Find the quadratic function that is the best fit for f(x) defined by the table below. 2 4 398 1603 X f(x) 0 0 6 3602 8 10 6402 9997 2 The quadratic function is y=x x²+x+· (Type an integer or decimal rounded to two decimal places as needed.)
Find the quadratic function that is the best fit for f(x) defined by the table below. 2 4 398 1603 X f(x) 0 0 6 3602 8 10 6402 9997 2 The quadratic function is y=x x²+x+· (Type an integer or decimal rounded to two decimal places as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Title: Finding the Best-Fit Quadratic Function**
**Objective:**
Determine the quadratic function that best fits a given set of data points.
**Data Table:**
| x | 0 | 2 | 4 | 6 | 8 | 10 |
|-----|----|-----|-----|-----|-----|-----|
| f(x)| 0 | 398 | 1603| 3602| 6402| 9997|
**Task:**
Identify the quadratic function of the form \( y = ax^2 + bx + c \) that matches the data. Enter the values of \( a \), \( b \), and \( c \) as integers or decimals rounded to two decimal places.
**Instructions:**
1. Analyze the data table provided to observe the relationship between \( x \) and \( f(x) \).
2. Fit a quadratic model to these points using methods such as least squares fitting or another appropriate method.
3. Once the best-fit quadratic function is determined, fill in the coefficients for \( a \), \( b \), and \( c \) in the function \( y = ax^2 + bx + c \).
**Example:**
If the data comfortably fits a quadratic trend, you may find coefficients such as:
- \( a \) = 1.23
- \( b \) = 4.56
- \( c \) = 7.89
The resulting quadratic function would be:
\[ y = 1.23x^2 + 4.56x + 7.89 \]
**Note:**
Use technology or mathematical techniques to derive the most accurate coefficients. Ensure your answer is rounded to two decimal places for precision.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F556745e6-02d3-4800-a933-7dc40a753d9d%2F3bad7222-ca43-4a99-a787-c2c0707c414f%2Fdzmrhfg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Best-Fit Quadratic Function**
**Objective:**
Determine the quadratic function that best fits a given set of data points.
**Data Table:**
| x | 0 | 2 | 4 | 6 | 8 | 10 |
|-----|----|-----|-----|-----|-----|-----|
| f(x)| 0 | 398 | 1603| 3602| 6402| 9997|
**Task:**
Identify the quadratic function of the form \( y = ax^2 + bx + c \) that matches the data. Enter the values of \( a \), \( b \), and \( c \) as integers or decimals rounded to two decimal places.
**Instructions:**
1. Analyze the data table provided to observe the relationship between \( x \) and \( f(x) \).
2. Fit a quadratic model to these points using methods such as least squares fitting or another appropriate method.
3. Once the best-fit quadratic function is determined, fill in the coefficients for \( a \), \( b \), and \( c \) in the function \( y = ax^2 + bx + c \).
**Example:**
If the data comfortably fits a quadratic trend, you may find coefficients such as:
- \( a \) = 1.23
- \( b \) = 4.56
- \( c \) = 7.89
The resulting quadratic function would be:
\[ y = 1.23x^2 + 4.56x + 7.89 \]
**Note:**
Use technology or mathematical techniques to derive the most accurate coefficients. Ensure your answer is rounded to two decimal places for precision.
Expert Solution
Step 1
let quadratic function is y= ax2+bx+c ----(1)
from table (x,y)= (2,398),(4,1603),(6,3602)
at (2,398) from (1) 4a+2b+c=398 ----(2)
at (4,1603) from (1) 16a+4b+c=1603 -----(3)
at (6,3602) from (1) 36a+6b+c=3602 -----(4)
from (1) c= 398-4a-2b
put value of c in (3), we get
16a+4b+398-4a-2b=1603
12a+2b= 1205 ----(5)
put value of c in (4), we get
36a+6b+398-4a-2b=3602
32a+4b=3204 ---(6)
multiply (5) by 2 and subtract from (6) ,we get
24a+4b-32a-4b= 2410-3204
-8a=-794
a= 99.25
put a in (5), we get
12(99.25)+2b=1205
1191+2b=1205
b=7
now c= 398-4(99.25)-2(7)= 398-397-14=13
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