Mehlala ea Lipotso Puisano Ho rarolla Mathata ka Mesebetsi ea Quadratic
Sehloohong sena, re tla ithuta mokhoa oa ho rarolla mathata ka ho sebelisa mesebetsi ea quadratic ka ho fana ka mehlala le mehato e qaqileng ea puisano. Mosebetsi oa quadratic ke mosebetsi oa polynomial oa boemo ba bobeli o nang le sebopeho se akaretsang \( ax^2 + bx + c \), moo \( a \), \( b \), le \( c \) e leng li-constant le \( a \neq 0 \). Mesebetsi ea quadratic maemong a fapaneng hangata e hlaha fisiks, moruo le boenjiniere, e leng se etsang hore e be sehlooho sa bohlokoa haholo ho se tseba.
A re qaleng ka ho buisana ka mehopolo e meng ea motheo ebe re tla kena mathateng a mang a mohlala.
Mehopolo ea Motheo ea Mesebetsi ea Quadratic
1. Sebopeho se Akaretsang: Mosebetsi wa quadratic o hlaloswa jwalo ka \( f(x) = ax^2 + bx + c \).
2. Metso e Sekwere: Metso ea equation ea quadratic \( ax^2 + bx + c = 0 \) e ka fumanoa ho sebelisoa foromo ea quadratic, e leng:
\[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
\]
3. Khethollo: Khethollo ea equation ea quadratic ke \( D = b^2 – 4ac \). Boleng ba khethollo bo bontša mofuta oa metso ea equation ea quadratic:
– Haeba \( D > 0 \), e na le metso e 'meli e fapaneng ea 'nete.
– Haeba \( D = 0 \), e na le motso o le mong wa nnete (motso o mobedi).
– Haeba \( D < 0 \), e na le metso e 'meli e kopaneng e rarahaneng. 4. Ntlheng ea Parabola: Likhokahano tsa ntlha ea parabola e entsoeng ke mosebetsi oa quadratic li ka fumanoa ho sebelisoa mokhoa ona: \[ x = -\frac{b}{2a} \] Bakeng sa boleng ba \( y \) ho ntlha ea vertex, e ka baloa ka ho nkela \( x \) sebaka mosebetsing oa quadratic.
– Haeba \( a < 0 \), parabola e buleha ho ya tlase. Ka mehopolo ena yohle ya motheo kelellong, ha re boneng hore na re ka e sebedisa jwang ho rarolla mathata. Mohlala Bothata ba 1: Ho Fumana Metso ya Mosebetsi wa Quadratic Bothata: Fumana metso ya equation ya quadratic \( 2x^2 - 3x - 2 = 0 \). Tharollo: Ho fumana metso ya equation ya quadratic, re ka sebedisa foromo ya quadratic. Mehato ke e latelang: 1. Hlalosa di-coefficients \( a \), \( b \), le \( c \): \[ a = 2, \quad b = -3, \quad c = -2 \] 2. Bala karohano: \[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25 \] 3. Kaha \( D > 0 \), re tla ba le metso e mmedi ya nnete e fapaneng. Tswela pele ka ho bala metso ena:
\[
x_{1,2} = \frac{-(-3) \pm \sqrt{25}}{2 \cdot 2} = \frac{3 \pm 5}{4}
\]
4. Bala boleng bo habeli ba \( x \):
\[
x_1 = \frac{3 + 5}{4} = 2 \quad \text{and} \quad x_2 = \frac{3 – 5}{4} = -\frac{1}{2}
\]
Kahoo, metso ea equation \( 2x^2 – 3x – 2 = 0 \) ke \( x = 2 \) le \( x = -\frac{1}{2} \).
Mohlala oa Potso ea 2: Ho Fumana Likhokahano tsa Ntlheng ea Parabola
Potso:
Fumana likhokahano tsa vertex ea mosebetsi oa quadratic \( f(x) = 3x^2 – 6x + 2 \).
Puisano:
Ho fumana likhokahano tsa tlhōrō, sebelisa foromo ea khokahano ea tlhōrō:
1. Hlalosa di-coefficient \( a \) le \( b \):
\[
a = 3, \quad b = -6
\]
2. Bala \( x \) holimo:
\[
x = -\frac{b}{2a} = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1
\]
3. Bala \( y \) ka ho kenya \( x = 1 \) sebakeng sa mosebetsi \( f(x) \):
\[
f(1) = 3(1)^2 – 6(1) + 2 = 3 – 6 + 2 = -1
\]
Kahoo, dikhokahano tsa vertex tsa mosebetsi \( f(x) = 3x^2 – 6x + 2 \) ke \( (1, -1) \).
Mohlala oa Potso ea 3: Ho Fumana Tataiso ea ho Bula ea Parabola
Potso:
Fumana tataiso ea ho buloa ha parabola ea mosebetsi oa quadratic \( f(x) = -x^2 + 4x – 7 \).
Puisano:
Ho fumana tataiso ea ho buloa ha parabola, re sheba feela letšoao la coefficient \( a \):
1. Hlalosa coefficient \( a \):
\[
a = -1
\]
2. Kaha \( a < 0 \), parabola e buleha ho ya tlase. Kahoo, tataiso ya ho buleha ha parabola ya mosebetsi \( f(x) = -x^2 + 4x - 7 \) e ya tlase. Mohlala wa 4: Ho sebedisa Mesebetsi ya Quadratic Maemong a Bophelo ba Sebele
Potso: Bolo e lahlelwa fatshe ka equation ya quadratic \( h(t) = -5t^2 + 20t \), moo \( h \) e leng bophahamo ba bolo ka dimetara mme \( t \) e le nako ka metsotswana. Ho nka nako e kae hore bolo e fihle bophahamong ba yona bo phahameng, mme bophahamo ba yona bo phahameng ke bofe? Puisano: 1. Fumana nako eo bophahamo bo phahameng bo fihlellehang ka yona (dikhokahano tsa tlhoro): \[ a = -5, \quad b = 20 \] \[ t = -\frac{b}{2a} = -\frac{20}{2(-5)} = \frac{20}{10} = 2 \quad \text{seconds} \] 2. Bala bophahamo bo boholo ka ho kenya \( t \) sebakeng sa equation \( h(t) \): \[ h(2) = -5(2)^2 + 20(2) = -5(4) + 40 = -20 + 40 = 20 \quad \text{meters} \] Kahoo, nako e nkiloeng ke bolo ho fihlela bophahamo bo boholo ke metsotsoana e 2, mme bophahamo ba yona bo boholo ke limithara tse 20. Sephetho Sehloohong sena, re buisane ka dikarolo tse fapaneng tsa bohlokwa tsa mesebetsi ya quadratic hammoho le mokhoa wa ho rarolla mathata a amanang le mesebetsi ya quadratic ka mehlala e mmalwa. Ho buisana ka metso ea li-quadratic equation, ho fumana li-coordinate tsa vertex, ho fumana tataiso ea ho buloa ha parabola, le ho sebelisa mesebetsi ea quadratic maemong a lefats'e la 'nete, joalo ka ho hlalosa motsamao oa lintho. Ka kutloisiso e tiileng ea likhopolo tsena tsa motheo, u tla khona ho atamela mathata a fapaneng a lipalo le saense a amanang le mesebetsi ea quadratic ka kholiseho e kholoanyane. Mesebetsi ea quadratic ha e bohlokoa feela khopolo-taba empa hape e na le thuso e kholo lits'ebetsong tsa lefats'e la 'nete le ho rarolla mathata masimong a mangata.