Explain the relationship between a function rule and its table of values and the graph of the function.

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A function or a map is defined on a set of values called domain. The set of values produced by the function is called range. A  function uniquely relates elements from domain with elements from range such that for any x value in domain it is associated only one y value in range.

The domain and the range of a function form a set of ordered pairs generated by the rule y = f(x).

If there is known only the set of generated ordered pairs but the rule is unknown, you can draw only points in a system of coordinates.

If the rule that generates the set of ordered pairs is known, then you may graph the curve that links the points resulted from the ordered pairs.

f(x) = x^2 + 2x + 1

Table of values:

x = -2 => f(-2) = 4 - 4 + 1 = 1 => pair (-2,1)

x = -1 => f(-1) = 1 - 2 + 1 = 0 => pair (-1,0)

x = 0 => f(0) = 1 => pair (0,1)

x = 1 => f(1) = 4 => pair (1,4)

x = 2 => f(2) = 9 => pair (2,9)

Set of ordered pairs generated by the rule f(x) = x^2 + 2x + 1: {(-2,1) ; (-1,0) ; (0,1) ;(1,4) ;  (2,9)}

The graph is parabola that joins the ordered pairs above.

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