find the Function rule

Question

x: -2,-1,0,1,2
y: -1,1,3,5,7

Answer 1

y = 2 x + 3

Answer 2

Average rate of change of linear function is slope.

m = [ f(b) - f(a) ] / ( b - a )

For a linear function, the slope is constant.

In this case:

x1 = - 2 , y1 = - 1

x2 = - 1 , y2 = 1

x3 = 0 , y3 = 3

x4 = 1 , y4 = 5

x5 = 2 , y5 = 7

( y2 - y1 ) / ( x2 - x1 ) = [ 1 - ( - 1 ) ] / [ ( - 1) - ( - 2 ) ] = ( 1 + 1 ) / ( - 1 + 2 ) = 2 / 1 = 2

( y3 - y2 ) / ( x3 - x2 ) = ( 3 - 1 ) / [ 0 - ( - 1 ) ] = 2 / ( 0 + 1 ) = 2 / 1 = 2

( y4 - y3 ) / ( x4 - x3 ) = ( 5 - 3 ) / ( 1 - 0 ) ] = 2 / 1 = 2

( y5 - y4 ) / ( x5 - x4 ) = ( 7 - 5 ) / ( 2 - 1 ) ] = 2 / 1 = 2

So the slope:

m = 2

Put this value for any pair of coordinates in poit-slope fotmula for example for x4 = 1 , y4 = 5

y = m x + b

y = 2 x + b

5 = 2 ∙ 1 + b

5 = 2 + b

5 - 2 = b

3 = b

b = 3

y = 2 x + 3