write the equation in standard form ofo the parabola containing the following point (0,1) (1,-3) (-1,-9)
To find the standard form equation of a parabola, we need to determine the values of the coefficients a, b, and c.
The general form of the equation of a parabola is:
y = ax^2 + bx + c
Given the points (0,1), (1,-3), and (-1,-9), we can substitute these values into the equation to form a system of equations to solve for a, b, and c.
1 = a(0)^2 + b(0) + c
-3 = a(1)^2 + b(1) + c
-9 = a(-1)^2 + b(-1) + c
Simplifying these equations, we have:
1 = c
-3 = a + b + c
-9 = a + (-b) + c
Since we know c = 1, we can substitute this into the second and third equations:
-3 = a + b + 1
-9 = a - b + 1
Rearranging these equations:
a + b = -4 (equation 1)
a - b = -10 (equation 2)
Adding equation 1 and equation 2, we eliminate b:
2a = -14
a = -7
Substituting the value of a back into equation 1:
-7 + b = -4
b = -4 + 7
b = 3
Therefore, the values of a and b are -7 and 3, respectively. Now we can write the equation of the parabola in standard form:
y = -7x^2 + 3x + 1
To find the equation of a parabola in standard form, you need three points on the parabola. Let's use the given three points (0,1), (1,-3), and (-1,-9).
The standard form of a parabola equation is given by:
y = ax^2 + bx + c
To find the values of a, b, and c, we can substitute the x and y-values of the given points into the equation.
Using the point (0,1):
1 = a(0)^2 + b(0) + c
1 = c
Using the point (1,-3):
-3 = a(1)^2 + b(1) + c
-3 = a + b + c
Using the point (-1,-9):
-9 = a(-1)^2 + b(-1) + c
-9 = a - b + c
Now we have a system of three equations with three unknowns:
1 = c
-3 = a + b + c
-9 = a - b + c
We can substitute the value of c from the first equation into the second and third equations:
-3 = a + b + 1
-9 = a - b + 1
Simplifying these equations, we get:
a + b = -4
a - b = -10
To solve this system, we can add the two equations:
2a = -14
a = -7
Substitute this value back into one of the equations, for example:
-7 - b = -10
b = -3
Now we have the values of a and b. Substituting these values back into the first equation, we have:
1 = (-7)(0)^2 + (-3)(0) + 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, you need to use the general equation of a parabola: y = a*x^2 + b*x + c. We will substitute each of the given points into this equation and create a system of three equations using the coordinates (0,1), (1,-3), and (-1,-9).
1. For the point (0,1):
Substitute x = 0 and y = 1:
1 = a*0^2 + b*0 + c
1 = c
Therefore, we have c = 1.
2. For the point (1,-3):
Substitute x = 1 and y = -3:
-3 = a*1^2 + b*1 + c
-3 = a + b + c
We can use the value of c from the previous equation, which is c = 1, to simplify this equation:
-3 = a + b + 1
Rearrange it to get:
a + b = -4 ... Equation (1)
3. For the point (-1,-9):
Substitute x = -1 and y = -9:
-9 = a*(-1)^2 + b*(-1) + c
-9 = a - b + c
Again, use the value of c, which is 1:
-9 = a - b + 1
Rearrange it to get:
a - b = -10 ... Equation (2)
Now we have a system of two equations with two variables—Equations (1) and (2) above:
a + b = -4 ... Equation (1)
a - b = -10 ... Equation (2)
By solving this system of equations, we can find the values of a and b. Adding Equation (1) and Equation (2) eliminates the b term:
(a + b) + (a - b) = -4 + (-10)
2a = -14
Divide both sides by 2:
a = -7
Substituting the value of a = -7 into Equation (1):
-7 + b = -4
Rearrange it to solve for b:
b = -4 + 7
b = 3
Now that we have the values of a and b, we can write the equation of the parabola in standard form:
y = -7x^2 + 3x + 1
Therefore, the equation of the parabola passing through the points (0,1), (1,-3), and (-1,-9) in standard form is y = -7x^2 + 3x + 1.