Question 124962
Since f(0) = -2, we have the first point (0,-2). Also, because f(4) = 5, we have the second point (4,5)



So let's find the equation of the line that goes through (0,-2) and (4,5)


First lets find the slope through the points ({{{0}}},{{{-2}}}) and ({{{4}}},{{{5}}})


{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula (note: *[Tex \Large \left(x_{1},y_{1}\right)] is the first point ({{{0}}},{{{-2}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{4}}},{{{5}}}))


{{{m=(5--2)/(4-0)}}} Plug in {{{y[2]=5}}},{{{y[1]=-2}}},{{{x[2]=4}}},{{{x[1]=0}}}  (these are the coordinates of given points)


{{{m= 7/4}}} Subtract the terms in the numerator {{{5--2}}} to get {{{7}}}.  Subtract the terms in the denominator {{{4-0}}} to get {{{4}}}

  

So the slope is

{{{m=7/4}}}


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Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
{{{y-y[1]=m(x-x[1])}}} where {{{m}}} is the slope, and *[Tex \Large \left(\textrm{x_{1},y_{1}}\right)] is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


{{{y--2=(7/4)(x-0)}}} Plug in {{{m=7/4}}}, {{{x[1]=0}}}, and {{{y[1]=-2}}} (these values are given)



{{{y+2=(7/4)(x-0)}}} Rewrite {{{y--2}}} as {{{y+2}}}



{{{y+2=(7/4)x+(7/4)(0)}}} Distribute {{{7/4}}}


{{{y+2=(7/4)x+0}}} Multiply {{{7/4}}} and {{{0}}} to get {{{0/4}}}. Now reduce {{{0/4}}} to get {{{0}}}


{{{y=(7/4)x+0-2}}} Subtract {{{2}}} from  both sides to isolate y


{{{y=(7/4)x-2}}} Combine like terms {{{0}}} and {{{-2}}} to get {{{-2}}} 

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Answer:



So the equation of the line which goes through the points ({{{0}}},{{{-2}}}) and ({{{4}}},{{{5}}})  is: {{{y=(7/4)x-2}}} (note: in function notation the equation is {{{f(x)=(7/4)x-2}}})


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=7/4}}} and the y-intercept is {{{b=-2}}}


Notice if we graph the equation {{{y=(7/4)x-2}}} and plot the points ({{{0}}},{{{-2}}}) and ({{{4}}},{{{5}}}),  we get this: (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)


{{{drawing(500, 500, -7, 11, -7.5, 10.5,
graph(500, 500, -7, 11, -7.5, 10.5,(7/4)x+-2),
circle(0,-2,0.12),
circle(0,-2,0.12+0.03),
circle(4,5,0.12),
circle(4,5,0.12+0.03)
) }}} Graph of {{{y=(7/4)x-2}}} through the points ({{{0}}},{{{-2}}}) and ({{{4}}},{{{5}}})


Notice how the two points lie on the line. This graphically verifies our answer.