Question

determine which regression model- quadratic, cubic, or quartic- best fits each table. write the function of the regression model for each table.
x | m(x)
-1 | 0
0 | -1
1 | 0
2 | 3
3 | 8
4 | 15

Explanation:

Step1: Analyze the data points

We have the points \((-1, 0)\), \((0, -1)\), \((1, 0)\), \((2, 3)\), \((3, 8)\), \((4, 15)\). Let's check the differences or try to fit a polynomial.

Step2: Test quadratic model (\(y = ax^2+bx + c\))

Using \((0, -1)\), we get \(c=-1\). Using \((-1, 0)\): \(a - b-1 = 0\). Using \((1, 0)\): \(a + b-1 = 0\). Solving these, we get \(a = 1\), \(b = 0\). So quadratic model is \(y=x^2 - 1\). Let's check other points:

• For \(x = 2\): \(y=4 - 1=3\) (matches).
• For \(x = 3\): \(y=9 - 1=8\) (matches).
• For \(x = 4\): \(y=16 - 1=15\) (matches).

Step3: Verify if cubic or quartic is needed

Since the quadratic model fits all points, we don't need higher - degree models (cubic or quartic).

Answer:

The best - fitting regression model is the quadratic model, and its function is \(M(x)=x^{2}-1\)