Thursday, September 17, 2009

Fungsi Kuadrat

Salah satu contoh grafik fungsi kuadrat adalah lintasan bola yang dilempar yang membentuk sebuah lintasan parabola.

sumber gambar: http://www.gurumuda.com/gerak-parabola/

A. Bentuk Umum
Secara umum fungsi kuadrat ditulis dengan rumus fungsi f(x) = ax² + bx + c, dengan a tidak boleh sama dengan nol. Huruf-huruf a, b, dan c disebut koefisien, koefisien dari x² adalah a, koefisien dari x adalah b, dan c adalah sebuah konstanta.

Fungsi kuadrat, apabila digambar pada bidang cartesius (x,y) maka akan membentuk sebuah kurva parabola, karena memang bentuknya mirip dengan parabola.

B.Menggambar grafik fungsi kuadrat secara umum
Pada umumnya, grafik fungsi kuadrat dapat digambar pada bidang cartesius apabila telah diketahui 3 titik yang dilaluinya. Hal khusus, jika melalui titik puncak, maka cukup diketahui dua titik saja, mengapa demikian?!.. karena sifat simetri dari parabola itu sendiri, persamaan sumbu simetri melaui titik puncak bukan?!

Berikut akan diberikan langkah2 secara umum untuk menggambar grafik fungsi kuadrat:
langkah 1:
a. menentukan titik potong dengan sumbu - x, syarat fungsi f(x) = 0;
sehingga diperoleh ax² + bx + c = 0,
selanjutnya untuk mendapatkan nilai x1 dan x2 dapat menggunakan cara memfaktorkan atau dengan rumus kuadrat
b. menentukan titik potong dengan sumbu - y, syarat x = 0
sehingga diperoleh y = c, maka titik potong dengan sumbu - y adalah (0, c).

langkah 2:
menentukan titik puncak
xp = -b/2a
yp = -D/4a, dimana D = b² - 4ac

langkah 3:
meletakkan titik2 yang diperoleh pada bidang xy/bidang cartesius lalu menghubungkannya

C. menganalisis grafik fungsi kuadrat
a. arti nilai koefisien a, b, dan c.
1. koefisien a,
menentukan seberapa cekung atau cembung kurva parabolanya,
disisi lain apabial a benilai positif maka kurva terbuka ke atas dan a bernilai negatif maka kurva terbuka kebawah.
positive quadratic y = x2 negative quadratic y = –x2 I'm feeling positive! I'm feeling negative. smily graph frowny graph
sumber gambar: www.purplemath.com

2. koefisien b,
menentukan dimana letak puncak parabola, apabila ilai b = 0 maka sudah dapat dipastikan bahwa puncak parabola memtotong sumbu - y.

3. koefisien c,
menentukan letak titik potong parabola terhadap sumbu - y.

b. nilai diskriminan
Nilai diskriminan suatu fungsi kuadrat atau disingkat dengan D adalah b² - 4ac.
Suatu fungsi kuadrat hanya memiliki akar riil, jika nilai D adalah nol atau positif. jika nilai D adalah negatif maka fungsi kuadrat tersebut tidak memiliki nilai x yang riil, apabila digambarkan, parabola memotong sumbu x di satu titik maupun di dua titik.

sumber gambar: www.wikipedia.com
nah... apakah sudah mempunyai gambaran secara umum untuk menggambar grafik fungsi kuadrat?

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