SOLUTION: What is the equation of the parabola passing through the points A(1,1) , B(2,2) , C(-1,5) and axis parallel to the y-axis

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: What is the equation of the parabola passing through the points A(1,1) , B(2,2) , C(-1,5) and axis parallel to the y-axis      Log On


   

Question 1060914: What is the equation of the parabola passing through the points A(1,1) , B(2,2) , C(-1,5) and axis parallel to the y-axis
Found 2 solutions by KMST, MathTherapy:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The equation would be of the form y=ax%5E2%2Bbx%2Bc .
Substituting the coordinates of A%281%2C1%29 , we get
1=a%2A1%5E2%2Bb%2A1%2Bc --> a%2Bb%2Bc=1 .
Substituting the coordinates of B%282%2C2%29 , we get
2=a%2A2%5E2%2Bb%2A2%2Bc --> 4a%2B2b%2Bc=2 .
Substituting the coordinates of C%28-1%2C5%29 , we get
5=a%2A%28-1%29%5E2%2Bb%2A%28-1%29%2Bc --> a-b%2Bc=5 .
The 3 equations above give us the system
system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc=2%2Ca-b%2Bc=5%29
Subtracting the first equation from each ot the other two equations, we get the equivalent system
system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc-a-b-c=2-1%2Ca-b%2Bc-a-b-c=5-1%29--->system%28a%2Bb%2Bc=1%2C3a%2Bb=1%2C-2b=4%29-->system%28a%2Bb%2Bc=1%2C3a%2Bb=1%2Cb=4%2F%28-2%29%29-->system%28a%2Bb%2Bc=1%2C3a%2Bb=1%2Chighlight%28b=-2%29%29 .
Substituting -2 for b in 3a%2Bb=1 we get
3a-2=1 --> 3a=1%2B2 --> 3a=3 --> a=3%2F3 --> highlight%28a=1%29 .
Then substituting -2 for b and 1 for a in a%2Bb%2Bc=1 , we get
1-2%2Bc=1 --> -1%2Bc=1 --> c=1%2B1 --> highlight%28c=2%29 .
So, the equation of the parabola is
highlight%28y=x%5E2-2x%2B2%29 .

We could also write it as
y=x%5E2-2x%2B1%2B1 --> y=%28x%5E2-2x%2B1%29%2B1 --> highlight%28y=%28x-1%29%5E2%2B1%29 .

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
What is the equation of the parabola passing through the points A(1,1) , B(2,2) , C(-1,5) and axis parallel to the y-axis
Standard form equation of a vertical parabola:  

A (1, 1)
a%281%29%5E2+%2B+b%281%29+%2B+c+=+1 ------ Substituting A (1, 1)
a + b + c = 1 --------- eq (i)

B (2, 2)
a%282%29%5E2+%2B+b%282%29+%2B+c+=+2 ------ Substituting B (2, 2)
4a + 2b + c = 2 ------- eq (ii)

C (- 1, 5)
a%28-+1%29%5E2+%2B+b%28-+1%29+%2B+c+=+5 --- Substituting C (- 1, 5)
a - b + c = 5 --------- eq (iii)

2b = - 4 ------- Subtracting eq (iii) from eq (i)
highlight%28matrix%281%2C5%2C+b%2C+%22=%22%2C+%28-+4%29%2F2%2C+or%2C+-+2%29%29

a - 2 + c = 1 ------ Substituting - 2 for b in eq (i) 
a + c = 3 ---------- eq (iv)

4a - 4 + c = 2 ----- Substituting - 2 for b in eq (ii)  
4a + c = 6 --------- eq (v)
3a = 3 ------------- Subtracting eq (iv) from eq (v)
highlight%28matrix%281%2C5%2C+a%2C+%22=%22%2C+3%2F3%2C+or%2C+1%29%29

1 - 2 + c = 1 ------ Substituting 1 for a, and - 2 for b in eq (i)
- 1 + c = 1
highlight%28matrix%281%2C5%2C+c%2C%22=%22%2C+1+%2B+1%2C+or%2C+2%29%29

y+=+ax%5E2+%2B+bx+%2B+c 
highlight_green%28y+=+x%5E2+-+2x+%2B+2%29 ----- Substituting 1 for a, - 2 for b, and 2 for c