Question

Find the quadratic model for the following sets of values.
Please show work.

Answer 1

`Quadratic\ function:\ f(x)=ax^2+bx+c.\\\\\text{From a table we have the points}\\\\(-1,\ 1),\ (0,\ -1)\ and\ (2,\ 7).\\\\Substitute\ the\ coordinates\ of\ the\ points\ to \ the\ equation\ of\ f(x):\\\\For\ (0,\ -1)\\\\-1=a(0)^2+b(0)+c\to \boxed{c=-1}\\\\Thereofre\ we\ have\ f(x)=ax^2+bx-1.\\\\For\ (-1,\ 1)\\\\1=a(-1)^2+b(-1)-1\\1=1a-1b-1\qquad\text{add 1 to both sides}\\2=a-b\qquad\text{add b to both sides}\\a=2+b\qquad(*)`

`For\ (2,\ 7)\\\\7=a(2)^2+b(2)-1\\7=4a+2b-1\qquad\text{add 1 to both sides}\\8=4a+2b\qquad\text{divide both sides by 2}\\4=2a+b\qquad\text{substitute from}\ (*)\\\\4=2(2+b)+b\qquad\text{use distributive property}\\4=4+2b+b\qquad\text{subtract 4 from both sides}\\0=3b\qquad\text{divide both sides by 3}\\\boxed{b=0}\\\\Substitute\ the\ value\ of\ b\ to\ (*):\\\\a=2+0\\\boxed{a=2}\\\\Answer:\ \boxed{f(x)=2x^2-1}`

Answer 2

`f(x)=2x^2-1`

Explanation:

To determine a quadratic function fully, three known points are needed. Each point can be used to determine one parameter of the function. A general quadratic is as follows:

`f(x)=ax^2+bx+c`

where a,b, and c are to be determined.

Use the individual points from the table to set up an equation system with the unknowns a,b, and c, then solve:

`1=a(-1)^2+b(-1)+c\implies 1=a-b+c\\-1=a0+b0+c\implies -1=c\\7=a2^2+b2+c\implies 7=4a+2b+c\\\\\mbox{solve:}\\\\1=a-b-1\implies a=2+b\\7=4(2+b)+2b-1\implies b=0\implies a=2\\\\\mbox{solution:}\\a=2, b=0, c=-1\\f(x)=2x^2-1`