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Lukislah DHP dari setiap SPtKKDV di bawah ini. 7. ⎩⎨⎧​4x2+y2≤4y≤1y>x2−1​

Pertanyaan

Lukislah DHP dari setiap SPtKKDV di bawah ini.

7. open curly brackets table attributes columnalign left end attributes row cell 4 x squared plus y squared less or equal than 4 end cell row cell y less or equal than 1 end cell row cell y greater than x squared minus 1 end cell end table close    

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Cari DHP dari 4 x squared plus y squared less or equal than 4 terlebih dahulu

Kurva pembatas dari 4 x squared plus y squared less or equal than 4 adalah 4 x squared plus y squared equals 4   

Ubah bentuk persamaan tersebut menjadi seperti berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 x squared plus y squared end cell equals 4 row cell fraction numerator 4 x squared plus y squared over denominator 4 end fraction end cell equals cell 4 over 4 end cell row cell fraction numerator 4 x squared over denominator 4 end fraction plus y squared over 4 end cell equals 1 row cell x squared over 1 plus y squared over 4 end cell equals 1 row cell x squared over 1 squared plus y squared over 2 squared end cell equals 1 end table  

x squared over a squared plus y squared over b squared equals 1 dengan a less than b merupakan bentuk persamaan elips vertikal dengan titik pusat left parenthesis 0 comma space 0 right parenthesis dengan left parenthesis a comma space 0 right parenthesis dan left parenthesis negative a comma space 0 right parenthesis adalah titik potong sumbu x dan left parenthesis 0 comma space b right parenthesis dan left parenthesis 0 comma space minus b right parenthesis adalah titik potong sumbu y maka dari persamaan x squared over 1 squared plus y squared over 2 squared equals 1 diketahui bahwa

table attributes columnalign right center left columnspacing 0px end attributes row pusat equals cell left parenthesis 0 comma space 0 right parenthesis end cell row cell tipot space sb space x end cell equals cell left parenthesis 1 comma space 0 right parenthesis space dan space left parenthesis negative 1 comma space 0 right parenthesis end cell row cell tipot space sb space y end cell equals cell left parenthesis 0 comma space 2 right parenthesis space dan space left parenthesis 0 comma space minus 2 right parenthesis end cell end table  

Untuk mengetahui letak DHP lakukan uji titik

untuk left parenthesis 0 comma space 0 right parenthesis 

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 x squared plus y squared end cell less or equal than 4 row cell 4 left parenthesis 0 right parenthesis squared plus 0 squared end cell less or equal than 4 row 0 less or equal than cell 4 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 4 x squared plus y squared less or equal than 4.

Selanjutnya cari DHP dari y less or equal than 1 

Garis pembatas dari y less or equal than 1 adalah y equals 1 

Untuk mengetahui DHP lakukan uji titik

untuk left parenthesis 0 comma space 0 right parenthesis 

y less or equal than 1 0 less or equal than 1 space left parenthesis benar right parenthesis 

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 less or equal than 1.

Selanjutnya cari DHP dari y greater than x squared minus 1   

Kurva pembatas dari y greater than x squared minus 1 adalah y equals x squared minus 1      

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

a equals 1 b equals 0 c equals negative 1   

Cari titik puncak dari bentuk persamaan y equals a x squared plus b x plus c menggunakan rumus left parenthesis x subscript p comma space y subscript p right parenthesis equals 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 

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 0 over denominator 2 left parenthesis 1 right parenthesis end fraction comma space minus fraction numerator 0 squared minus 4 left parenthesis 1 right parenthesis left parenthesis negative 1 right parenthesis over denominator 4 left parenthesis 1 right parenthesis end fraction right parenthesis end cell row blank equals cell left parenthesis 0 comma space minus 4 over 4 right parenthesis end cell row blank equals cell left parenthesis 0 comma space minus 1 right parenthesis end cell end table    

Cari titik potong sumbu xy equals 0 

 table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x squared minus 1 end cell row 0 equals cell x squared minus 1 end cell row 1 equals cell x squared end cell row blank blank cell x equals 1 logical or x equals negative 1 end cell row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis 1 comma space 0 right parenthesis end cell row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis negative 1 comma space 0 right parenthesis end cell end table 

Cari titik potong sumbu yx equals 0 

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

Untuk mengetahui letak DHP lakukan uji titik

untuk left parenthesis 0 comma space 0 right parenthesis    

table attributes columnalign right center left columnspacing 0px end attributes row y greater than cell x squared minus 1 end cell row 0 greater than cell 0 squared minus 1 end cell row 0 greater than cell negative 1 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 greater than x squared minus 1.  

DHP dari open curly brackets table attributes columnalign left end attributes row cell 4 x squared plus y squared less or equal than 4 end cell row cell y less or equal than 1 end cell row cell y greater than x squared minus 1 end cell end table close adalah irisan dari DHP 4 x squared plus y squared less or equal than 4, DHP y less or equal than 1, dan DHP y greater than x squared minus 1.

Jadi, DHP dari open curly brackets table attributes columnalign left end attributes row cell 4 x squared plus y squared less or equal than 4 end cell row cell y less or equal than 1 end cell row cell y greater than x squared minus 1 end cell end table closeyaitu

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