what is the equation in standard form of parabola that contains the following points
(-2-20) (0,-4) (4,-20)
A y=-2.5^2+5x
B -x^2+4x-4
C y=-2x^2+4x-4
D y=-2.25x^2+4.5x-2
C y=-2x^2+4x-4
To find the equation of a parabola in standard form, we need three points on the parabola. Let's use the given points (-2,-20), (0,-4), and (4,-20) to determine the equation.
Step 1: Write the equation: y = ax^2 + bx + c.
Step 2: Substitute the coordinates of each point into the equation to create three equations.
(-2, -20): -20 = a(-2)^2 + b(-2) + c
(0, -4): -4 = a(0)^2 + b(0) + c
(4, -20): -20 = a(4)^2 + b(4) + c
Step 3: Simplify the equations.
-20 = 4a - 2b + c
-4 = c
-20 = 16a + 4b + c
Step 4: Substitute c = -4 into the other equations.
-20 = 4a - 2b - 4
-20 = 16a + 4b - 4
Step 5: Simplify further.
-16 = 4a - 2b
-16 = 16a + 4b
Step 6: Solve the system of equations.
Multiplying the first equation by 2 and adding the second equation, we get:
-32 = 8a - 4b
-16 = 16a + 4b
-16 + (-32) = 16a + 4b + 8a - 4b
-48 = 24a
a = -2
Substituting the value of a into one of the previous equations:
-16 = 16(-2) + 4b
-16 = -32 + 4b
4b = 16
b = 4
Step 7: Substitute the values of a, b, and c back into the equation y = ax^2 + bx + c.
So, the equation in standard form is:
y = -2x^2 + 4x - 4
Therefore, the correct answer is C: y = -2x^2 + 4x - 4.
To find the equation of the parabola in standard form that contains the given points, you can follow these steps:
Step 1: Plug in the coordinates of each point into the general equation of a parabola, which is y = ax^2 + bx + c.
Point 1 (-2, -20):
-20 = a(-2)^2 + b(-2) + c
Point 2 (0, -4):
-4 = a(0)^2 + b(0) + c
Point 3 (4, -20):
-20 = a(4)^2 + b(4) + c
Simplifying each equation, we have:
1. 4a - 2b + c = -20
2. c = -4
3. 16a + 4b + c = -20
Step 2: Substitute the value of c from equation 2 into equations 1 and 3:
1. 4a - 2b - 4 = -20
2. c = -4
3. 16a + 4b - 4 = -20
Step 3: Simplify equations 1 and 3 further:
1. 4a - 2b = -16
3. 16a + 4b = -16
Step 4:
To eliminate the b term, you can multiply equation 1 by 2 and equation 3 by 1:
2(4a - 2b) = 2(-16)
16a + 4b = -16
Simplifying:
8a - 4b = -32
16a + 4b = -16
The b term cancels out, leaving you with:
24a = -48
Dividing both sides by 24:
a = -2
Step 5: Substitute the value of a back into equation 1 to solve for b:
4(-2) - 2b = -16
-8 - 2b = -16
-2b = -16 + 8
-2b = -8
b = 4
Step 6: Plug the values of a, b, and c back into the general equation y = ax^2 + bx + c:
y = (-2)x^2 + 4x - 4
Therefore, the correct answer is option C: y = -2x^2 + 4x - 4.