Question 254011
Since f(1)=8 and f(7)=-10, this means that the line goes through the points (1,8) and (7,-10)





First let's find the slope of the line through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(7,-10\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(1,8\right)]. So this means that {{{x[1]=1}}} and {{{y[1]=8}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(7,-10\right)].  So this means that {{{x[2]=7}}} and {{{y[2]=-10}}}.



{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(-10-8)/(7-1)}}} Plug in {{{y[2]=-10}}}, {{{y[1]=8}}}, {{{x[2]=7}}}, and {{{x[1]=1}}}



{{{m=(-18)/(7-1)}}} Subtract {{{8}}} from {{{-10}}} to get {{{-18}}}



{{{m=(-18)/(6)}}} Subtract {{{1}}} from {{{7}}} to get {{{6}}}



{{{m=-3}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(7,-10\right)] is {{{m=-3}}}



Now let's use the point slope formula:



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-8=-3(x-1)}}} Plug in {{{m=-3}}}, {{{x[1]=1}}}, and {{{y[1]=8}}}



{{{y-8=-3x+-3(-1)}}} Distribute



{{{y-8=-3x+3}}} Multiply



{{{y=-3x+3+8}}} Add 8 to both sides. 



{{{y=-3x+11}}} Combine like terms. 



{{{y=-3x+11}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(7,-10\right)] is {{{y=-3x+11}}}



This means that the function is {{{f(x)=-3x+11}}} (replace 'y' with f(x))



 Notice how the graph of {{{y=-3x+11}}} goes through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(7,-10\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-3x+11),
 circle(1,8,0.08),
 circle(1,8,0.10),
 circle(1,8,0.12),
 circle(7,-10,0.08),
 circle(7,-10,0.10),
 circle(7,-10,0.12)
 )}}} 
 
 Graph of {{{y=-3x+11}}} through the points *[Tex \LARGE \left(1,8\right)] and *[Tex \LARGE \left(7,-10\right)]