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

Задание

Заполните таблицу значений гиперболы \(\displaystyle y=\frac{1}{x}{\small :}\)

\(\displaystyle \small x\)\(\displaystyle \small -2\)\(\displaystyle \small -1{,}25\)\(\displaystyle \small -1\)\(\displaystyle \small -0{,}8\)\(\displaystyle \small -0{,}5\)\(\displaystyle \small -0{,}4\)
\(\displaystyle \small y=\frac{1}{x}\)

 

\(\displaystyle \small x\)\(\displaystyle \small 0{,}4\)\(\displaystyle \small 0{,}5\)\(\displaystyle \small 0{,}8\)\(\displaystyle \small 1\)\(\displaystyle \small 1{,}25\)\(\displaystyle \small 2\)
\(\displaystyle \small y=\frac{1}{x}\)


Мысленно постройте график гиперболы \(\displaystyle y=\frac{1}{x} \) по полученным точкам:

Решение

Заполним таблицу значений гиперболы \(\displaystyle y=\frac{1}{x}{\small :}\)

\(\displaystyle \small x\)\(\displaystyle \small -2\)\(\displaystyle \small -1{,}25\)\(\displaystyle \small -1\)\(\displaystyle \small -0{,}8\)\(\displaystyle \small -0{,}5\)\(\displaystyle \small -0{,}4\)
\(\displaystyle \small y=\frac{1}{x}\)\(\displaystyle \small \frac{1}{-2}\)\(\displaystyle \small \frac{1}{-1{,}25}\)\(\displaystyle \small \frac{1}{-1}\)\(\displaystyle \small \frac{1}{-0{,}8}\)\(\displaystyle \small \frac{1}{-0{,}5}\)\(\displaystyle \small \frac{1}{-0{,}4}\)

 

\(\displaystyle \small x\)\(\displaystyle \small 0{,}4\)\(\displaystyle \small 0{,}5\)\(\displaystyle \small 0{,}8\)\(\displaystyle \small 1\)\(\displaystyle \small 1{,}25\)\(\displaystyle \small 2\)
\(\displaystyle \small y=\frac{1}{x}\)\(\displaystyle \small \frac{1}{0{,}4}\)\(\displaystyle \small \frac{1}{0{,}5}\)\(\displaystyle \small \frac{1}{0{,}8}\)\(\displaystyle \small \frac{1}{1}\)\(\displaystyle \small \frac{1}{1{,}25}\)\(\displaystyle \small \frac{1}{2}\)


Вычисляем значения:

\(\displaystyle \small x\)\(\displaystyle \small -2\)\(\displaystyle \small -1{,}25\)\(\displaystyle \small -1\)\(\displaystyle \small -0{,}8\)\(\displaystyle \small -0{,}5\)\(\displaystyle \small -0{,}4\)
\(\displaystyle \small y=\frac{1}{x}\)\(\displaystyle \small -0{,}5\)\(\displaystyle \small -0{,}8\)\(\displaystyle \small -1\)\(\displaystyle \small -1{,}25\)\(\displaystyle \small -2\)\(\displaystyle \small -2{,}5\)

 

\(\displaystyle \small x\)\(\displaystyle \small 0{,}4\)\(\displaystyle \small 0{,}5\)\(\displaystyle \small 0{,}8\)\(\displaystyle \small 1\)\(\displaystyle \small 1{,}25\)\(\displaystyle \small 2\)
\(\displaystyle \small y=\frac{1}{x}\)\(\displaystyle \small 2{,}5\)\(\displaystyle \small 2\)\(\displaystyle \small 1{,}25\)\(\displaystyle \small 1\)\(\displaystyle \small 0{,}8\)\(\displaystyle \small 0{,}5\)


Построим точки на плоскости:


Построим график гиперболы \(\displaystyle y=\frac{1}{x}\) по полученным точкам, добавляя еще точки, если это необходимо:


Замечание / комментарий

Построение по точкам

Если построить по оси \(\displaystyle ОХ\) много точек с координатами от \(\displaystyle -2 \) до \(\displaystyle 2{\small , } \) то получаем следующую картинку графика: