Write the equation, in standard form, of the parabola containing the following points: (0, 1), (1, -3), (-1, -9).
You must set up a system of three equations in your work in order to get full credit for this question.
A. y=-7x²+3x+1
B. y=-3x²+7x+1
C. y=3x²-5x-1
D. y=-x²+5x+3
To find the equation of the parabola containing the given points, we can use the standard form of a parabolic equation which is y = ax² + bx + c.
Now, we'll substitute each point into the equation to form a system of equations.
For the point (0, 1):
1 = a(0)² + b(0) + c
1 = c
For the point (1, -3):
-3 = a(1)² + b(1) + c
-3 = a + b + c
For the point (-1, -9):
-9 = a(-1)² + b(-1) + c
-9 = a - b + c
Now we have the following system of equations:
1 = c
-3 = a + b + c
-9 = a - b + c
We can solve this system of equations to find the values of a, b, and c.
From equation 1, c = 1.
Substitute c = 1 into equations 2 and 3:
-3 = a + b + 1
-9 = a - b + 1
Rearranging equation 2, we get:
b = -a - 4
Substituting b = -a - 4 into equation 3:
-9 = a - (-a - 4) + 1
-9 = a + a + 4 + 1
-14 = 2a
a = -7
Substituting a = -7 into b = -a - 4:
b = -(-7) - 4
b = 7 - 4
b = 3
So, the values of a, b, and c are -7, 3, and 1, respectively.
Therefore, the equation of the parabola is:
y = -7x² + 3x + 1
Hence, the correct option is A.