Kvadratické útvary

Příklad 1 - KRUŽNICE

V kartézské soustavě souřadnic zobrazte graf relace pro níž platí:

(a) (x+|x|)^2+(y+|y|)^2=4 (b) (x-|x|)^2+(y-|y|)^2=4
(c) (|x|-1)^2+(y-2)^2=4 (d) (|x+1|-1)^2+(|y+2|-2)^2=4
(e) \left(x-\dfrac{|x|}{x}\right)^2+\left(y-\dfrac{|y|}{y}\right)^2=4 (f) \left(x-2\dfrac{|x|}{x}\right)^2+\left(y-2\dfrac{|y|}{y}\right)^2=5
(g) \left(x-\dfrac{|x|}{x}+\dfrac{|y|}{y}\right)^2+\left(y-\dfrac{|x|}{x}-\dfrac{|y|}{y}\right)^2=4 (h) \left(x-\dfrac{|x|}{x}-\dfrac{|y|}{y}\right)^2+\left(y-\dfrac{|x|}{x}-\dfrac{|y|}{y}\right)^2=4
(i) \dfrac{|x|}{|y|}=\dfrac{2-y}{2-x} (j) \dfrac{|x|}{|y|}=\dfrac{y-2}{2-x}    
(k) |y^2-4|=x^2+4x (l) |x^2-1|=2y-y^2
(m) x|x|+y|y|=x+y (n) x|x|+y^2=x+|y|
(o) y^2-x+1=|x^2-x+2y| (p) x^2-y=|y^2+y-2x|
(q) |x+y^2|=|x^2-|y|| (r) ||x|-y^2|=|x^2-y|
(s) (||x|-2|-1)^2+y^2=1 (t) (|x|-2)^2+y^2=1

Příklad 2 - OSTATNÍ KUŽELOSEČKY

V kartézské soustavě souřadnic zobrazte graf relace pro níž platí:

(a) x^2=|y-x^2| (b) |x|+|y|=|y^2+x|
(c) xy=|x|+|y| (d) xy=|x|-|y|
(e) y^2=|y-x^2| (f) |x^2-4|+|y^2-1|=4
(g) \left[y-\sqrt{1-(|x|-1)^2}\right]\cdot\left[y+3\sqrt{1-\dfrac{|x|}{2}}\right]=0 (h) |y|=(2-x)|x|        
(i) |y|=x|x-2| (j) x^2-y=|x|+|y|
(k) x^2+y=|x|-|y| (l) \dfrac{x^2-1}{x+y}=\left|1-\dfrac{x}{y}\right|
(m) \dfrac{x^2-1}{x-y}=\left|1+\dfrac{x}{y}\right| (n) \dfrac{x^2-1}{x+y}=\dfrac{|x-y|}{y}
(o) \dfrac{x^2-1}{x-y}=\dfrac{|x+y|}{y} (p) |y|-x^2+x=y+|x^2-x|
(q) |x|+y^2=1 (r) y^2-\dfrac{x(x-|x|)}2=4
(s) y^2-x|x|=4 (t) 1-(x^2-|y|)^2=0

Příklad 3 - Nerovnice

V kartézské soustavě souřadnic zobrazte graf relace pro níž platí:

(a) (x^2-y-1)(y+x^2+1)\le 0 (b) 1-|x|\le y\le (|x|-1)^2-4
(c) x\ge|x^3+xy^2| (d) y\ge|y^3+x^2y|
(e) (2-x^2-y^2)(|x|+|y|-2)\ge0 (f) (|x|+|y|-1)(x^2+y^2-1)\le0
(g) 2|x-y-2|\le3-x^2-y^2 (h) \dfrac32-x^2-y^2\ge|x+y-1|
(i) x^2+y\le|x-y| (j) x^2-y\ge|y-x|
(k) -x^2+4x-3\le y\le-\dfrac32|2x-3|+x+\dfrac32 (l) x+\dfrac y{2x}\ge0
(m) 1-(x^2+y)^2\ge0 (n) |x-xy|<1
(o) |xy|\le1 (q) |x^2+y^2-2|\le1

Příklad 4 - Soustavy

V kartézské soustavě souřadnic zobrazte graf relace pro níž platí:

(a) \begin{cases}x^2+y^2-4x-2y+1\le0\\ x+2y-4\le0\end{cases} (b) \begin{cases}x^2+y^2+2y-3\le0\\-1\le y\le x+1\end{cases}
(c) \begin{cases}x^2+y^2=x+y\\ x+y\le1\end{cases} (d) \begin{cases}x^2-y^2=x-y\\ x+y\ge1\end{cases}
(e) \begin{cases}|x|+|y-1|\le2\\ x^2+y^2\ge1-\frac{2}{\sqrt3}y\end{cases} (f) \begin{cases}|x+y|+|y|\le2\\ x^2+y^2-1\le2\sqrt3x\end{cases}
(g) \begin{cases}|x+y|\ge2\\ x^2+y^2\le2(1+x+y)\end{cases} (h) \begin{cases}|x-y|\le3\\ x^2+y^2\le2(2-x+2y)\end{cases}
(i) \begin{cases}x^2+y^2\le6y\\ |x|\le|y-3|\end{cases} (j) \begin{cases}x^2+y^2\le4x\\ |y|\ge|2-x|\end{cases}
(k) \begin{cases}|x-y|\le1\\ x^2+y^2+2(x+y)\le0\end{cases} (l) \begin{cases}|x+y|\le1\\ x^2+y^2\ge x+y\end{cases}
(m) \begin{cases}x^2-2x-y\le 0\\ 2x-6y+3\ge0\end{cases} (n) \begin{cases}4x^2+9y^2\le36\\y^2\le x\end{cases}
(o) \begin{cases}y\le5-x^2\\xy\le2\end{cases} (p) \begin{cases}x^2+y^2\le25\\x^2-y<5\end{cases}
(q) \begin{cases}4-x^2\ge0\\ y^2-9\ge0\end{cases} (r) \begin{cases}x^2-y^2\ge9\\ x^2+2x+y^2\le15\end{cases}