Question

determine which regression model- quadratic, cubic, or quartic- best fits each table. write the function of the regression model for each table.

x	h(x)
-1	0
0	-1
1	0
2	3
3	8
4	15

Explanation:

Step1: Analyze the data pattern

Let's list the \( x \) and \( h(x) \) values: \( (-1, 0) \), \( (0, -1) \), \( (1, 0) \), \( (2, 3) \), \( (3, 8) \), \( (4, 15) \). Notice that \( h(x) = x^2 - 1 \) for \( x=-1,0,1 \) (since \( (-1)^2 - 1 = 0 \), \( 0^2 - 1 = -1 \), \( 1^2 - 1 = 0 \)), and for \( x=2 \), \( 2^2 - 1 = 3 \), \( x=3 \), \( 3^2 - 1 = 8 \), \( x=4 \), \( 4^2 - 1 = 15 \). So the pattern fits a quadratic model.

Step2: Formulate the quadratic function

A quadratic function is of the form \( h(x)=ax^2 + bx + c \). Using \( x = 0 \), \( h(0)=-1 \), so \( c=-1 \). Using \( x = 1 \), \( h(1)=0=a(1)^2 + b(1)-1\Rightarrow a + b=1 \). Using \( x=-1 \), \( h(-1)=0=a(-1)^2 + b(-1)-1\Rightarrow a - b=1 \). Solving \( a + b = 1 \) and \( a - b = 1 \), we add the equations: \( 2a = 2\Rightarrow a = 1 \), then \( b = 0 \). So the function is \( h(x)=x^2 - 1 \).

Answer:

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