Materi : Bentuk dan Persamaan Kuadrat

Bentuk Umum : f(x) = ax² + bx + c

Soal Nomor 1

Tentukan rumus fungsi kuadrat jika grafik fungsi kuadrat memotong sb. x di (-6,0) dan (1,0) serta melalui titik (0,-6)

f(-6) = 36a - 6b + c = 0 ( 36a - 6b = 6 )

f(1) = a + b + c = 0 ( a + b = 6 )

f(0) = [ c = -6 ]

{ Eliminasi ( b ) }

36a - 6b = 6

- 6a - 6b = -36

___________-

42a = 42

[ a = 1 ]

---

b = 6 - a

b = 6 - 1

[ b = 5 ]

Hasil / Result

{ a = 1 , b = 5 , c = -6 }

Bentuk Fungsi : f(x) = x² + 5x - 6

Soal Nomor 2

Tentukan rumus fungsi kuadrat jika grafik fungsi kuadrat memotong sb. X di (4,0) dan melalui titik (3,2)

f(4) = 16a + 4b + c = 0

f(3) = 9a + 3b + c = 2

SPLDV

16a + 4b = 0 ( 4a + b = 0 )

9a + 3b = 2

[ Eliminasi ( b ) ]

12a + 3b = 0

9a + 3b = 2

__________-

3a = -2

[ a = -⅔ ]

---

b = -4a

b = -4(-⅔)

[ b = 8/3 ]

---

9a + 3b + c = 2

9(-⅔) + 3(8/3) + c = 2

-6 + 8 + c = 2

c = 2 + 6 - 8

[ c = 0 ]

Hasil / Result :

{ a = -⅔ , b = 8/3 , c = 0 }

Bentuk Fungsi [ ×3 ] : f(x) = - 2x² + 8x

Soal Nomor 3

Tentukan rumus fungsi kuadrat jika grafik mempunyai titik puncak (-4,-25) dan melalui titik (2,11)

y = a( x - Xp )² + Yp

11 = a( 2 + 4 )² - 25

11 + 25 = a(6)²

36 = a(36)

[ a = 1 ]

---

f(2) = 4a + 2b + c = 11 [ 2b + c = 7 ]

f(-4) = 16a - 4b + c = -25 [ - 4b + c = -41 ]

SPLDV

c = 7 - 2b

- 4b + 7 - 2b = -41

- 6b = -48

[ b = 8 ]

---

c = 7 - 2b

c = 7 - 2(8)

c = 7 - 16

[ c = -11 ]

Hasil / Result :

{ a = 1, b = 8, c = -11 }

Bentuk Fungsi : f(x) = x² + 8x - 11

Soal Nomor 4

Tentukan rumus fungsi kuadrat jika grafik fungsi melalui titik (1,12), (-2,-3), dan (3,42)

f(1) = a + b + c = 12

f(-2) = 4a - 2b + c = -3

f(3) = 9a + 3b + c = 42

SPLDV

4a - 2b = -3

9a + 3b = 42

[ Eliminasi ( b ) ]

- 12a + 6b = 9

18a + 6b = 84

___________-

-30a = -75

a = 75/30

[ a = 5/2 ]

---

2b = 4a + 3

2b = 4(5/2) + 3

2b = 20/2 + 3

2b = 10 + 3

2b = 13

[ b = 13/2 ]

---

a + b + c = 12

5/2 + 13/2 + c = 12

18/2 + c = 12

c + 9 = 12

c = 12 - 9

[ c = 3 ]

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