Asosiy tarkibga o'tish

1.10-bo'lim : Umumiy grafiklar

7. Grafik kalkulyatordan foydalanmasdan \(W\left( x \right) = {{\bf{e}}^{x + 2}} - 3\) grafigini chizing.

Yechim

Ushbu bo'limning birinchi bir nechta muammolarida biz ishlatgan algebraik o'zgarishlarni bir vaqtning o'zida grafikni yuqoriga/pastga va o'ngga/chapga siljitish uchun birlashtirish mumkin. Agar biz \(g\left( x \right)\) ning grafigini bilsak, \(g\left( {x + c} \right) + k\) grafigi oddiygina \(g\left( x \right)\) ning grafigi, agar \(c < 0\) bo‘lsa, \(c\) birliklari bilan o‘ngga siljiydi yoki \(c > 0\) bo‘lsa, \(c\) birliklari bilan chapga siljiydi va \(k\) birliklari pastga siljiydi10 \(k\) birliklari, agar \(k < 0\).

Shunday qilib, bizning holatlarimizda \(g\left( x \right) = {{\bf{e}}^x}\), biz buni ko'rishimiz mumkin,

\[W\left( x \right) = {{\bf{e}}^{x + 2}} - 3 = g\left( {x + 2} \right) - 3\]

va shuning uchun bizdan eskiz qilishimiz so'ralayotgan grafik \(g\left( x \right) = {{\bf{e}}^x}\) ning 2 birlikka chapga va 3 birlikka pastga siljigan grafigi.

Mana \(W\left( x \right) = {{\bf{e}}^{x + 2}} - 3\) grafigi va shuni yodda tutingki, transformatsiyani ko'rishga yordam berish uchun biz \(g\left( x \right) = {{\bf{e}}^x}\) grafigida ham chizganmiz.

The graph of \(W\left( x \right) = {{\bf{e}}^{x + 2}} - 3\), drawn as a solid red curve, along with the graph of \(g\left( x \right) = {{\bf{e}}^x}\), drawn as a dashed gray curve, for \(-4 \le x \le 2\). The graph of \(g\left( x \right)\) is just above the \(x\)-axis on the left and increases through (0,1) to about 7.4 at \(x = 2\). The graph of \(W\left( x \right)\) is just above \(y = -3\) on the left, crosses the \(x\)-axis near \(x = -0.9\) and stops at about (0,4.4), so only the part of it with \(x \le 0\) is shown.

Bu holda, \(g\left( x \right)\) grafigini siljitish orqali olingan \(W\left( x \right)\) eskizi haqiqatdan ham eng yaxshisi emas, chunki u \(x = 0\) da deyarli uzilib qoladi, shuning uchun bu holda biz \(W\left( x \right)\) grafigini biroz kengaytirishimiz kerak. Bu erda grafikning yaxshiroq eskizi.

The same graphs as the previous figure except that the graph of \(W\left( x \right) = {{\bf{e}}^{x + 2}} - 3\) has been extended past \(x = 0\). The solid red curve for \(W\left( x \right)\) now continues increasing past (0,4.4) and leaves the top of the graph at \(y = 8\) near \(x = 0.4\). The dashed gray curve for \(g\left( x \right) = {{\bf{e}}^x}\) is unchanged.