Determining increase/decrease intervals for ax^2+bx+c

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SUMMARY

The discussion focuses on determining the intervals of increase and decrease for the quadratic function y=ax^2+bx+c without using differentiation. The key conclusion is that the function is increasing when x > -b/(2a), derived from the condition that for any x1 and x2 in the interval, f(x1) > f(x2) holds true. The participants also suggest using the method of completing the square to facilitate this analysis, which provides a clearer understanding of the function's behavior.

PREREQUISITES
  • Understanding of quadratic functions and their standard form (y=ax^2+bx+c).
  • Knowledge of inequalities and their implications on function behavior.
  • Familiarity with the concept of limits, particularly Δ approaching 0.
  • Basic skills in algebraic manipulation, including completing the square.
NEXT STEPS
  • Study the method of completing the square for quadratic functions.
  • Research the properties of parabolas and their vertex forms.
  • Explore the implications of the vertex on the intervals of increase and decrease.
  • Learn about the relationship between coefficients a, b, and the direction of the parabola.
USEFUL FOR

Students studying algebra, particularly those tackling quadratic functions, as well as educators looking for effective methods to teach function behavior without calculus.

peripatein
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Hello,

Homework Statement



I am trying to determine on what intervals the parabola y=ax^2+bx+c increases/decreases, without resorting to differentiation.

Homework Equations





The Attempt at a Solution



For the function to be increasing on a certain interval f(x1)>f(x2) for any x1 and x2 on that interval such that x1>x2. Hence, ax1^2+bx1+c>ax2^2+bx2+c. That yields, (x1+x2)>-b/a. How do I derive the expected x>-b/2a from that? That result could be obtained for x1=x2, hence Δ=0, but why is that and why ought it to be used in order to obtain the correct answer?
 
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Let x2=x1+Δ and then consider the limes of Δ->0?
 
peripatein said:
Hello,

Homework Statement



I am trying to determine on what intervals the parabola y=ax^2+bx+c increases/decreases, without resorting to differentiation.

Homework Equations





The Attempt at a Solution



For the function to be increasing on a certain interval f(x1)>f(x2) for any x1 and x2 on that interval such that x1>x2. Hence, ax1^2+bx1+c>ax2^2+bx2+c. That yields, (x1+x2)>-b/a. How do I derive the expected x>-b/2a from that? That result could be obtained for x1=x2, hence Δ=0, but why is that and why ought it to be used in order to obtain the correct answer?

How about completing the square?
 

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