What is the equation of a parabola containing the points (0,0),(1,-2)and (-1,-4)?
y = ax^2 + bx + c
0 = a(0^2) + b(0) + c ... 0 = 0 + 0 + c
-2 = a(1^2) + b(1) + c ... -2 = a + b + c
-4 = a(-1^2) + b(-1) + c ... -4 = a - b + c
solve the system for a, b, and c
let the equation be
y = ax^2 + bx + c
since (0,0) is a point on it
0 = 0+ 0 + c, so c = 0, and we can go with
y = ax^2+ bx
If (1, -2) lies on it ---> -2 = a+b
if (-1,-4) lies on it ---> -4 = a - 4b
subtract them:
2 = 5b
b = 2/5
sub back in to get a and write your equation
made an error, go with Damon and Scott
To find the equation of a parabola, we need to use the general form of a quadratic equation, which is y = ax^2 + bx + c. Since we are given three points on the parabola, we can substitute the x and y values of each point into the equation to form a system of three equations.
Let's start by substituting the first point (0,0) into the equation:
0 = a(0)^2 + b(0) + c
0 = 0 + 0 + c
0 = c
So, we now know that c = 0.
Next, we substitute the second point (1, -2) into the equation:
-2 = a(1)^2 + b(1) + c
-2 = a + b
Lastly, we substitute the third point (-1, -4) into the equation:
-4 = a(-1)^2 + b(-1) + c
-4 = a - b
Now, we have a system of two equations:
-2 = a + b
-4 = a - b
To solve this system, we can add the two equations together:
-2 + -4 = (a + b) + (a - b)
-6 = 2a
Dividing by 2, we find that a = -3.
Now, substitute the value of a (-3) into one of the original equations to find the value of b:
-4 = (-3) - b
-4 + 3 = -b
-1 = -b
b = 1
Finally, we have determined that a = -3, b = 1, and c = 0. The equation of the parabola is therefore:
y = -3x^2 + x + 0
Simplifying, we get the equation:
y = -3x^2 + x
y =a x^2 + b x +c
0 = a*0 + b*0 + c so c = 0
y = a x^2 + b x
-2 = a + b
y = a x^2 + b x
-4 = a - b
add the two
-6 = 2 a
a = -3
then b = 1
so
y = -3 x^2 + x