I have tried several times to solve this. I have 3 points (0,-5), (3, -6), (-3,-7). I'm trying to get the quadratic equation. I keep getting y=(-1/6)x^2+ (1/6)x +0. When i plug in other numbers to check it's not the parabola facing down like my image. Please help!
let y = ax^2 + bx + c
for (0,-5)
-5 = 0 + 0 + c, so c = -5
for (3,-6)
-6 = 9a + 3b - 5
9a + 3b = -1 , #1
for (-3,-7)
9a -3b -5 = -7
9a - 3b = -2 , #2
add #1 and #2
18a = -2
a = -1/+
sub into #1
9(-1/6) + 3b = -1
-3/2 + 3b = -1
-3 + 6b = -2
6b = 1
b = 1/6
so f(x) = (-1/6)x^2 + (1/6)x - 5
All 3 points satisfy this equation, you have no constant in your equation
You seem to have made the error in finding the constant (which was the easiest part)
THANK YOU! THANK YOU! THANK YOU!!!
To find the quadratic equation that represents a parabola passing through three given points, you can use the process of interpolation. The general form of a quadratic equation is y = ax^2 + bx + c.
Let's start by substituting the coordinates (0,-5) into the equation: -5 = a(0)^2 + b(0) + c. This simplifies to c = -5.
Next, substitute the coordinates (3,-6) into the equation: -6 = a(3)^2 + b(3) - 5. This simplifies to 9a + 3b = -1.
Finally, substitute the coordinates (-3,-7) into the equation: -7 = a(-3)^2 + b(-3) - 5. This simplifies to 9a - 3b = -2.
Now you have a system of equations:
9a + 3b = -1
9a - 3b = -2
To solve this system, you can eliminate b by adding the two equations together. The 3b and -3b terms will cancel out, leaving you with 18a = -3. Dividing both sides by 18 gives you a = -1/6.
Now you can substitute this value of a back into one of the equations to solve for b. Let's use the first equation: 9(-1/6) + 3b = -1. Simplifying this gives -3/2 + 3b = -1. Adding 3/2 to both sides gives 3b = 1/2, and dividing by 3 gives b = 1/6.
So now you know that a = -1/6 and b = 1/6. And we already found c = -5.
Therefore, the quadratic equation is y = (-1/6)x^2 + (1/6)x - 5.
If you are getting a different equation, there may be an error in your arithmetic or calculation. Double-check your calculations and make sure you apply the correct signs when substituting the given points.