Iklan

Iklan

Pertanyaan

Grafik fungsi f ( x ) = − 2 x 2 − 5 x + 1 melalui titik A ( 3 , − 8 ) . Tentukan persamaan garis singgung di titik A tersebut!

Grafik fungsi  melalui titik . Tentukan persamaan garis singgung di titik A tersebut!space

Iklan

A. Allamah

Master Teacher

Mahasiswa/Alumni Universitas Brawijaya

Jawaban terverifikasi

Jawaban

persamaan garis singgung fungsi melalui titik adalah dan .

persamaan garis singgung fungsi f open parentheses x close parentheses equals negative 2 x squared minus 5 x plus 1 melalui titik straight A open parentheses 3 comma space minus 8 close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 15 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 45 end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 30 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 85 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 84 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 55 end table.

Iklan

Pembahasan

Diketahui fungsi dantitik . Substitusikan pada fungsi kuadrat. Karena , maka: Cek kedudukan titik dengan cara substitusikan pada fungsi kuadrat, yaitu: Karena hasilnya , maka titik berada di luar kurva. Persamaan garis singgung jika diketahui titik dan gradien adalah: Substitusikan titik pada . Substitusikan pada persamaan kuadrat. Maka, diperoleh: Garis dan kurva bersinggungan, maka . Sehingga diperoleh koefisien-koefisien dari persamaan kuadrat di atas yaitu: Gunakan rumus kuadratik untuk menentukan nilai . Rumus kuadratik. Maka diperoleh: Sehingga, persamaan garis singgung untuk . Persamaan garis singgung untuk . Jadi, persamaan garis singgung fungsi melalui titik adalah dan .

Diketahui fungsi f open parentheses x close parentheses equals negative 2 x squared minus 5 x plus 1 dan titik straight A open parentheses 3 comma space minus 8 close parentheses. Substitusikan x equals 3 pada fungsi kuadrat.

Karena y equals f left parenthesis x right parenthesis, maka:

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 2 x squared minus 5 x plus 1 end cell row blank blank blank end table

Cek kedudukan titik dengan cara substitusikan x equals 3 pada fungsi kuadrat, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell right parenthesis squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 5 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis 3 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table y table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 18 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 15 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 32 end table 

Karena hasilnya y not equal to negative 8, maka titik straight A open parentheses 3 comma space minus 8 close parentheses berada di luar kurva.

Persamaan garis singgung jika diketahui titik open parentheses x subscript 1 comma space y subscript 1 close parentheses dan gradien m adalah:

y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses  

Substitusikan titik straight A open parentheses 3 comma space minus 8 close parentheses pada y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses.

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m open parentheses x minus x subscript 1 close parentheses end cell row cell y plus 8 end cell equals cell m open parentheses x minus 3 close parentheses end cell row y equals cell m x minus 3 m minus 8 end cell end table 

Substitusikan table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank m end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank m end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 8 end table pada persamaan kuadrat.

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 2 x squared minus 5 x plus 1 end cell row cell m x minus 3 m minus 8 end cell equals cell negative 2 x squared minus 5 x plus 1 end cell row cell 2 x squared plus 5 x plus m x minus 3 m minus 8 minus 1 end cell equals 0 row cell 2 x squared plus open parentheses 5 plus m close parentheses x plus open parentheses negative 3 m minus 9 close parentheses end cell equals 0 end table 

Maka, diperoleh:

a equals 2 comma space b equals 5 plus m comma space c equals negative 3 m minus 9 

Garis dan kurva bersinggungan, maka D equals 0

table attributes columnalign right center left columnspacing 0px end attributes row D equals 0 row cell b squared minus 4 a c end cell equals 0 row cell open parentheses 5 plus m close parentheses squared minus 4 times 2 times open parentheses negative 3 m minus 9 close parentheses end cell equals 0 row cell 25 plus 10 m plus m squared plus 24 m plus 72 end cell equals 0 row cell m squared plus 34 m plus 97 end cell equals 0 end table 

Sehingga diperoleh koefisien-koefisien dari persamaan kuadrat di atas yaitu:

p equals 1 comma space q equals 34 comma space r equals 97 

Gunakan rumus kuadratik untuk menentukan nilai m

Rumus kuadratik.

m table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank subscript 1 comma 2 end subscript end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative q plus-or-minus square root of q squared minus 4 p r end root over denominator 2 p end fraction end cell end table

Maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript 1 comma 2 end subscript end cell equals cell fraction numerator negative 34 plus-or-minus square root of 34 squared minus 4 times 1 times 97 end root over denominator 2 times 1 end fraction end cell row blank equals cell fraction numerator negative 34 plus-or-minus square root of 768 over denominator 2 end fraction end cell row blank equals cell fraction numerator negative 34 plus-or-minus 27 comma 7 over denominator 2 end fraction end cell row cell m subscript 1 end cell equals cell fraction numerator negative 34 plus 27 comma 7 over denominator 2 end fraction end cell row cell m subscript 1 end cell equals cell negative 3 comma 15 end cell row cell m subscript 2 end cell equals cell fraction numerator negative 34 minus 27 comma 7 over denominator 2 end fraction end cell row cell m subscript 2 end cell equals cell negative 30 comma 85 end cell end table 

Sehingga, persamaan garis singgung untuk m subscript 1 equals negative 3 comma 15.

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m open parentheses x minus x subscript 1 close parentheses end cell row cell y plus 8 end cell equals cell negative 3 comma 15 open parentheses x minus 3 close parentheses end cell row y equals cell negative 3 comma 15 x plus 9 comma 45 minus 8 end cell row y equals cell negative 3 comma 15 x plus 1 comma 45 end cell end table 

Persamaan garis singgung untuk m subscript 2 equals negative 30 comma 85.

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m open parentheses x minus x subscript 1 close parentheses end cell row cell y plus 8 end cell equals cell negative 30 comma 85 open parentheses x minus 3 close parentheses end cell row y equals cell negative 30 comma 85 x plus 92 comma 55 minus 8 end cell row y equals cell negative 30 comma 85 x plus 84 comma 55 end cell end table 

Jadi, persamaan garis singgung fungsi f open parentheses x close parentheses equals negative 2 x squared minus 5 x plus 1 melalui titik straight A open parentheses 3 comma space minus 8 close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 15 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 45 end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 30 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 85 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 84 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 55 end table.

Perdalam pemahamanmu bersama Master Teacher
di sesi Live Teaching, GRATIS!

44

Iklan

Iklan

Pertanyaan serupa

Tentukan gradien dan kemudian persamaan garis singgung setiap kurva berikut ini pada titik yang diketahui: a. y = x 2 − 4 x di titik ( − 1 , 6 )

14

5.0

Jawaban terverifikasi

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Roboguru

Coba GRATIS Aplikasi Ruangguru

Download di Google PlayDownload di AppstoreDownload di App Gallery

Produk Ruangguru

Hubungi Kami

Ruangguru WhatsApp

+62 815-7441-0000

Email info@ruangguru.com

info@ruangguru.com

Contact 02140008000

02140008000

Ikuti Kami

©2024 Ruangguru. All Rights Reserved PT. Ruang Raya Indonesia