What is the equation, in standard form, of a parabola that contains the following points?
(-2, -20), (0, -4), (4, -20)
a x^2 + b x + c
(1) a(4) + b(-2) + c = -20
(2) a(0) + b(0) + c = -4
(3) a(16) + b(4) + c = -20
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eqn (2) tells us that c = -4
so c = -20
(1) a(4) + b(-2) = -16
(2) a(0) + b(0) = 0
(3) a(16) + b(4) = -16
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(1) -8 a + 4 b = 32
(3) 16 a + 4 b = -16
---------------------- subtract
-24 a = 48
a = - 2
you can take it from there
so c = -20
I do not know why I said that, typo while copying and pasting
How do I find b and c?
To find the equation of a parabola in standard form that passes through the given points, we can first determine the general form of the equation for a parabola.
The general form of a parabolic equation is y = ax^2 + bx + c.
To find the specific equation, we need to substitute the coordinates of the given points into this form to create a system of equations.
Let's start with the point (-2, -20):
-20 = a(-2)^2 + b(-2) + c
Next, substitute the coordinates of the point (0, -4):
-4 = a(0)^2 + b(0) + c
And, finally, use the last given point (4, -20):
-20 = a(4)^2 + b(4) + c
Now we have a system of three equations that we can solve to find the values of a, b, and c. So, we have:
1) 4a - 2b + c = -20
2) c = -4
3) 16a + 4b + c = -20
From equation 2, we get the value of c = -4.
Substituting -4 for c in equations 1 and 3, we have:
1) 4a - 2b - 4 = -20
2) -4 = -4
3) 16a + 4b - 4 = -20
Simplifying equations 1 and 3 further, we get:
1) 4a - 2b = -16
3) 16a + 4b = -16
Now, we can either solve this system of equations using substitution, elimination, or matrix methods. Let's use elimination.
Multiplying equation 1 by 2 and subtracting it from equation 3, we eliminate b:
(16a + 4b) - (8a - 4b) = -16 - (-16)
8a + 8b = 0
8a = -8
a = -1
Now that we know a = -1, we can substitute this value back into equation 1 to solve for b:
4(-1) - 2b = -16
-4 - 2b = -16
-2b = -16 + 4
-2b = -12
b = 6
We now have a = -1, b = 6, and c = -4.
So, the equation, in standard form, of the parabola that passes through the given points (-2, -20), (0, -4), and (4, -20) is:
y = -1x^2 + 6x - 4