The parabola with vertex (4,-4) and passing through the point (8,6) is given be the equation y=ax^2+Bx+C?

Question

need to solve for a b and c. How do i do this

Answer 1

you could start at...
y + 4 = a(x-4)^2

then since you have the point... (8,6)
you could substitute that to get a.

then rearrange and you have your values...





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Answer 2

The axis of symmetry goes through the vertex. you can use that to find a third point.
If y =ax^2+Bx+C, is the equation you will be using you will need three points to get solutions for a, B and C.
The axis of symmetry is x=4 four units to the right of that you have a point on the parabola. Reflecting that point on the axis there must be a point at (0,6)
The equations are by substituting the points:
6 = 0 + 0 + c
-4 = 16a + 4B +C
6 = 64a + 8B + C

since we already know C=6
we have:
-10 = 16a + 4B
0 = 64a + 8B

-20 = -32 a
a = 5/8

8B = - 40
B = -5

Your parabola's equation is:
y = 5/8x^2 -5x + 6

Answer 3

You have 3 unknowns and need 3 equations. Your first 2 equations can come from pluging in the information given:

-4 = a*(4^2) + b*(4) + c = 14a + 4b + c
6 = a(8^2) + b*8 + c = 64a + 8b + c

Your 3rd equation can come from recognizing that at the vertex of a parabola, the derivative is 0.

dy/dx = 0 = 2ax + b

Since this occurs when x = 4, then

0 = 2a*4 + b = 8a + b.

Now you just have to solve for a, b, & c using these three equations:

0 = 8a + b
-4 = 16a + 4b + c
6 = 64a + 8b +c

Hope this helps

Answer 4

Since the vertex is at (4, -4), the equation can be written...

(y - -4) = Q(x - 4)^2

Then we can plug in the other point we know to find a value for Q:

(6 + 4) = Q(8 - 4)^2
10 = Q 16
Q = 5/8

So...
(y+4) = 5/8 (x - 4)^2

Now just put that into the form you want:
y+4 = 5/8 (x^2 - 8x +16)
y+4 = 5/8x^2 - 5x + 10
y = 5/8x^2 - 5x + 6

Check to make sure it works for the two points we know:
Does it work for (8, 6) ?
6 = 5/8 (64) - 5(8) + 6
6 = 40 - 40 + 6
6 = 6 Good.

Does it work for (4, -4) ?
-4 = 5/8 (16) - 5(4) + 6
-4 = 10 - 20 + 6
-4 = -4 Good.

So...
a = 5/8 = 0.625
b = -5
c = 6

Answer 5

(x-4)^2 +2p(y+4) = 0
To find p substitute x=8 and y = 6 in above equation
(8-4)^2 +2p(6+4) = 0
16 +20p = 0 --> p = -16/20 = -.8
So equation is (x-4)^2 -1.6(y+4) =0
x^2-8x +16 -1.6y -6.4 = 0
x^2 -8x +9.6 =1.6y
y = .625x^2 - 5x + 6
So a = .625, b= -5, and c = 6