To find the linear function with the greatest unit rate, we need to calculate the rate of change (slope) of each of the given points.
A. (2,16) and (5,10), Rate of change= [tex] \frac{10-16}{5-2}=\frac{-6}{3}= -2 [/tex]
B. (2,14) and (6,12), Rate of change= [tex] \frac{12-14}{6-2}=\frac{-2}{4}=\frac{-1}{2} [/tex]
C. (2,-24) and (5,-15), Rate of change=[tex] \frac{-15+24}{5-2}=\frac{9}{3}=3 [/tex]
D. (2,-12) and (6,-16), Rate of change= [tex] \frac{-16+12}{6-2}=\frac{-4}{2}=-2 [/tex]
E. (2,-19) and (6,-17), Rate of change=[tex] \frac{-17+19}{6-2}=\frac{2}{4}=\frac{1}{2} [/tex]
Since, it can be seen that the greatest rate of change is 3 which is of the points given in Option C, therefore, it is the linear function with greatest unit rate(increasing).
For any two given points [tex](x_{1} ,y_{1} ) and(x_{2},y_{2})[/tex]
Rate of change =[tex]\frac{y_{2} -y_{1} }{x_{2} -x_{1} }.[/tex]
To find linear function with the greatest unit rate we find the rate of change of each points.
A) The given points are(2, 16) and (5, 10 )
Rate of change =[tex]\frac{10-16}{5-2}=\frac{-6}{2}=-3.[/tex]
B) The given points are :(2,14) and (6,12)
Rate of change=[tex]\frac{12-14}{6-2}=\frac{-2}{4}=\frac{-1}{2}.[/tex]
C) Points are (2,-24) and (5,-15)
Rate of change = [tex]\frac{-15+24}{5-2}=\frac{9}{3}= 3.[/tex]
D) Points are :(2,-12) and (6,-16)
Rate of change =[tex]\frac{-16+12}{6-2}=\frac{-4}{4}=-1.[/tex]
E) Points are (2, -19 ) and (6, -17)
Rate of change =[tex]\frac{-17+19}{6-2}=\frac{-2}{4}=\frac{-1}{2}.[/tex]
Option C has the greatest rate of change.