Find the equation of the curve

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SUMMARY

The discussion focuses on deriving the equation of a curve from a specific numerical sequence: (11.25, 10, 9.25, 9, 9.25, 10, 11.25, 13, 15.25, 18). Participants identified that the sequence exhibits constant second differences, indicating a quadratic relationship. The curve is likely a parabola, and the vertex can be determined from the sequence. Additionally, the concept of Difference Equations is suggested as a relevant mathematical area for further exploration.

PREREQUISITES
  • Understanding of quadratic functions and parabolas
  • Knowledge of graphing sequences and interpreting coordinates
  • Familiarity with calculating differences in sequences
  • Basic concepts of Difference Equations
NEXT STEPS
  • Study how to derive the equation of a parabola from given points
  • Learn about calculating first and second differences in sequences
  • Explore the application of Difference Equations in sequence analysis
  • Practice graphing quadratic functions using software tools like Desmos
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Students in mathematics, educators teaching algebra, and anyone interested in sequence analysis and curve fitting techniques.

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Homework Statement


From the given sequence of numbers (11.25, 10, 9.25, 9, 9.25, 10, 11.25, 13, 15.25, 18 ...)
A) Graph the sequence
B) Find the equation of the curve
C) Determine the 100th term of the sequence


Homework Equations





The Attempt at a Solution


I know how to graph the sequence by setting the given numbers as y coordinates and their order as x coordinate but beyond there I'm stuck. Help please!
 
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aerin8 said:

Homework Statement


From the given sequence of numbers (11.25, 10, 9.25, 9, 9.25, 10, 11.25, 13, 15.25, 18 ...)
A) Graph the sequence
B) Find the equation of the curve
C) Determine the 100th term of the sequence


Homework Equations





The Attempt at a Solution


I know how to graph the sequence by setting the given numbers as y coordinates and their order as x coordinate but beyond there I'm stuck. Help please!
There is a certain pattern that jumps out here. Look at the difference of successive terms in this sequence.

Also, the second differences are constant.
 
I can see the pattern but I don't know how to apply it in the equation of the curve. Is there a formula I can plug the data into to find the equation?
 
is the curve the same thing as a parabola?
 
The curve might be a parabola. If so, you know where the vertex is.

In case it's not a parabola, there's an area of mathematics called Difference Equations (see http://en.wikipedia.org/wiki/Recurrence_relation) that might be helpful. See especially the section titled "Relationship to difference equations narrowly defined."
 

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