Fungsi dan persamaan quadratik
Quadratic functions and equations
1.0 Nota
Fungsi dan persamaan quadratik
Quadratic functions and equations
Fungsi quadratik
Persamaan kuadratik
𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Keadaan:
Condition:
1. 𝑎 ≠ 0
2. Kuasa tertinggi adalah 2
Mode - Mode - Eqn - > - Degree – 2
4𝑥 2 + 8𝑥 + 16 = 0
(𝑥 + 1)(𝑥 + 1) = 0
𝑥+1=0
𝑥 = −1
Mode - Mode - Eqn - > - Degree – 2
𝑓(𝑥) = 4𝑥 2 + 8𝑥 + 16
Bagi 𝑓(𝑥) = 0,
Let 𝑓(𝑥) = 0,
(𝑥 + 1)(𝑥 + 1) = 0
𝑥+1=0
𝑥 = −1
Graph fungsi quadratik
Graph of quadratic function
1. Analisasi fungsi quadratik
Analyse the quadratic function
𝑎
Analisasi:
Analyse:
Jika 𝑎 > 0:
If 𝑎 > 0:
Jika 𝑎 < 0:
If 𝑎 < 0:
.
1
𝑐
𝑐 adalah pintasan 𝑦 ataupun pintasan 𝑓(𝑥)
𝑐 is the 𝑦 intercept or 𝑓(𝑥) intercept
Atau
Or
Pintasan 𝑥
𝑦-intercept
Nilai 𝑓(𝑥) when 𝑥 = 0
Value of 𝑓(𝑥) when 𝑥 = 0
Pintasan 𝑥
𝑥-intercept
Punca fungsi quadratik
The roots of the function quadratic
o
Dua punca berbeza
Two distinct roots
o
Satu punca
One root
2
o
Tiada punca sebenar
No real roots
Atau
Or
Nilai 𝑥 when 𝑓(𝑥) = 0
Value of 𝑥 when 𝑓(𝑥) = 0
Titik
maxima/
minima
Maximum/
minimum
point
Titik maxima
Maximum point
Titik minima
Minimum point
.
3
2.0 Soalan praktis
Soalan 1
Lakarkan graph berikut:
Sketch the graph of:






𝑦 = (𝑥 + 1)(−𝑥 + 3)
𝑦 = −𝑥 2 + 2𝑥 + 6
𝑦 = 𝑥 2 + 2𝑥 + 4
𝑦 = 4𝑥 2 + 6𝑥 + 12
𝑓(𝑥) = 2𝑥 2 + 4𝑥 + 6
𝑓(𝑥) = (−𝑥 + 4)(−𝑥 + 3)
You may use graphic calculator to validate your answers
https://www.geogebra.org/graphing?lang=en
4
3.0 KBAT
Soalan 1
Rajah berikut menunjukkan graph 𝑦 = 𝑥 𝑎 + 𝑏𝑥 + 2.
The diagram below shows the graph 𝑦 = 𝑥 𝑎 + 𝑏𝑥 + 2.
Cari nilai 𝑎 + 𝑏.
Find the value of 𝑎 + 𝑏.
(𝑥 − 1)(𝑥 − 2) = 0
𝑥 2 − 2𝑥 − 𝑥 + 2 = 0
𝑥 2 − 3𝑥 + 2 = 0
Bandingkan dengan 𝑥 𝑎 + 𝑏𝑥 + 2,
𝑎=2
𝑏 = −3
𝑎 + 𝑏 = −1
5
Soalan 2
Rajah berikut menujukkan suatu reka bentuk dengan luas 145 𝑐𝑚2.
The diagram below shows a shape with area 145 𝑐𝑚2.
Cari nilai 𝑥.
Find the value of 𝑥.
𝑥(13 + 𝑥) + 𝑥(11 + 𝑥) = 145
13𝑥 + 𝑥 2 + 11𝑥 + 𝑥 2 = 145
2𝑥 2 + 24𝑥 − 145 = 0
Dari kalkulator
𝑥 = 4.416 dan 𝑥 = 16.416 (ditolak)
Nilai 𝑥 ialah 4.416
6
Soalan 3
Sekumpulan pelajar daripada kelas 4A ingin membeli hadiah kepada guru mereka. Kos hadiah
tersebut ialai RM 60 dan ia akan ditanggung secara rata antara pelajar. Terdapat 4 pelajar dari kelas
4B juga ingin menanggung kos tersebut, menjadikan semua pelajar akan membayar RM 0.50 lebih
kurang. Cari jumlah keseluruhan perlajar daripada kelas 4A dan 4B yang menanggung kos hadiah
tersebut.
A group of students from class 4A wanted to buy a gift for their teacher. The cost of present is RM 60
and will be shared equally by the group of students. There are 4 students from class 4B who wanted
to share the cost of present as well. Each students will pay RM 0.50 less. Determine the total number
of students sharing the present.
60
60
−
= 0.5
𝑥 𝑥+4
60 −
60𝑥
= 0.5𝑥
𝑥+4
60(𝑥 + 4) − 60𝑥 = 0.5𝑥(𝑥 + 4)
60𝑥 + 240 − 60𝑥 = 0.5𝑥 2 + 2𝑥
0.5𝑥 2 + 2𝑥 − 240 = 0
(𝑥 − 20)(𝑥 + 24 ) = 0
𝑥 = 20 dan 𝑥 = −24 (ditolak)
Keseluruhan pelajar di 4A dan 4B ialah 20 pelajar
7