SOLUTION: Given the graph with the points: (1,-2)(5,0)(10,1),write the radical function for f(x)

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Question 1091208: Given the graph with the points: (1,-2)(5,0)(10,1),write the radical function for f(x)
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
the radical function would be:
f%28x%29=a%2Asqrt%28x-b%29%2Bc
use the points
(1,-2),
(5,0),
(10,1) to find a,b, and c
f%28x%29=a%2Asqrt%28x-b%29%2Bc...........(1,-2)
-2=a%2Asqrt%281-b%29%2Bc....solve for c

c=-2-a%2Asqrt%281-b%29.......eq.1
f%28x%29=a%2Asqrt%28x-b%29%2Bc..........(5,0)
0=a%2Asqrt%285-b%29%2Bc..........solve for c
c=-a%2Asqrt%285-b%29.......eq.2

f%28x%29=a%2Asqrt%28x-b%29%2Bc..........(10,1)
1=a%2Asqrt%2810-b%29%2Bc..........solve for c
c=1-a%2Asqrt%2810-b%29.......eq.3
from eq.1 and eq.2 we have:
-2-a%2Asqrt%281-b%29=-a%2Asqrt%285-b%29.....both sides multiply by -1
2%2Ba%2Asqrt%281-b%29=a%2Asqrt%285-b%29............square both sides
%282%2Ba%2Asqrt%281-b%29%29%5E2=%28a%2Asqrt%285-b%29%29%5E2
4%2B2a%2Asqrt%281-b%29%2Ba%5E2%281-b%29=a%5E2%285-b%29
2a%2Asqrt%281-b%29=a%5E2%285-b%29-a%5E2%281-b%29-4
2a%2Asqrt%281-b%29=5a%5E2-a%5E2b-a%5E2%2Ba%5E2b-4
2a%2Asqrt%281-b%29=4a%5E2-4
sqrt%281-b%29=4a%5E2%2F2a-4%2F2a
sqrt%281-b%29=2a-2%2Fa
sqrt%281+-+b%29+=+%282+%28a+-+1%29+%28a+%2B+1%29%29%2Fa.......square both sides
1+-+b+=+%28%282+%28a+-+1%29+%28a+%2B+1%29%29%2Fa%29%5E2....solve for b
b=1+-+%284+%28a+-+1%29%5E2+%28a+%2B+1%29%5E2%29%2Fa%5E2..............eq.b
from eq.2 and eq.3 we have:
1-a%2Asqrt%2810-b%29=-a%2Asqrt%285-b%29
1-asqrt%2810-b%29%2Basqrt%285-b%29=0
1%2Ba%28sqrt%285-b%29-sqrt%2810-b%29%29=0..........solve for b
b+=+%28-25+a%5E4+%2B+30+a%5E2+-+1%29%2F%284+a%5E2%29...........eq.b1
from .eq.b and .eq.b1:
......solve for a

to make it short, you will have
highlight%28a=1%29 or highlight%28a=-1%29

go to b=1+-+%284+%28a+-+1%29%5E2+%28a+%2B+1%29%5E2%29%2Fa%5E2..............eq.b, plug in a=1
b=1+-+%284+%281+-+1%29%5E2+%281%2B+1%29%5E2%29%2F1%5E2
b=1+-+%284+%2A0+%2A4%29%2F1
highlight%28b=1+%29
b=1+-+%284+%28a+-+1%29%5E2+%28a+%2B+1%29%5E2%29%2Fa%5E2..............eq.b, plug in a=-1
b=1+-+%284+%28-1+-+1%29%5E2+%28-1%2B+1%29%5E2%29%2F%28-1%29%5E2
b=1+-+%284+%2A4+%2A0%29%2F1
highlight%28b=1+%29; so, there is one double solution for b


now find c:
c=-a%2Asqrt%285-b%29.......eq.2...............highlight%28b=1%29 and highlight%28a=1%29
c=-1%2Asqrt%285-1%29
c=-sqrt%284%29
highlight%28c=-2%29
c=-a%2Asqrt%285-b%29.......eq.2...............highlight%28b=1%29 and highlight%28a=-1%29
c=-%28-1%29%2Asqrt%285-1%29
c=1sqrt%284%29
highlight%28c=2%29
your solutions are:
a=1 , b=1, c=-2 or
a=-1 , b=1, c=2
the radical function would be:
f%28x%29=a%2Asqrt%28x-b%29%2Bc .....if a=1 , b=1, c=-2
highlight%28f%28x%29=sqrt%28x-1%29-2%29
or
f%28x%29=a%2Asqrt%28x-b%29%2Bc .....if a=-1 , b=1, c=2
f%28x%29=-sqrt%28x-1%29%2B2->
since given point (1,-2), the value of x=1 will give us sqrt%281-1%29=sqrt%280%29 and y=2
so, disregard f%28x%29=-sqrt%28x-1%29%2B2