Let g(x)=x2+2(a+b)x+2(a2+b2),where a and b are constants. b. Use the result in part (a) to explain why the graph of g has no x -intercepts unless a=ba = b, in which  case there is only one x-intercept.

Pertanyaan

Let g open parentheses x close parentheses equals x squared plus 2 open parentheses a plus b close parentheses x plus 2 open parentheses a squared plus b squared close parentheses,where a and b are constants.

b. Use the result in part (a) to explain why the graph of g has no x -intercepts unless a equals ba = b, in which  case there is only one x-intercept.

I. Sutiawan

Master Teacher

Mahasiswa/Alumni Universitas Pasundan

Jawaban terverifikasi

Jawaban

dapat disimpulkan ketika nilai a equals b, maka grafik g open parentheses x close parentheses equals x squared plus 2 open parentheses a plus b close parentheses x plus 2 open parentheses a squared plus b squared close parentheses melalui sumbu x di satu titik (menyinggung sumbu x).

Pembahasan

Diketahui g open parentheses x close parentheses equals x squared plus 2 open parentheses a plus b close parentheses x plus 2 open parentheses a squared plus b squared close parentheses, maka untuk menentukan vertex atau titik puncaknya kita dapat menggunakan rumus left parenthesis x subscript p comma space y subscript p right parenthesis equals open parentheses negative fraction numerator b over denominator 2 a end fraction comma space minus fraction numerator D over denominator 4 a end fraction close parentheses, sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript p end cell equals cell negative fraction numerator b over denominator 2 a end fraction end cell row blank equals cell negative fraction numerator 2 left parenthesis a plus b right parenthesis over denominator 2 left parenthesis 1 right parenthesis end fraction end cell row blank equals cell negative fraction numerator 2 left parenthesis a plus b right parenthesis over denominator 2 end fraction end cell row blank equals cell negative left parenthesis a plus b right parenthesis end cell row blank blank blank row cell y subscript p end cell equals cell negative fraction numerator D over denominator 4 a end fraction end cell row blank equals cell negative fraction numerator b squared minus 4 a c over denominator 4 a end fraction end cell row blank equals cell negative fraction numerator left parenthesis 2 left parenthesis a plus b right parenthesis right parenthesis squared minus 4 left parenthesis 1 right parenthesis left parenthesis 2 left parenthesis a squared plus b squared right parenthesis right parenthesis over denominator 4 left parenthesis 1 right parenthesis end fraction end cell row blank equals cell negative fraction numerator 4 left parenthesis a squared plus 2 a b plus b squared right parenthesis minus 8 a squared minus 8 b squared over denominator 4 end fraction end cell row blank equals cell negative fraction numerator 4 a squared plus 8 a b plus 4 b squared minus 8 a squared minus 8 b squared over denominator 4 end fraction end cell row blank equals cell negative fraction numerator negative 4 a squared plus 8 a b minus 4 b squared over denominator 4 end fraction end cell row blank equals cell negative fraction numerator up diagonal strike 4 left parenthesis negative a squared plus 2 a b minus b squared right parenthesis over denominator up diagonal strike 4 end fraction end cell row blank equals cell a squared minus 2 a b plus b squared end cell row blank equals cell left parenthesis a minus b right parenthesis squared end cell end table

Dari perhitungan di atas, titik puncak dari  fungsi kuadrat g open parentheses x close parentheses equals x squared plus 2 open parentheses a plus b close parentheses x plus 2 open parentheses a squared plus b squared close parentheses adalah left parenthesis negative left parenthesis a plus b right parenthesis comma space left parenthesis a minus b right parenthesis squared right parenthesis.

Ketika a equals b, maka titik puncak menjadi:

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 left parenthesis b plus b right parenthesis comma space left parenthesis b minus b right parenthesis right parenthesis end cell row blank equals cell left parenthesis negative b comma space 0 right parenthesis end cell end table 

Karena ordinat titik puncak bernilai 0, artinya grafik menyinggung sumbu x ketika titik puncak. Dalam parabola, jika grafik memotong sumbu x di titik puncak, maka grafik tersebut hanya menyinggung dan tidak mungkin memotong di dua titik.

Sehingga dapat disimpulkan ketika nilai a equals b, maka grafik g open parentheses x close parentheses equals x squared plus 2 open parentheses a plus b close parentheses x plus 2 open parentheses a squared plus b squared close parentheses melalui sumbu x di satu titik (menyinggung sumbu x).

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