Question

Find a parabola with equation y = ax2 + bx + c that has slope 5 at x = 1, slope −11 at x = −1, and passes through the point (2, 18).

Answer 1

By "slope" I assume you mean slope of the tangent line to the parabola.

For any given value of x, the slope of the tangent to the parabola is equal to the derivative of y :

$y=ax^2+bx+c\implies y'=2ax+b$

The slope at x = 1 is 5:

$2a+b=5$

The slope at x = -1 is -11:

$-2a+b=-11$

We can already solve for a and b :

$\begin{cases}2a+b=5\\-2a+b=-11\end{cases}\implies 2b=-6\implies b=-3$

$2a-3=5\implies 2a=8\implies a=4$

Finally, the parabola passes through the point (2, 18); that is, the quadratic takes on a value of 18 when x = 2:

$4a+2b+c=18\implies2(2a+b)+c=10+c=18\implies c=8$

So the parabola has equation

$\boxed{y=4x^2-3x+8}$