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. (1 point) show all of your work
The equation of a parabola in standard form is given by: y = ax^2 + bx + c.
We can set up a system of three equations using the given points:
1) (0, 1): 1 = a(0)^2 + b(0) + c, which simplifies to: 1 = c.
2) (1, -3): -3 = a(1)^2 + b(1) + c, which simplifies to: -3 = a + b + 1.
3) (-1, -9): -9 = a(-1)^2 + b(-1) + c, which simplifies to: -9 = a - b + 1.
From equation 1, we know that c = 1. Substituting this value into equations 2 and 3, we get:
-3 = a + b + 1,
-9 = a - b + 1.
Subtracting the second equation from the first, we eliminate a: 6 = 2b, which simplifies to: b = 3.
Substituting this value back into equation 2, we get: -3 = a + 3 + 1,
-3 = a + 4,
a = -7.
Therefore, the equation of the parabola is: y = -7x^2 + 3x + 1.
To find the equation of the parabola, we can start by assuming the equation is of the form: y = ax^2 + bx + c.
We can substitute the given points into this equation and 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 three equations:
1 = c (Equation 1)
-3 = a + b + c (Equation 2)
-9 = a - b + c (Equation 3)
To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of substitution:
From Equation 1, we have c = 1. Substitute this value into Equations 2 and 3:
-3 = a + b + 1 (Equation 2')
-9 = a - b + 1 (Equation 3')
Rearranging Equation 2' and Equation 3', we get:
a + b = -4 (Equation 4)
a - b = -10 (Equation 5)
Now, we have a system of two equations (Equation 4 and Equation 5) with two variables (a and b). We can solve this system to find the values of a and b.
Adding Equation 4 and Equation 5, we eliminate the variable b:
2a = -14
a = -14/2
a = -7
Substitute the value of a = -7 into Equation 4:
-7 + b = -4
b = -4 + 7
b = 3
Now, we have the values of a = -7 and b = 3.
Substitute these values into Equation 1 to find the value of c:
1 = c
c = 1
Therefore, the equation of the parabola in standard form is:
y = -7x^2 + 3x + 1
To find the equation of a parabola in standard form, we need to use the general equation of a parabola: y = ax^2 + bx + c.
We will set up a system of three equations using the given points: (0, 1), (1, -3), and (-1, -9).
For point (0, 1):
1 = a(0)^2 + b(0) + c
1 = c -----> Equation 1
For point (1, -3):
-3 = a(1)^2 + b(1) + c
-3 = a + b + c -----> Equation 2
For point (-1, -9):
-9 = a(-1)^2 + b(-1) + c
-9 = a - b + c -----> Equation 3
Now we have a system of three equations:
1 = c -----> Equation 1
-3 = a + b + c -----> Equation 2
-9 = a - b + c -----> Equation 3
To solve this system of equations, we can use substitution or elimination. Let's solve it using elimination:
Adding Equation 2 and Equation 3, we get:
(-3) + (-9) = (a + b + c) + (a - b + c)
-12 = 2a + 2c
Dividing both sides of the equation by 2, we have:
-6 = a + c -----> Equation 4
Substituting Equation 4 into Equation 2, we get:
-3 = (a + c) + b
-3 = -6 + b
b = -3 + 6
b = 3
Substituting b = 3 into Equation 3, we have:
-9 = a - 3 + c
-9 = a + c - 3
Now, let's solve for a and c. Adding 3 to both sides of the equation, we get:
-6 = a + c
Since we already have a = -6 - c from Equation 4, we can substitute it into the previous equation:
-6 = (-6 - c) + c
Simplifying, we get:
-6 = -6
This equation is true, indicating that a and c can have any value. Let's assign a value to c to find the values of a:
Let c = 0, then we have:
a = -6 - c
a = -6 - 0
a = -6
The values of a, b, and c are -6, 3, and 0, respectively.
Therefore, the equation of the parabola in standard form is:
y = -6x^2 + 3x + 0 (or simply y = -6x^2 + 3x).