Find a parabola with equation y=ax2+bx+c that has slope 8 at x = 1, slope -16 at x = -1 and passes through the point (2, 2).

Step-by-step explanation:

We are given the slopes (the derivatives) at and .

So let's find the derivative of our equation:

At we have :

At we have :

So we have the system:

We can solve this by elimination. I'm going to add the equations allowing me to solve for .

Divide both sides by 2:

Let's find .

with :

Add 4 on both sides:

Divide both sides by 2:

So we have the equation so far is:

.

We can find by using the point (2,2):

Subtract 16 on both sides:

So the equation is .

Let's do a final check.

Check to see if we have (1,8) and (-1,-16) satisfies this. (Note: )

Check for :

which is true.

Check for :

which is true.

Let's check to see (2,2) is on :

which is true.

The equation has been verified with the given conditions.