Answer: 1. y = 2(x + 4)² - 3
[tex]\bold{2.\quad y=-\dfrac{1}{3}(x+8)^2-7}[/tex]
[tex]\bold{3.\quad y=-\dfrac{1}{2}(x-7)^2+1}[/tex]
Step-by-step explanation:
Notes: The vertex form of a parabola is 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}=(-4,-3)\qquad \text{Directrix}:y=-\dfrac{25}{8}\\\\\text{Given}:(h, k)=(-4, 3)\\\\p=\dfrac{-24}{8}-\dfrac{-25}{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) = (-4, -3) into the equation y = a(x - h)² + k
y = 2(x + 4)² - 3
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2)
[tex]\text{Vertex}=(-8,-7)\qquad \text{Directrix}:y=-\dfrac{-25}{4}\\\\\text{Given}:(h, k)=(-8, -7)\\\\p=\dfrac{-28}{4}-\dfrac{-25}{4}=\dfrac{-3}{4}}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-3}{4})}=\dfrac{1}{-3}=-\dfrac{1}{3}[/tex]
Now input a = -1/3 and (h, k) = (-8, -7) into the equation y = a(x - h)² + k
[tex]\bold{y=-\dfrac{1}{3}(x+8)^2-7}[/tex]
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3)
[tex]\text{Focus}=\bigg(7,\dfrac{1}{2}\bigg)\qquad \text{Directrix}:y=\dfrac{3}{2}[/tex]
The midpoint of the focus and directrix is the y-coordinate of the vertex:
[tex]\dfrac{focus+directrix}{2}=\dfrac{\frac{1}{2}+\frac{3}{2}}{2}=\dfrac{\frac{4}{2}}{2}=\dfrac{2}{2}=1[/tex]
The x-coordinate of the vertex is given in the focus as 7
(h, k) = (7, 1)
Now let's find the a-value:
[tex]p=\dfrac{2}{2}-\dfrac{3}{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) = (7, 1) into the equation y = a(x - h)² + k
[tex]\bold{y=-\dfrac{1}{2}(x-7)^2+1}[/tex]
