac{1}{-\frac{1}{2}}=-2$

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)

$\text{Vertex}=(5,0)\qquad \text{Directrix}:x=\dfrac{21}{4}\\\\\text{Given}: (h,k)=(5,0)\\\\\\p=vertex-directrix=\dfrac{20}{4}-\dfrac{21}{4}=\dfrac{-1}{4}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{4})}=\dfrac{1}{-1}=-1$

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)

$\text{Vertex}=(-1,2)\qquad \text{Directrix}:y=\dfrac{41}{20}\\\\\text{Given}: (h,k)=(-1,2)\\\\\\p=vertex-directrix=\dfrac{40}{20}-\dfrac{41}{20}=\dfrac{-1}{20}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{20})}=\dfrac{1}{-\frac{1}{5}}=-5$

Now input a = -5 and (h, k) = (-1, 2) into the equation y = a(x - h)² + k

y = -5(x + 1)² + 2