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Lukislah daerah himpunan penyelesaian (DHP) dari masing-masing sistem pertidaksamaan di bawah ini.  c. {y≥x2+2xy2−x2≤9​

Pertanyaan

Lukislah daerah himpunan penyelesaian (DHP) dari masing-masing sistem pertidaksamaan di bawah ini. 

c. open curly brackets table attributes columnalign left end attributes row cell y greater or equal than x squared plus 2 x end cell row cell y squared minus x squared less or equal than 9 end cell end table close   

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Cari DHP dari y greater or equal than x squared plus 2 x 

Kurva pembatas dari y greater or equal than x squared plus 2 x adalah y equals x squared plus 2 x 

Berdasarkan bentuk umum fungsi kuadrat y equals a x squared plus b x plus c diketahui

a equals 1 b equals 2 c equals 0 

Cari titik puncak pada bentuk persamaan kuadrat y equals a x squared plus b x plus c seperti berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis x subscript p comma space y subscript p right parenthesis end cell equals cell left parenthesis negative fraction numerator b over denominator 2 a end fraction comma space minus fraction numerator b squared minus 4 a c over denominator 4 a end fraction right parenthesis end cell row blank equals cell left parenthesis negative fraction numerator 2 over denominator 2 left parenthesis 1 right parenthesis end fraction comma space minus fraction numerator 2 squared minus 4 left parenthesis 1 right parenthesis left parenthesis 0 right parenthesis over denominator 4 left parenthesis 1 right parenthesis end fraction right parenthesis end cell row blank equals cell left parenthesis negative 2 over 2 comma space minus 4 over 4 right parenthesis end cell row blank equals cell left parenthesis negative 1 comma space minus 1 right parenthesis end cell end table  

Cari titik-titik pembentuk kurva tersebut

Untuk x equals 1  

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x squared plus 2 x end cell row y equals cell 1 squared plus 2 left parenthesis 1 right parenthesis end cell row y equals cell 1 plus 2 end cell row y equals 3 row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis 1 comma space 3 right parenthesis end cell end table  

Untuk x equals negative 2 

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x squared plus 2 x end cell row y equals cell left parenthesis negative 2 right parenthesis squared plus 2 left parenthesis negative 2 right parenthesis end cell row y equals cell 4 minus 4 end cell row y equals 0 row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis negative 2 comma space 0 right parenthesis end cell end table      

Untuk mengetahui DHP lakukan uji titik

untuk left parenthesis 0 comma space 1 right parenthesis  

y greater or equal than x squared plus 2 x 1 greater or equal than 0 squared plus 2 left parenthesis 0 right parenthesis 1 greater or equal than 0 plus 0 1 greater or equal than 0 space left parenthesis benar right parenthesis   

karena titik left parenthesis 0 comma space 1 right parenthesis benar maka daerah yang terdapat titik left parenthesis 0 comma space 1 right parenthesis merupakan DHP dari y greater or equal than x squared plus 2 x 

Cari DHP dari y squared minus x squared less or equal than 9  

Kurva pembatas dari y squared minus x squared less or equal than 9 adalah y squared minus x squared less or equal than 9  

Ubah bentuk persamaan tersebut menjadi seperti berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell y squared minus x squared end cell equals 9 row cell fraction numerator y squared minus x squared over denominator 9 end fraction end cell equals 1 row cell y squared over 9 minus x squared over 9 end cell equals 1 row cell y squared over 3 squared minus x squared over 3 squared end cell equals 1 end table      

y squared over a squared minus x squared over b squared equals 1 merupakan bentuk persamaan hiperbola dengan titik pusat left parenthesis 0 comma space 0 right parenthesis dan left parenthesis 0 comma space a right parenthesisleft parenthesis 0 comma space minus a right parenthesis sebagai titik puncak. Maka dari persamaan y squared over 3 squared minus x squared over 3 squared equals 1 diketahui bahwa titik pusatnya adalah left parenthesis 0 comma space 0 right parenthesis dan a equals 3 sehingga titik puncaknya adalah left parenthesis 0 comma space 3 right parenthesis dan left parenthesis 0 comma space minus 3 right parenthesis.

Cari letak DHP dengan uji titik left parenthesis 0 comma space 0 right parenthesis 

table attributes columnalign right center left columnspacing 0px end attributes row cell y squared minus x squared end cell less or equal than 9 row cell 0 squared minus 0 squared end cell less or equal than 9 row 0 less or equal than cell 9 space left parenthesis benar right parenthesis end cell end table   

karena titik left parenthesis 0 comma space 0 right parenthesis benar maka daerah yang terdapat titik left parenthesis 0 comma space 0 right parenthesis merupakan DHP dari y squared minus x squared less or equal than 9 . 

DHP dari open curly brackets table attributes columnalign left end attributes row cell y greater or equal than x squared plus 2 x end cell row cell y squared minus x squared less or equal than 9 end cell end table close adalah irisan dari DHP y greater or equal than x squared plus 2 x dan DHP y squared minus x squared less or equal than 9.

Jadi, DHP dari open curly brackets table attributes columnalign left end attributes row cell y greater or equal than x squared plus 2 x end cell row cell y squared minus x squared less or equal than 9 end cell end table close yaitu

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