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

Задание

Заполните таблицу значений гиперболы \(\displaystyle y= -\frac{1}{2x}{:}\)

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

 

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

 

Мысленно постройте график гиперболы \(\displaystyle y=-\frac{1}{2x}\) по полученным точкам:

Решение

Заполним таблицу значений квадратичной функция \(\displaystyle y=-\frac{1}{2x}{\small :}\)

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

 

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


Вычисляем значения:

\(\displaystyle \small x\)\(\displaystyle \small -2\)\(\displaystyle \small -1{,}25\)\(\displaystyle \small -1\)\(\displaystyle \small -0{,}8\)\(\displaystyle \small -0{,}5\)\(\displaystyle \small -0{,}4\)
\(\displaystyle \small y=-\frac{1}{2x}\)\(\displaystyle \small 0{,}25\)\(\displaystyle \small 0{,}4\)\(\displaystyle \small 0{,}5\)\(\displaystyle \small 0{,}625\)\(\displaystyle \small 1\)\(\displaystyle \small 1{,}25\)

 

\(\displaystyle \small x\)\(\displaystyle \small 0{,}4\)\(\displaystyle \small 0{,}5\)\(\displaystyle \small 0{,}8\)\(\displaystyle \small 1\)\(\displaystyle \small 1{,}25\)\(\displaystyle \small 2\)
\(\displaystyle \small y=-\frac{1}{2x}\)\(\displaystyle \small -1{,}25\)\(\displaystyle \small -1\)\(\displaystyle \small -0{,}625\)\(\displaystyle \small -0{,}5\)\(\displaystyle \small -0{,}4\)\(\displaystyle \small -0{,}25\)


Построим точки на плоскости:


 

Построим график гиперболы \(\displaystyle y=-\frac{1}{2x}\) по полученным точкам, добавляя еще точки если это необходимо:


 

Замечание / комментарий

Постороение по точкам

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