Question 146541
f (3) =6 tells us that when x=3, then y=6. So we know that the equation goes through the point (3,6)



f (-4) =2 tells us that when x=-4, then y=2. So we know that the equation goes through the point (-4,2)



So let's find the equation of the line that goes through the points (3,6) and (-4,2)





First let's find the slope through the points *[Tex \LARGE \left(3,6\right)] and *[Tex \LARGE \left(-4,2\right)]



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



{{{m=(2-6)/(-4-3)}}} Plug in {{{y[2]=2}}}, {{{y[1]=6}}}, {{{x[2]=-4}}}, {{{x[1]=3}}}, , 



{{{m=(-4)/(-4-3)}}} Subtract {{{6}}} from {{{2}}} to get {{{-4}}}



{{{m=(-4)/(-7)}}} Subtract {{{3}}} from {{{-4}}} to get {{{-7}}}



{{{m=4/7}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(3,6\right)] and *[Tex \LARGE \left(-4,2\right)] is {{{m=4/7}}}



Now let's use the point slope formula:



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



{{{y-6=(4/7)(x-3)}}} Plug in {{{m=4/7}}}, {{{x[1]=3}}}, and {{{y[1]=6}}}



{{{y-6=(4/7)x+(4/7)(-3)}}} Distribute



{{{y-6=(4/7)x-12/7}}} Multiply



{{{y=(4/7)x-12/7+6}}} Add 6 to both sides. 



{{{y=(4/7)x+30/7}}} Combine like terms. note: If you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>.



{{{y=(4/7)x+30/7}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(3,6\right)] and *[Tex \LARGE \left(-4,2\right)] is {{{y=(4/7)x+30/7}}}



 Notice how the graph of {{{y=(4/7)x+30/7}}} goes through the points *[Tex \LARGE \left(3,6\right)] and *[Tex \LARGE \left(-4,2\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,(4/7)x+30/7),
 circle(3,6,0.08),
 circle(3,6,0.10),
 circle(3,6,0.12),
 circle(-4,2,0.08),
 circle(-4,2,0.10),
 circle(-4,2,0.12)
 )}}} Graph of {{{y=(4/7)x+30/7}}} through the points *[Tex \LARGE \left(3,6\right)] and *[Tex \LARGE \left(-4,2\right)]