Asosiy tarkibga o'tish

5.3-bo'lim : Ko'p nomlilarni grafik qilish

3. Quyidagi ko‘phadning har birining grafigini chizing.

\[h\left( x \right) = {x^4} + {x^3} - 12{x^2} + 4x + 16 = {\left( {x - 2} \right)^2}\left( {x + 1} \right)\left( {x + 4} \right)\]
Yechim

Birinchi qadam ko'phadning nollarini va har bir nolning ko'paytmasini aniqlashdir. Bu muammo uchun ko'phad allaqachon faktorlangan.

Bizda nol va ko'paytmalarning quyidagi ro'yxati mavjud.

\[\begin{align*}x & = - 4 :{\mbox{ multiplicity 1}}\\ x & = - 1 :{\mbox{ multiplicity 1}}\\ x & = 2\,\,:{\mbox{ multiplicity 2}}\end{align*}\]

\(x = - 4\) va \(x = - 1\) ko'pligi g'alati bo'lgani uchun biz bu nuqta \(x\) o'qini kesib o'tadigan \(x\) kesishmalariga to'g'ri kelishini bilamiz va \(x = 2\) juft ko'plikka ega va shuning uchun biz bilamizki, bu nuqta \(x\) kesmasiga to'g'ri keladi, lekin faqat \(x\) kesmasiga mos keladi. \(x\) o'qi.

2-qadam

Funktsiya uchun \(y\)-kesishmasi: \(\left( {0,h\left( 0 \right)} \right) = \left( {0,16} \right)\).

3-qadam

Koeffitsient 4th daraja atamasi ijobiy va shuning uchun biz bilamizki, ko'phad ikkala uchida ham chegaralanmagan holda ortadi.

4-qadam

Keyinchalik, grafik uchun bir nechta nuqtalarni tanlashimiz kerak. Biz chap va o'ng tomonda bitta nuqtani tanlaymiz va bu erda qaysi nuqtani tanlash muhim emas, shuning uchun ikkala tomondan nolga yaqin birini tanlang. Shuningdek, biz har bir nol juftligi o'rtasida nuqta tanlaymiz va yana qaysi nuqtani tanlash muhim emas, shuning uchun biz har bir nol juftining o'rtasiga yaqin nuqtalarni tanlaymiz.

Mana bu ko'phad uchun funksiya baholashlari. Baholarni tekshirishni sizga qoldiramiz.

\[h\left( { - 5} \right) = 196\hspace{0.25in}h\left( { - 3} \right) = - 50\hspace{0.25in}h\left( 1 \right) = 10\hspace{0.25in}h\left( 3 \right) = 28\]
5-qadam

Va nihoyat, bu erda ko'phadning qisqacha eskizi.

The graph of \(h\left( x \right) = {x^4} + {x^3} - 12{x^2} + 4x + 16 = {\left( {x - 2} \right)^2}\left( {x + 1} \right)\left( {x + 4} \right)\). The graph comes down steeply from the upper left, crosses the x-axis at \(\left( { - 4,0} \right)\), drops to a valley near \(\left( { - 3, - 50} \right)\), rises back up and crosses the x-axis at \(\left( { - 1,0} \right)\), reaches a small peak, comes back down to just touch the x-axis at \(\left( {2,0} \right)\) without crossing it, and then turns back up to the upper right. The extra points that were computed are marked and labeled and they are \(\left( { - 5,196} \right)\), \(\left( { - 3, - 50} \right)\), \(\left( {1,10} \right)\) and \(\left( {3,28} \right)\).

E'tibor bering, sizning grafikingiz bunga o'xshamasligi mumkin.

Bu kompyuter tomonidan yaratilgan grafik va shuning uchun juda aniq. Faqat algebra texnikasidan foydalanish bunday aniqlikka yaqinlashish juda qiyin bo'lishi mumkin. Grafikingizning asosiy harakati to'g'ri bo'lsa va sizda qo'shimcha nuqtalar to'g'ri chizilgan bo'lsa, sizda tezkor eskiz uchun ishlash uchun haqiqiy grafikga etarlicha yaqin bo'lgan narsa bo'lishi kerak.

Qiziqqanlar uchun, ko'phadning aniqroq eskizini yaratishga imkon beruvchi Calculus sinfidan bir nechta texnikalar mavjud!