Question 325447

We can see that the equation {{{y=x+4}}} has a slope {{{m=1}}} and a y-intercept {{{b=4}}}.



Now to find the slope of the perpendicular line, simply flip the slope {{{m=1}}} to get {{{m=1/1}}}. Now change the sign to get {{{m=-1}}}. So the perpendicular slope is {{{m=-1}}}.



Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope {{{m=-1}}} and the coordinates of the given point *[Tex \LARGE \left\(0,-1\right\)].



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



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



{{{y+1=-1(x-0)}}} Rewrite {{{y--1}}} as {{{y+1}}}



{{{y+1=-x-1(-0)}}} Distribute



{{{y+1=-x+0}}} Multiply



{{{y=-x+0-1}}} Subtract 1 from both sides. 



{{{y=-x-1}}} Combine like terms. 



So the equation of the line perpendicular to {{{y=x+4}}} that goes through the point *[Tex \LARGE \left\(0,-1\right\)] is {{{y=-x-1}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,x+4,-x-1)
circle(0,-1,0.08),
circle(0,-1,0.10),
circle(0,-1,0.12))}}}


Graph of the original equation {{{y=x+4}}} (red) and the perpendicular line {{{y=-x-1}}} (green) through the point *[Tex \LARGE \left\(0,-1\right\)].