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Теория: 02 Построение графика гиперболы \(\displaystyle \small y=\frac{k}{x}, k<0\)

Задание

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 \(\displaystyle y=-\frac{1}{x}{\small :}\)

\(\displaystyle \rm I\)\(\displaystyle \rm II\)
\(\displaystyle \rm III\)\(\displaystyle \rm IV\)
 
Решение

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:  \(\displaystyle \rm I {\small .}\)