Plotting Graphs of Functions

Plot the graph of the function $y = x^{3}-3a^{2}x$.

Plot the graph of the function $y = \dfrac {ax^{2}}{a^{2}+x^{2}}$.

Plot the graph of the function $y = (x+1)^{2}(x-2)$.

Plot the graph of the function $y = \dfrac {x-1}{x^{2}-3x-4}$.

Plot the graph of the function $y = \dfrac {a^{3}}{a^{2}+x^{2}}$.

Plot the graph of the function $y = Ae^{-x^{2}/a^{2}}$, $A > 0$.

Plot the graph of the function $y = x^{n}e^{-x/a}$, $n > 0$, $a > 0$.

Plot the graph of the function $y = x^{2}e^{-x^{2}/a^{2}}$.

Plot the graph of the function $y = \dfrac {1}{x^{2}}-\dfrac {1}{(x-1)^{2}}$.

Plot the graph of the function $y = \dfrac {x^{2}}{2}+\dfrac {1}{x}$.

Plot the graph of the function $y = \dfrac {1}{x}+\dfrac {2x}{x^{2}-1}$.

Plot the graph of the function $y = \dfrac {2x-1}{(x-1)^{2}}$.

Plot the graph of the function $y = \dfrac {x^{2}+1}{x^{2}-4x+3}$.

Plot the graph of the function $y = \pm \sqrt{\dfrac {x-1}{x+1}}$.

Plot the graph of the function $y = \dfrac {10\sqrt[3]{(x-1)^{2}}}{x^{2}+9}$.

Plot the graph of the function given by the equation $2y = \sqrt{x^{2}+x+1}+\sqrt{x^{2}-x+1}$.

Plot the graph of the function $y = \sqrt{x^{2}+x+1}-\sqrt{x^{2}-x+1}$.

Plot the graph of the function given by the equation $y = \pm x^{2}\sqrt{x+1}$.

Plot the graph of the function $y = (x+1)^{3}\sqrt[3]{x^{2}}$.

Plot the graph of the function $y = \sqrt[3]{(x+1)^{2}}+\sqrt[3]{(x-1)^{2}}$.

Plot the graph of the function $y = \sqrt[3]{(x+1)^{2}}-\sqrt[3]{(x-1)^{2}}$.

Plot the graph of the function $y = \sqrt{x^{2}+1}-\sqrt{x^{2}-1}$.

Plot the graph of the function given by the equation $y = \pm \dfrac {x}{\sqrt{x+1}}$.

Plot the graph of the function given by the equation $y = \pm \sqrt{\dfrac {1-x}{x}}$.

Plot the graph of the function given by the equation $y = \pm \dfrac {x}{\sqrt{1-x^{2}}}$.

Plot the graph of the function given by the equation $y = \pm \dfrac {x\sqrt{1-x}}{1+x}$.

Plot the graph of the function given by the equation $ay = \pm x\sqrt{x(a-x)}$, $a > 0$.

Plot the graph of the function given by the equation $y^{2} = x^{3}+px+q$ for $4p^{3}+27q^{2} > 0$.

Plot the graph of the function given by the equation $y^{2} = x^{3}+px+q$ for $4p^{3}+27q^{2} < 0$.

Plot the graph of the function given by the equation $y^{2} = x^{3}+px+q$ for $4p^{3}+27q^{2} = 0$.

Plot the graph of the function $y = \cos^{3}x+\sin^{3}x$.

Plot the graph of the function $y = \cos^{4}x+\sin^{4}x$.

Plot the graph of the function $y = \cos x+\dfrac {1}{2}\cos 2x+\dfrac {1}{3}\cos 3x$.

Plot the graph of the function $y = \sin x+\dfrac {1}{2}\sin 2x+\dfrac {1}{3}\sin 3x$.

Plot the graph of the function $y = \dfrac {\cos 2x}{\cos x}$.

Plot the graph of the function $y = \dfrac {\tg 3x}{\tg x}$.

Plot the graph of the function $y = \dfrac {\sin x}{x}$.

Plot the graph of the function $y = \sin x^{2}$.

Plot the graph of the function $y = x\ln x$.

Plot the graph of the function $y = x^{2}\ln x$.

Plot the graph of the function $y = \ln (x^{2}-1)$.

Plot the graph of the function $y = \dfrac {1}{2}\ln \dfrac {1+x}{1-x}$.

Plot the graph of the function $y = (1+x)^{1/x}$ for $x \geq -1$.

Determine the number of real roots of the equation $12x^{4}-14x^{3}-3x^{2}-5=0$.

Determine the number of real roots of the equation $x^{4}-4ax^{3}-2=0$.

Determine the number of real roots of the equation $2x^{3}-3ax^{2}+1=0$.

Determine the number of real roots of the equation $x\ln x=a$.

Determine the number of real roots of the equation $\ln x=ax$.

Prove that the equation $x^{3}+px+q=0$ with real $p$ and $q$ has one real root when $4p^{3}+27q^{2} > 0$ and three real roots when $4p^{3}+27q^{2} < 0$.

Prove that for $a > 1$ the equation $a^{x}=bx$ has two real roots when $b > e\ln a$, has no real roots when $e\ln a > b > 0$, and has one root when $b < 0$.

Determine for which value of the parameter $m$ the equation $3x^{4}+4x^{3}-6x^{2}-12x+m=0$ has two distinct roots.

Determine for which value of the parameter $m$ the equation $2x^{3}-13x^{2}-20x+m=0$ has one root.

Determine for which value of the parameter $m$ the equation $3x^{4}-14x^{3}-45x^{2}+m=0$ has four distinct roots.

Determine for which value of the parameter $m$ the equation $2x^{3}-4x^{2}-30x+m=0$ has two coincident roots and one simple root.

Determine for which value of the parameter $m$ the equation $x^{2}+x+e^{-x}+m=0$ has two coincident roots.

Determine for which value of the parameter $m$ the equation $x^{2}-x-\ln x+m=0$ has no roots at all.

Determine for which value of the parameter $m$ the equation $6\arctg x-x^{3}+m=0$ has three roots, two of which coincide.