Parabola General to standard form

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SUMMARY

The discussion focuses on converting the equation x^2 - 4x + 8y + 12 = 0 into standard form and finding the focus of the resulting parabola. The equation simplifies to (x - 2)^2 = -8(y - 8), indicating a vertex at (2, 8) and a downward-opening parabola. The relationship between the focus and the standard form is clarified, confirming that the focus is determined by the parameter 1/4a, where a equals -2.

PREREQUISITES
  • Understanding of quadratic equations and their forms
  • Familiarity with parabolic equations and their properties
  • Knowledge of vertex form of a parabola
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of parabolas, including focus and directrix
  • Learn about the derivation of the standard form of parabolic equations
  • Explore graphing techniques for quadratic functions
  • Investigate the implications of the parameter 'a' in the context of conic sections
USEFUL FOR

Students studying algebra, particularly those focusing on quadratic functions and conic sections, as well as educators teaching these concepts in mathematics courses.

markwjak
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Homework Statement


write the standard form of the equation
x^2-4x+8y+12=0
find the focus

Homework Equations


The Attempt at a Solution


i have it down to (x-2)^2=-8(y-8)
i'm not sure if you have to multiply the -8 on the right side of the equations by 4 since the focus is 1/4a
 
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markwjak said:

Homework Statement


write the standard form of the equation
x^2-4x+8y+12=0
find the focus

Homework Equations





The Attempt at a Solution


i have it down to (x-2)^2=-8(y-8)
i'm not sure if you have to multiply the -8 on the right side of the equations by 4 since the focus is 1/4a

(x-2)^2=-8(y-8)

let x' = x-2
let y' = y-8

So, x^2 = -8y'
or y' = -(1/8)*x^2

This is it in standard form. To plot it, the vertex of the parabola will be (2, 8) and it will look like an upside down 'U'.
 

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