Needs to find equation for each
Decide which of the two given functions is linear and find its equation. x=0,3,6,10,15 f(x)=0,3,47,77,114 g(x)=-1,26,53,89,134 g(x)
x   f(x)   g(x)   ∆x  ∆f    ∆g  ∆f/∆x  ∆g/∆x


0      0       -1


                         3     3    27       1       9


3      3      26


                         3   44   27     44/3   9


6    47      53


                         4   30  36      15/2   9


10  77      89


                         5   37  45      37/5   9


15 114    134


The table shows the value x and the corresponding values of f(x) and g(x). The ∆x column shows the difference between the x values from the preceding row and the next row. ∆f and ∆g columns show the corresponding differences for each of the two functions. A linear function has a constant gradient, and the gradients for f and g are given by the division ∆f/∆x and ∆g/∆x in the last two columns. We can see that g(x) has a constant gradient 9, so it must be a linear function. ∆f/∆x is clearly not linear because the gradients are not constant.


If we choose a point (0,-1) we can write the equation knowing the gradient: g(x)-(-1)=9x. So g(x)+1=9x and g(x)=9x-1.

