What is the equation, in standard form, of a parabola that contains the following points?
(-2,-20), (0, -4), (4, -20)
(1 point)
Responses
y equals negative 2.5 x squared plus 5 x
Image with alt text: y equals negative 2.5 x squared plus 5 x
y equals negative x squared plus 4 x minus 4
Image with alt text: y equals negative x squared plus 4 x minus 4
y equals negative 2 x squared plus 4 x minus 4
Image with alt text: y equals negative 2 x squared plus 4 x minus 4
y equals negative 2.25 x squared plus 4.5 x minus 2
Image with alt text: y equals negative 2.25 x squared plus 4.5 x minus 2
The correct equation, in standard form, of the parabola that contains the given points is:
y equals negative 2 x squared plus 4 x minus 4
To find the equation of a parabola in standard form, we need to use the equation:
y = ax^2 + bx + c
To determine the values of a, b, and c, we can use the given points (-2, -20), (0, -4), and (4, -20). We will substitute the x and y coordinates of each point into the equation and solve the resulting system of equations.
Substituting (-2, -20):
-20 = a(-2)^2 + b(-2) + c
-20 = 4a - 2b + c (equation 1)
Substituting (0, -4):
-4 = a(0)^2 + b(0) + c
-4 = c (equation 2)
Substituting (4, -20):
-20 = a(4)^2 + b(4) + c
-20 = 16a + 4b + c (equation 3)
Using equation 2, we find that c = -4. Now we can substitute this value into equations 1 and 3.
-20 = 4a - 2b - 4
-20 = 4a - 2b - 4
-20 - 4 = 4a - 2b
-24 = 4a - 2b (equation 4)
-20 = 16a + 4b - 4
-20 = 16a + 4b - 4
-20 + 4 = 16a + 4b
-16 = 16a + 4b (equation 5)
Simplifying equations 4 and 5, we get:
-12 = 2a - b
-4 = 4a + b
We can solve this system of equations by adding them together:
-12 + (-4) = 2a - b + 4a + b
-16 = 6a
a = -16/6 = -8/3
Now we can substitute the value of a into one of the original equations, let's use equation 4:
-24 = 4(-8/3) - 2b
-24 = -32/3 - 2b
Multiplying by 3 to eliminate the fraction:
-72 = -32 - 6b
-72 + 32 = -6b
-40 = -6b
b = -40/-6 = 20/3
Finally, we can substitute the values of a, b, and c back into the general equation of a parabola:
y = (-8/3)x^2 + (20/3)x + (-4)
Multiplying through by 3 to eliminate fractions:
3y = -8x^2 + 20x - 12
Therefore, the equation of the parabola in standard form is:
-8x^2 + 20x - 3y - 12 = 0
To find the equation in standard form (y = ax^2 + bx + c) of a parabola that contains the given points (-2,-20), (0, -4), and (4, -20), we can use the method of substitution.
Let's start by substituting the coordinates of the points into the equation y = ax^2 + bx + c.
For the point (-2, -20):
-20 = a(-2)^2 + b(-2) + c
-20 = 4a - 2b + c ------(1)
For the point (0,-4):
-4 = a(0)^2 + b(0) + c
-4 = c ------(2)
For the point (4,-20):
-20 = a(4)^2 + b(4) + c
-20 = 16a + 4b + c ------(3)
Now, we have a system of three equations with three variables (a, b, and c). We can solve this system to find the values of a, b, and c.
Using equations (2) and (3), we can substitute c = -4 into equation (3):
-20 = 16a + 4b - 4
Rearranging the equation:
16a + 4b = -16 ------(4)
Now, we can substitute c = -4 into equation (1) and simplify:
-20 = 4a - 2b - 4
-20 = 4a - 2b - 4
Simplifying further:
4a - 2b = -16 ------(5)
Now we have a system of two equations with two variables (equations 4 and 5). We can solve this system to find the values of a and b.
Multiplying equation (5) by 2, we can eliminate the b terms:
8a - 4b = -32 ------(6)
Adding equations (4) and (6) together:
16a + 4b + 8a - 4b = -16 - 32
24a = -48
Dividing both sides by 24:
a = -2
Substitute this value for a into equation (4):
16(-2) + 4b = -16
Simplifying:
-32 + 4b = -16
4b = 16
b = 4
Now, we have found the values of a and b. Substituting these values into equation (2), we can find c:
-4 = c
Putting it all together, the equation in standard form of the parabola that contains the points (-2,-20), (0, -4), and (4, -20) is:
y = -2x^2 + 4x - 4