Write the equation, in standard form, of the parabola containing the following points: (0, 1), (1, -3), (-1, -9).
To find the equation of a parabola, let's assume it has the form y = ax^2 + bx + c.
By substituting the three given points into this equation, we can form three equations:
For point (0, 1): 1 = a(0)^2 + b(0) + c
1 = c
For point (1, -3): -3 = a(1)^2 + b(1) + c
-3 = a + b + c
For point (-1, -9): -9 = a(-1)^2 + b(-1) + c
-9 = a - b + c
Now, we have a system of three equations with three variables (a, b, and c):
1 = c
-3 = a + b + c
-9 = a - b + c
From the first equation, we know that c = 1. Substituting this into the other equations gives us:
-3 = a + b + 1
-9 = a - b + 1
Rearranging these equations:
a + b = -4
a - b = -10
Adding the two equations together, we get:
2a = -14
a = -7
Substituting this value of a into one of the equations, we can solve for b:
-7 - b = -10
b = 3
Now that we have the values of a, b, and c, we can write the equation in standard form:
y = ax^2 + bx + c
y = -7x^2 + 3x + 1