Дана гипербола \(\displaystyle y=-\frac{1}{x}{\small .}\) Вычислите значения функции в точках:
| \(\displaystyle \small x\) | \(\displaystyle \small -4\) | \(\displaystyle \small -2\) | \(\displaystyle \small -1{,}6\) | \(\displaystyle \small -1\) | \(\displaystyle \small -0{,}4\) | \(\displaystyle \small -0{,}25\) |
| \(\displaystyle \small y=-\frac{1}{x}\) |
| \(\displaystyle \small x\) | \(\displaystyle \small 0{,}25\) | \(\displaystyle \small 0{,}4\) | \(\displaystyle \small 1\) | \(\displaystyle \small 1{,}6\) | \(\displaystyle \small 2\) | \(\displaystyle \small 4\) |
| \(\displaystyle \small y=-\frac{1}{x}\) |
Выберите правильный график с точками, лежащими на этой гиперболе \(\displaystyle y=-\frac{1}{x}{\small :}\)
| \(\displaystyle \rm I\) | \(\displaystyle \rm II\) |
 |  |
| \(\displaystyle \rm III\) | \(\displaystyle \rm IV\) |
 |  |
Вычислим значение гиперболы \(\displaystyle y=-\frac{1}{x}\) в заданных точках:
| \(\displaystyle \small x\) | \(\displaystyle \small -4\) | \(\displaystyle \small -2\) | \(\displaystyle \small -1{,}6\) | \(\displaystyle \small -1\) | \(\displaystyle \small -0{,}4\) | \(\displaystyle \small -0{,}25\) |
| \(\displaystyle \small y=-\frac{1}{x}\) | \(\displaystyle \small 0{,}25\) | \(\displaystyle \small 0{,}5\) | \(\displaystyle \small 0{,}625\) | \(\displaystyle \small 1\) | \(\displaystyle \small 2{,}5\) | \(\displaystyle \small 4\) |
| \(\displaystyle \small x\) | \(\displaystyle \small 0{,}25\) | \(\displaystyle \small 0{,}4\) | \(\displaystyle \small 1\) | \(\displaystyle \small 1{,}6\) | \(\displaystyle \small 2\) | \(\displaystyle \small 4\) |
| \(\displaystyle \small y=-\frac{1}{x}\) | \(\displaystyle \small -4\) | \(\displaystyle \small -2{,}5\) | \(\displaystyle \small -1\) | \(\displaystyle \small -0{,}625\) | \(\displaystyle \small -0{,}5\) | \(\displaystyle \small -0{,}25\) |
Отметим точки
\(\displaystyle (-4;\, 0{,}25),\, (-2;\, 0{,}5),\, (-1{,}6;\, 0{,}625),\, (-1;\, 1),\, (-0{,}4;\, 2{,}5),\, (-0{,}25;\, 4),\,\)
\(\displaystyle (0{,}25;\, -4),\,(0{,}4;\, -2{,}5),\, (1;\, -1),\, (1{,}6;\, -0{,}625),\,(2;\, -0{,}5),\,(4;\, -0{,}25)\)
на координатной плоскости:
Сравним с данными графиками:
| \(\displaystyle \rm I\) | \(\displaystyle \rm II\) |
| |
| \(\displaystyle \rm III\) | \(\displaystyle \rm IV\) |
| |
Видим, что график \(\displaystyle \rm I\) соответствует нашей гиперболе \(\displaystyle y=-\frac{1}{x}{\small .}\)
Ответ: \(\displaystyle \rm I {\small .}\)