Question #0b94e

Aug 22, 2016

$y \left(x\right) = - 2 {x}^{2} + x + 14$

Explanation:

To arrive at $\frac{\mathrm{dy}}{\mathrm{dx}}$ we have differentiated $y \left(x\right)$ with respect to $x$.

To obtain an expression for $y \left(x\right)$ we take the anti-derivative or integral of $\frac{\mathrm{dy}}{\mathrm{dx}}$. This comes from the fundamental theorem of calculus.

$y \left(x\right) = \int \frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{dx} = \int \left(1 - 4 x\right) \mathrm{dx} = x - 2 {x}^{2} + C$

$y \left(x\right) = - 2 {x}^{2} + x + C$

Using the given point we can work out the value of our arbitrary constant:

$y \left(- 2\right) = 4 = - 2 {\left(- 2\right)}^{2} + \left(- 2\right) + C$

$\implies 4 = C - 10 \implies C = 14$

$\therefore y \left(x\right) = - 2 {x}^{2} + x + 14$