I'm having a hard time figuring out a quadratic model for this data.
x y
----------
1 3
2 13
3 31
4 57
5 91
6 133
How do I go about doing this?
Is there a website that could help me?
First show that this is a quadratic relationship by taking differences and second differences:
y dy d2y
3 10 8
13 18 8
31 26 8
57 34 8
91 42 8
133
If the second difference is constant, it is a quadratic relationship.
Assume the quadratic model as:
f(x)=ax2 + bx + c
then
f(x+1)=a(x+1)^2+b(x+1)+c
and
f(x+1)-f(x)=2ax+b+a
Similarly,
f(x+2)-f(x+1)=2ax+b+3a
f(x+3)-f(x+2)=2ax+b+5a
....
Thus, the second difference = 2a
2a=8
a=4
By substituting values of x and a into one of the formulas, b can be found
for x=1,
dx=10
2ax+b+a=10
b=-2
Substitute a and b in f(1), we get
2(1)2-2(1)+c=3
c=1
Thus
f(x)=4x2-2x+1
check:
f(1)=3
f(3)=31
f(6)=133
To find a quadratic model for the given data, you can follow these steps:
Step 1: Write down the general quadratic equation. A quadratic equation can be written in the form of y = ax^2 + bx + c.
Step 2: Create a system of three equations using the given data points. Substituting the x and y values from the data into the equation, you will get three equations.
Step 3: Use the system of equations to solve for the values of a, b, and c.
To help you with this process, you can use various resources available online, such as websites, graphing calculators, or software. Here are a few websites that can aid you in finding the quadratic model for your data:
1. Wolfram Alpha (www.wolframalpha.com): This website provides a wide range of mathematical tools and solutions. You can enter your data points and navigate through the available options to find the quadratic model.
2. Desmos (www.desmos.com): Desmos is a powerful graphing calculator that allows you to plot your data points and explore different curve fitting options, including quadratic models.
3. Mathway (www.mathway.com): Mathway is an online math problem solver that can assist you in finding the quadratic model for your data. Simply input the data points and select the appropriate option to solve for the quadratic equation.
Remember that while these websites can provide valuable assistance, it is important to interpret and understand the steps involved in finding the quadratic model. Understanding the process will help you comprehend the solution more effectively.