Question 156168
The statement {{{f(4)=0}}} tells us that if {{{x=4}}}, then {{{y=0}}}. So the point (4,0) is on the line. Also, the statement {{{f(6) =6}}} tells us that if {{{x=6}}}, then {{{y=6}}}. So the point (6,6) is also on the line.



So let's find the equation of the line that goes through the two points (4,0) and (6,6)




To do that, we first need to find the slope of the line through the points *[Tex \LARGE \left(4,0\right)] and *[Tex \LARGE \left(6,6\right)]



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



{{{m=(6-0)/(6-4)}}} Plug in {{{y[2]=6}}}, {{{y[1]=0}}}, {{{x[2]=6}}}, and {{{x[1]=4}}}



{{{m=(6)/(6-4)}}} Subtract {{{0}}} from {{{6}}} to get {{{6}}}



{{{m=(6)/(2)}}} Subtract {{{4}}} from {{{6}}} to get {{{2}}}



{{{m=3}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(4,0\right)] and *[Tex \LARGE \left(6,6\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-0=3(x-4)}}} Plug in {{{m=3}}}, {{{x[1]=4}}}, and {{{y[1]=0}}}



{{{y-0=3x+3(-4)}}} Distribute



{{{y-0=3x-12}}} Multiply



{{{y=3x-12+0}}} Add 0 to both sides. 



{{{y=3x-12}}} Combine like terms. 



{{{y=3x-12}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(4,0\right)] and *[Tex \LARGE \left(6,6\right)] is {{{y=3x-12}}}



In function notation, the answer is {{{f(x)=3x-12}}}


Notice how {{{f(4)=3(4)-12=12-12=0}}} and {{{f(6)=3(6)-12=18-12=6}}}. So this also verifies our answer.




 Also, notice how the graph of {{{y=3x-12}}} goes through the points *[Tex \LARGE \left(4,0\right)] and *[Tex \LARGE \left(6,6\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,3x-12),
 circle(4,0,0.08),
 circle(4,0,0.10),
 circle(4,0,0.12),
 circle(6,6,0.08),
 circle(6,6,0.10),
 circle(6,6,0.12)
 )}}} Graph of {{{y=3x-12}}} through the points *[Tex \LARGE \left(4,0\right)] and *[Tex \LARGE \left(6,6\right)]