Question

consider the data set shown in the table.
use an online graphing tool to find the quadratic equation that models the data in this table. for help, see this worked example.
type the correct answer in each box. use numerals instead of words. round your answers to the nearest hundredth.
the quadratic equation that models the data in this table is ( y=square x^{2}+square x+square ).

Explanation:

Step1: Use an online graphing tool

We can use an online graphing tool such as Desmos. In Desmos, we input the data points \((x,y)\) where \(x = 0,1,\cdots,10\) and \(y=1.5,2.2,\cdots,2.8\). Then we use the quadratic regression feature.

Step2: Analyze the regression result

When we perform quadratic regression \(y = ax^{2}+bx + c\) on the data set, the formula for quadratic regression is \(y=\sum_{i = 1}^{n}(ax_{i}^{2}+bx_{i}+c - y_{i})^{2}\), and we minimize this sum with respect to \(a\), \(b\), and \(c\).
After using an online graphing - tool (e.g., Desmos) for quadratic regression on the given data set \(\{(0,1.5),(1,2.2),(2,2.7),(3,2.6),(4,3.5),(5,3.8),(6,4.0),(7,3.4),(8,3.2),(9,2.9),(10,2.8)\}\), we get:
For \(a\approx - 0.07\), \(b\approx0.76\), \(c\approx1.59\)

Answer:

The quadratic equation that models the data in this table is \(y=-0.07x^{2}+0.76x + 1.59\)