Rewriting f(x) as a Transformation of g(x)

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SUMMARY

The discussion focuses on rewriting the quadratic function f(x) = ax² + bx + c in vertex form to understand its transformation relative to the base function g(x) = x². Participants clarify the process of completing the square, resulting in f(x) = a(x + b/(2a))² + c - b²/(4a), where h = -b/(2a) and k = c - b²/(4a). This transformation highlights how the graph of f can be derived from g through translations and scaling. The key takeaway is the identification of h and k as the parameters that dictate the transformations.

PREREQUISITES
  • Understanding of quadratic functions and their standard form
  • Knowledge of completing the square technique
  • Familiarity with vertex form of quadratic equations
  • Basic graphing skills for quadratic functions
NEXT STEPS
  • Study the process of completing the square in depth
  • Learn about transformations of functions, specifically vertical and horizontal shifts
  • Explore the implications of the vertex form on graphing quadratic functions
  • Investigate the effects of varying coefficients a, b, and c on the graph of f(x)
USEFUL FOR

Students studying algebra, particularly those learning about quadratic functions and their transformations, as well as educators seeking to clarify these concepts for their students.

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Homework Statement



Can someone assist me with number 3 please...

Homework Equations





The Attempt at a Solution



I went ahead and rewrote f(x) in vertex form in terms of a,b, and c but I'm having a hard time writing down how f can be thought of as a transformation of g.
 

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What did you get? You have f(x) = ax2 + bx + c. Complete the square to get this into a form so that you can recognize the translations and other transformations relative to the graph of y = x2.
 


i got a[x+(b/2a)]^2 - [(b^2)/4a] + c where h=-(b/2a) and k=c-[(b^2)/4a]
 


halvizo1031 said:
i got a[x+(b/2a)]^2 - [(b^2)/4a] + c where h=-(b/2a) and k=c-[(b^2)/4a]
Shouldn't h be + b/(2a)?
So f(x) = a(x + b/(2a))2 + c - b2/(4a)

Or, f(x) = a(x + h)2 + k, with h = b/(2a) and k = c - b2/(4a).

You know what the graph of y = g(x) = x2 looks like, right? How would you need to transform the graph of g to get the graph of f?
 

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