Find equation of a quadratic with line of symmetry

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SUMMARY

The discussion focuses on deriving the equation of a quadratic function given its axis of symmetry and an x-intercept. The axis of symmetry is defined as x = -b/(2a), and with an x-intercept at (-2, 0), the equation can be expressed as 0 = a(-2)^2 + b(-2) + c. The participants derive relationships between coefficients b and c in terms of a, ultimately leading to the conclusion that if the axis of symmetry is x = 1.75, then b = 3.5a and further conditions yield b = 11 + 2a.

PREREQUISITES
  • Understanding of quadratic equations and their standard form (y = ax^2 + bx + c)
  • Knowledge of the concept of axis of symmetry in parabolas
  • Ability to solve algebraic equations involving multiple variables
  • Familiarity with x-intercepts and their significance in graphing quadratics
NEXT STEPS
  • Study the derivation of the quadratic formula from standard form
  • Learn how to graph parabolas using vertex and intercept forms
  • Explore the implications of changing coefficients a, b, and c on the shape of the parabola
  • Investigate real-world applications of quadratic equations in physics and engineering
USEFUL FOR

Students, educators, and anyone interested in mastering quadratic equations, particularly those focusing on algebra and geometry concepts.

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Here's something to start you off. The axis of symmetry for the parabola [math]y = ax^2 + bx + c[/math] is [math]x = -\dfrac{b}{2a}[/math] and if an x-intercept is at (-2, 0) then [math]0 = a(-2)^2 + b(-2) + c[/math]. That will give you both b and c as functions of a.

Can you finish?

-Dan
 
Equivalently, if the axis of symmetry is x= 1.75 then the equation can be written as $y= a(x- 1.75)^2+ c= ax^2+ 3.5ax+ 3.0625+ c= ax^2+ bx+ 22. We must have b= 3.5a.

If, in addition, y= 0 when x= -2, 0= 4a- 2b+ 22 so b= 11+ 2a.
 

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