Question 641379
We have two points (1,20) and (2,16). Let's find the equation of the line through them.




First let's find the slope of the line through the points *[Tex \LARGE \left(1,20\right)] and *[Tex \LARGE \left(2,16\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(1,20\right)]. So this means that {{{x[1]=1}}} and {{{y[1]=20}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(2,16\right)].  So this means that {{{x[2]=2}}} and {{{y[2]=16}}}.



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



{{{m=(16-20)/(2-1)}}} Plug in {{{y[2]=16}}}, {{{y[1]=20}}}, {{{x[2]=2}}}, and {{{x[1]=1}}}



{{{m=(-4)/(2-1)}}} Subtract {{{20}}} from {{{16}}} to get {{{-4}}}



{{{m=(-4)/(1)}}} Subtract {{{1}}} from {{{2}}} to get {{{1}}}



{{{m=-4}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(1,20\right)] and *[Tex \LARGE \left(2,16\right)] is {{{m=-4}}}



Now let's use the point slope formula:



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



{{{y-20=-4(x-1)}}} Plug in {{{m=-4}}}, {{{x[1]=1}}}, and {{{y[1]=20}}}



{{{y-20=-4x+-4(-1)}}} Distribute



{{{y-20=-4x+4}}} Multiply



{{{y=-4x+4+20}}} Add 20 to both sides. 



{{{y=-4x+24}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(1,20\right)] and *[Tex \LARGE \left(2,16\right)] is {{{y=-4x+24}}}


So the equation is {{{y=-4x+24}}} which can be written as the function {{{f(x)=-4x+24}}}


As a check, plug in each given x value. You should get the corresponding y values given.