Answer
$y=3x^2-2x+7$
Work Step by Step
A parabola $y=ax^2+bx+c$ satisfies the following conditions.
1) At $x = 1$, the slope is $4$.
2) At $x=-1$, the slope is $-8$.
3) It passes through $(2,15)$.
Find $y'$:
$y'=2ax+b$
Using the condition 1, we get $2a+b=4$.
Using the condition 2, we get $-2a+b=-8$.
Adding both equations, $2b=-4\Rightarrow b=-2$.
Meanwhile, substracting both equations, $4a=12\Rightarrow a=3$.
Now, we have $y=3x^2-2x+c$.
Using the condition 3, we get
$3\cdot 2^2-2\cdot 2+c=15$
$12-4+c=15$
$c=7$
Thus, the equation of the parabola is $y=3x^2-2x+7$.
