Question

For: https://socratic.org/users/jeish-g The points with coordinates (-1,4) and (1,-8) lie on a curve which has the rule of the form `y=aroot(3)(x)+b`. Find the values of `a and b`.?

Answer 1

`a=-6,b=-2`

Explanation:

We have two points for which we know the `x` and `y` coordinates. We can just plug these two points into the equation and solve for `a` and `b`:
`-8=aroot(3)(1)+b`

`4=aroot(3)(-1)+b`

We can subtract these two equations from each other to cancel the `b`'s:
`-8-4=aroot(3)(1)+b-(aroot(3)(-1)+b)`

`-12=a*1+b-(-a+b)`

`-12=a+b+a-b`

`-12=2a`

`a=-12/2=-6`

Now we can plug in `a=-6` and solve for `b`:
`4=-6root(3)(-1)+b`

`4=6+b`

`b=4-6=-2`

So, the final equation is `y=-6root(3)(x)-2`

Answer 2

Reworkded:

`a=-6`
`b=-2`

Explanation:

`color(brown)("Formatting tip - write "color(white)("d")" hash y = a root(3)(x)+b hash ")`
`color(brown)( "giving: ............."color(white)("ddddddddddddd") y=a root(3)(x)+b`

Have a look at: https://socratic.org/help/symbols
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Set point 1 as `P_1->(x_1,y_1)=(-1,4)`
Set point 2 as `P_2->(x_2,y_2)=(1,-8) color(red)( larr" +8 corrected to -8")`

Given:`color(white)("d") y=aroot(3)(x)+b`

`P_1->+4=aroot(3)(-1)+b" "...............Equation(1)" checked"`
`P_2->color(white)(".")color(red)(-8)=aroot(3)(+1)+b" "...............Equation(2)" corrected"`

`color(blue)("Determine the value of "b)`

Note that:
`(-1)xx(-1)xx(-1)color(white)("d")=color(white)("d")(-1)" so "root(3)(-1)=-1`

Note that `root(3)(+1)=1`

So we have:

`color(white)("d")+4=-a+b" "....................Equation(1_a)`
`ul(color(white)("d")-8=+a+b)" "....................Equation(2_a)`
`color(green)(color(white)("d")-4=color(white)(".dd")0+2b) larr" "Eqn(1_a)+Eqn(2_a)`

Divide both sides by `color(red)(2)`

`color(green)(-4/color(red)(2)=(2b)/color(red)(2))`

`-2=bcolor(white)("d")` more formally presented as: `b=-2`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the value of "a)`

Choosing `Equation(2_a)` substitute for `b` where `color(red)(b=-2)`

`color(green)(-8=+a+color(red)(b) color(white)("ddd") -> color(white)("ddd")-8=a+(color(red)(-2)))`

Add 2 to both sides

`color(green)(color(white)("dddddddddddddd")->color(white)("ddd")-6=a)`

More formally `a=-6`

color(white)("d")