How to sketch a cubic polynomial?

Question

How do you sketch $-4x^3+15x-1 = 0$? Any help would be appreciated!

Answer 1

As it's cubic, you determine the orientation - that it has $-4$ in front of the $x^3$ tells you that it will be positive for very negative values of $x$ and negative for very positive values of $x$. So it will start in the second quadrant, and end in the fourth quadrant.

Now, determine the $y$ intercept (let $x=0$) and the $x$ intercepts (solve the equation you provided). Then determine the turning points (find the $x$ where the derivative of the function is zero, then the $y$ value at that $x$). Then it's just a matter of plotting those points, joining them with something resembling a cubic.

Answer 2

Hope this help.

$f = -4x^3+15x-1$

$f' = -12x^2 + 15$

$f'=0 => x_{1,2} = \pm \sqrt5/2$

$f(-2) = 1, f'(-2) = -33$

$f(-1) = 18, f'(-1) = 3$

$f(0) = -1, f'(0) = 15$

$f(1) = 10, f'(1) = 3$

$f(2) = -3, f'(2) = -33$