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) Responses y=−7x2+3x+1 y is equal to negative 7 x squared plus 3 x plus 1 y=−3x2+7x+1 y is equal to negative 3 x squared plus 7 x plus 1 y=−x2+5x+3
Wrong again, Bot! Do you never check your solutions?
y = -7x^2 - 3x + 1 contains the points
(-1,-3), (0,1), (1,-9)
The correct equations to solve are
a-b+c = -9
c = 1
a+b+c = -3
which gives the answer of
y = -7x^2 + 3x + 1
Apologies for the mistake.
To find the equation of the parabola, we set up the following system of equations using the three given points:
(-1, -3): -3 = a(-1)^2 + b(-1) + c
-3 = a - b + c ----(1)
(0, 1): 1 = a(0)^2 + b(0) + c
1 = c ----(2)
(1, -9): -9 = a(1)^2 + b(1) + c
-9 = a + b + c ----(3)
From equation (2), we find that c = 1. Substituting this value back into equations (1) and (3):
-3 = a - b + 1
-9 = a + b + 1
Rearranging these equations:
a - b = -4 ----(4)
a + b = -10 ----(5)
To solve for a and b, we can add equations (4) and (5) to eliminate b:
2a = -14
a = -7
Substituting this value back into equation (4):
-7 - b = -4
b = -3
Thus, the equation of the parabola in standard form is:
y = -7x^2 + 3x + 1
To find the equation of the parabola, we can use the standard form of a quadratic equation, which is given by:
y = ax^2 + bx + c, where a, b, and c are constants.
We can set up a system of three equations using the three given points.
First, substitute the coordinates (0, 1) into the equation:
1 = a(0)^2 + b(0) + c
1 = c -- Equation 1
Next, substitute the coordinates (1, -3) into the equation:
-3 = a(1)^2 + b(1) + c
-3 = a + b + c -- Equation 2
Finally, substitute the coordinates (-1, -9) into the equation:
-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, subtract Equation 3 from Equation 2 to eliminate the variable "c":
-3 - (-9) = (a + b + c) - (a - b + c)
6 = 2b
Simplifying, we have:
2b = 6
b = 3
Now, substitute the value of b into Equation 2 to find the value of a:
-3 = a + 3 + c
a + c = -6
Substitute the values obtained for a and b into Equation 1:
1 = c
Therefore, a = -6 - c, and substituting this into Equation 1 gives:
1 = c
Now, we have:
a = -6 - c
b = 3
c = 1
Substituting these values into the standard form equation, we get:
y = a*x^2 + b*x + c
y = (-6 - c) * x^2 + 3 * x + 1
Simplifying further, we get the equation:
y = -6x^2 - cx + 3x + 1
So, the equation of the parabola in standard form is:
y = -6x^2 + (3 - c) * x + 1
To find the equation of a parabola in standard form, we need three points on the parabola. Let's consider the given points: (0, 1), (1, -3), and (-1, -9).
The standard form of a quadratic equation is y = ax^2 + bx + c.
We can set up a system of three equations using the coordinates of the points and solve for the coefficients a, b, and c.
1) Using the point (0, 1):
1 = a(0)^2 + b(0) + c
1 = c
2) Using the point (1, -3):
-3 = a(1)^2 + b(1) + c
-3 = a + b + c
3) Using the point (-1, -9):
-9 = a(-1)^2 + b(-1) + c
-9 = a - b + c
Now we have a system of three equations:
Equation 1: 1 = c
Equation 2: -3 = a + b + c
Equation 3: -9 = a - b + c
We can solve this system of equations to find the values of a, b, and c.
First, substitute Equation 1 into Equations 2 and 3:
Equation 2: -3 = a + b + 1
Equation 3: -9 = a - b + 1
Simplifying these equations:
Equation 2: a + b = -4
Equation 3: a - b = -10
Add Equations 2 and 3:
2a = -14
a = -7
Substitute the value of a back into Equation 2:
-7 + b = -4
b = -4 + 7
b = 3
Now we have the values of a = -7, b = 3, and c = 1. These values can be plugged into the standard form equation:
y = -7x^2 + 3x + 1
Therefore, the equation of the parabola in standard form, containing the points (0, 1), (1, -3), and (-1, -9), is:
y = -7x^2 + 3x + 1.
To find the equation of the parabola, we can use the standard form equation for a parabola:
y = ax^2 + bx + c
We need to find the values of a, b, and c. We can set up a system of equations using the three given points:
(0, 1): 1 = a(0)^2 + b(0) + c
1 = c
(1, -3): -3 = a(1)^2 + b(1) + c
-3 = a + b + c
(-1, -9): -9 = a(-1)^2 + b(-1) + c
-9 = a - b + c
Using the first equation, we find that c = 1. Substituting this into the second and third equations, we get:
-3 = a + b + 1
-9 = a - b + 1
Rearranging these equations:
a + b = -4
a - b = -10
Adding the two equations together, we eliminate b:
2a = -14
a = -7
Substituting this value back into one of the equations:
-7 - b = -10
b = -3
Thus, the equation of the parabola in standard form is:
y = -7x^2 - 3x + 1