Answer :
The ordered pairs are (1,4), (5,6) and (7,7) of the form (n,g)
The rate of change is the slope of the equation
[tex]\begin{gathered} \frac{g_2-g_1}{n_2-n_1}=\frac{6-4}{5-1}=\frac{2}{4}=\frac{1}{2} \\ m=\frac{1}{2} \end{gathered}[/tex]The equation of the linear function now becomes;
[tex]\begin{gathered} m=\frac{g-g_1}{n-n_1} \\ \frac{1}{2}=\frac{g-4}{n-1} \\ 2(g-4)=n-1 \\ 2g-8=n-1 \\ 2g=n-1+8 \\ 2g=n+7 \\ \text{Divide all through by 2} \\ g=\frac{1}{2}n+\frac{7}{2} \\ g=\frac{1}{2}n+3.5 \end{gathered}[/tex]Hence the equation of the linear function is g = 0.5n + 3.5
Another equation for a function that has twice the rate of change and the same initial value becomes;
initial rate of change is 0.5
hence for this, the rate of change is 0.5 x 2
m = 1
the initial value is at n = 0,
so g = 0.5(0) + 3.5
g = 3.5
The new equation then becomes g = mn + C
g = 1n + 3.5
Hence the new equation that has twice the rate of change and the same initial value is g = n + 3.5