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What is the correct standard form of the equation of the parabola?
Be sure to show each step of your work and how you got the equation

Given information:
Directrix: y= -6
Focus= (2,-4)
Vertex= (2,-5)

Distance to directrix: |y+6|
Distance to focus: (square root of) (x-2)^2+(y+4)^2

What Is The Correct Standard Form Of The Equation Of The Parabola Be Sure To Show Each Step Of Your Work And How You Got The Equation Given Information Directri class=

Sagot :

Intriguing456idn

Answer:

[tex]y= x^2 -x+6[/tex]

Step-by-step explanation:

The equation for a parabola defined as a conic section is:

[tex](x-h)^2=4P(y-k)[/tex]

where:

  • [tex](h,k)[/tex] is the vertex of the parabola
  • [tex]P[/tex] is the shortest distance from the vertex to the directrix (or the focus)

We can identify the following variable values from the given information:

  • [tex]h=2[/tex]
  • [tex]k=-5[/tex]
  • [tex]P=1[/tex]   (positive because the parabola opens upward)

Plugging these into the equation, we get:

[tex]\boxed{(x-2)^2=4(y+5)}[/tex]

This can be converted to standard form by solving for y:

[tex]x^2-4x+4=4y+20[/tex]

[tex]4y=4x^2-4x-16[/tex]

[tex]\boxed{y= x^2 -x-4}[/tex]

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