Question 153214
Notice how as x goes up by 1, y goes up by 3. So this implies a linear relationship. This means that the equation is a line.




So let's find equation of the line through the first two points (0,-2) and (1,1) (note: you can pick any pair of points)





First let's find the slope of the line through the points *[Tex \LARGE \left(0,-2\right)] and *[Tex \LARGE \left(1,1\right)]



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



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



{{{m=(3)/(1-0)}}} Subtract {{{-2}}} from {{{1}}} to get {{{3}}}



{{{m=(3)/(1)}}} Subtract {{{0}}} from {{{1}}} to get {{{1}}}



{{{m=3}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(0,-2\right)] and *[Tex \LARGE \left(1,1\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--2=3(x-0)}}} Plug in {{{m=3}}}, {{{x[1]=0}}}, and {{{y[1]=-2}}}



{{{y+2=3(x-0)}}} Rewrite {{{y--2}}} as {{{y+2}}}



{{{y+2=3x+3(-0)}}} Distribute



{{{y+2=3x+0}}} Multiply



{{{y=3x+0-2}}} Subtract 2 from both sides. 



{{{y=3x-2}}} Combine like terms. 



{{{y=3x-2}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(0,-2\right)] and *[Tex \LARGE \left(1,1\right)] is {{{y=3x-2}}}



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Answer:


So the function takes the x values and produces the y values is {{{f(x)=3x-2}}}



 Notice how the graph of {{{f(x)=3x-2}}} goes through the points (0,-2), (1,1), (2,4), (3,7) and (4,10). So this visually verifies our answer.



 {{{drawing( 500, 500, -10, 10, -10, 12,
 grid(1),
 graph( 500, 500, -10, 10, -10, 12,3x-2),
 circle(0,-2,0.08),
 circle(0,-2,0.10),
 circle(0,-2,0.12),
 circle(1,1,0.08),
 circle(1,1,0.10),
 circle(1,1,0.12),

 circle(2,4,0.08),
 circle(2,4,0.10),
 circle(2,4,0.12),

 circle(3,7,0.08),
 circle(3,7,0.10),
 circle(3,7,0.12),

 circle(4,10,0.08),
 circle(4,10,0.10),
 circle(4,10,0.12)
 )}}} Graph of {{{y=3x-2}}}