a parabola with turning point at (1,-2) and passing through the origin.find the equation of this parabola.
how do i do this? can someone please show me how to solve it step by step?thanks!
You know that at x = 0 the function is zero. Because the turning point is at
x = 1 must be in the middle of the two zeros of the function, the other zero is at x = 2.
Therefore the function is of the form:
y = A(x - 0) (x - 2) = Ax(x-2)
Solve for A by demanding that at x = 1 y = -2
To find the equation of the parabola, we can use the vertex form of a parabola equation, which is given by:
y = a(x - h)^2 + k
where (h, k) represents the vertex or turning point of the parabola.
In this case, the turning point is given as (1, -2), so h = 1 and k = -2. Substituting these values into the equation, we have:
y = a(x - 1)^2 - 2
Since the parabola passes through the origin (0, 0), we can substitute these values into the equation to solve for a:
0 = a(0 - 1)^2 - 2
0 = a(1)^2 - 2
0 = a - 2
2 = a
Now that we know the value of a, we can substitute it back into the equation to get the final equation of the parabola:
y = 2(x - 1)^2 - 2
Therefore, the equation of the parabola with a turning point at (1, -2) and passing through the origin is y = 2(x - 1)^2 - 2.
To solve the equation for A, substitute the values of x = 1 and y = -2 into the equation:
-2 = A(1)(1 - 2)
-2 = A(1)(-1)
-2 = -A
To isolate A, divide both sides by -1:
A = 2
Now you have the value of A.
To find the equation of the parabola, substitute A = 2 back into the equation:
y = 2x(x - 2)
So the equation of the parabola is y = 2x(x - 2).
Therefore, the parabola with a turning point at (1, -2) and passing through the origin is y = 2x(x - 2).