Question

Which quadratic function is represented by the table?

Answer 1

`\boxed{f(x)=3x^2-2x+5}`

Explanation:

Since we know that the function of the table is a quadratic function, we can start writing this function as follows:

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

Given that we have three variables
`a, \ b \ and \ c`, we can solve this problem by taking three points, say:

`(0,5), \ (1,6) \ and \ (-1,10)`

So these points must satisfy the equation of the function. Therefore:

`For \ (0,5) \\ \\5=a(0)^2+b(0)+c \\ \\ 5=c \rightarrow \boxed{c=5} \\ \\ \\ For \ (1,6) \\ \\ Solving \\ \\ 6=a(1)^2+b(1)+5 \\ \\ \boxed{a+b=1} \\ \\For \ (-1,10) \\ \\Solving \\ \\10=a(-1)^2+b(-1)+5 \\ \\\boxed{a-b=5}`

So we have two equations to solve for a and b:

`(1) \ a+b=1 \ and \ (2) \ a-b=5 \\ \\ From \ 1: \\ \\ a=1-b \\ \\ From \ 2:\\ \\ a=5+b \\ \\ \\Matching: \\ \\ 1-b=5+b \ Solving: \\ \\ 1-5=b+b \\ \\ \boxed{b=-2} \\ \\ Finally: \\ \\ a=1-(-2) \therefore \boxed{a=3}`

So the quadratic function represented by the table is:

`\boxed{f(x)=3x^2-2x+5}`