Ho Setšoantšo sa Mesebetsi ea Quadratic: Tataiso e Felletseng
Kerafo ea mosebetsi oa quadratic ke e 'ngoe ea lihlooho tsa motheo lipalo, haholo-holo algebra le geometry ea analytic. Mosebetsi oa quadratic o hlalositsoeng ka mokhoa oa \( f(x) = ax^2 + bx + c \), moo \( a \), \( b \), le \( c \) e leng li-constant, o hlahisa kerafo ea parabolic. Sengoloa sena se tla hlalosa ka botlalo ka kerafo ea mosebetsi oa quadratic, ho qala ka sebopeho sa parabola, mokhoa oa ho e taka, le lits'ebetso tse sebetsang lefatšeng la 'nete.
1. Sebopeho se Akaretsang sa Mosebetsi oa Quadratic
Mosebetsi oa quadratic o na le sebopeho se akaretsang se latelang:
\[ f(x) = selepe^2 + bx + c \]
Mona, \( a \), \( b \), le \( c \) ke di-constant, moo:
– \( a \) ke coefficient ya quadratic e etsang qeto ya tataiso le bophara ba parabola.
– \( b \) ke coefficient e otlolohileng e amang boemo ba axis ea symmetry ea parabola.
– \( c \) ke ntho e sa fetoheng e khethollang ntlha eo parabola e kopanang le mothapo wa y.
2. Thepa ea Likerafo tsa Mesebetsi ea Quadratic
Kerafo ea mosebetsi oa quadratic ke parabola e nang le litšobotsi tse 'maloa tsa bohlokoa, e leng:
– Tsela ea Parabola: E khethoa ke letšoao la coefficient \( a \).
– Haeba \( a > 0 \), parabola e buleha hodimo.
– Haeba \( a < 0 \), parabola e buleha ho ya tlase.
- Ntlheng ea Parabola: Ntlheng ea parabola e ka emeloa ke li-coordinates \((h, k)\), moo: \[ h = -\frac{b}{2a} \] \[ k = f(h) = f\left(-\frac{b}{2a}\right) \] Ntlheng ena ke ntlha e phahameng ka ho fetisisa kapa e tlase ka ho fetisisa ea parabola ho latela tataiso ea parabola. - Axis of Symmetry: Mola o otlolohileng o fetang vertex ea parabola mme o e arola ka litšoantšo tse peli tsa seipone, ka equation: \[ x = -\frac{b}{2a} \] - Ntlha ea Phapang le Axis: Ntlha ea phapang ea parabola le x-axis (metso ea equation ea quadratic) e fumanoa ka ho rarolla equation ea quadratic \( ax^2 + bx + c = 0 \) ho sebelisoa foromo ea quadratic: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Ntlha ea phapang le y-axis ke ha \( x = 0 \), ke hore \( y = c \). 3. Ho graphing Quadratic Functions Mohato oa 1: Ho fumana Vertex Coordinates Ho graph mosebetsi oa quadratic, mohato oa pele ke ho fumana li-vertex coordinates \((h, k)\) ho sebelisoa foromo e hlalositsoeng. Mohato oa 2: Ho Fumana Lintlha tse Eketsehileng Ntle le ntlha ea vertex, re hloka lintlha tse 'maloa tse eketsehileng ho taka parabola ka nepo haholoanyane. Lintlha tsena li ka fumanoa ka ho khetha boleng ba x le ho bala boleng ba y bo tsamaellanang. Mohato oa 3: Thala Axis ea Symmetry Thala axis ea symmetry ea parabola ho pholletsa le ntlha \( x = -\frac{b}{2a} \). Mohato oa 4: Thala Lintlha le Sebopeho sa Parabola Thala lintlha tsohle tse baliloeng ho kenyeletsoa le ntlha ea vertex le lintlha tse ling. Ebe, taka kobeho ea parabola ho pholletsa le lintlha tsena, u etse bonnete ba hore e lekana le axis ea symmetry.
4. Ts'ebeliso ea Mesebetsi ea Quadratic Mesebetsi ea Quadratic le li-graph tsa eona li na le lits'ebetso tse fapaneng bophelong ba letsatsi le letsatsi le thutong. Tse ling tsa lits'ebetso tsena ke tsena: 4.1. Fisiks Fisiks, mesebetsi ea quadratic hangata e hlaha liekateng tse amanang le motsamao oa parabolic, joalo ka tsela ea projectile. Mohlala, tsela ea ntho e lahleloang tlas'a tšusumetso ea matla a khoheli e latela kerafo ea mosebetsi oa quadratic, moo vertex e leng ntlha e phahameng ka ho fetisisa e fihletsoeng ke ntho. 4.2. Moruo Moruong, mesebetsi ea quadratic e sebelisoa ho etsa mohlala oa litšenyehelo le chelete e kenang. Mohlala, litšenyehelo tsohle \( C(x) \) hangata li hlahisoa ka mokhoa oa quadratic, moo \( x \) e leng palo ea li-unit tse hlahisoang kapa tse rekisoang. Mesebetsi ea quadratic e ka boela ea sebelisoa ho fumana lintlha tsa mantlha tsa khokahano lipakeng tsa mesebetsi e 'meli ea litšenyehelo kapa chelete e kenang bakeng sa tlhahlobo ea phaello. 4.3. Boenjiniere Boenjiniereng, mesebetsi ea quadratic e sebelisoa tlhahlobong ea sebopeho le ntlafatsong. Mohlala, moralong oa borokho kapa moaho, sebopeho sa parabola sa mosebetsi oa quadratic se ka thusa ho fumana kobeho e ntle ka ho fetisisa e fokotsang tšebeliso ea thepa ha ka nako e ts'oanang e boloka matla a sebopeho.
4.4. Lipalopalo Lipalong, ho khutlela morao ha quadratic ho sebelisoa ho fumana kamano e ntle ka ho fetisisa lipakeng tsa lihlopha tse peli tsa data. Mesebetsi ea quadratic e sebelisoa ho etsa mohlala oa ho itšetleha ho seng ho otlolohileng ho ke keng ha sebetsanoa le ho khutlela morao ho otlolohileng. 5. Mohlala oa Mathata le Litharollo Mohlala oa Bothata 1 Thala kerafo ea mosebetsi o latelang oa quadratic: \[ f(x) = 2x^2 - 4x + 1 \] Mohato oa 1: Fumana likhokahano tsa vertex \[ h = -\frac{b}{2a} = -\frac{-4}{2(2)} = 1 \] \[ k = f(1) = 2(1)^2 - 4(1) + 1 = -1 \] Kahoo, likhokahano tsa vertex ke (1, -1). Mohato oa 2: Fumana lintlha tse ling Mohlala, ha re khetheng \( x = 0 \) le \( x = 2 \): \[ f(0) = 2(0)^2 - 4(0) + 1 = 1 \] \[ f(2) = 2(2)^2 - 4(2) + 1 = 1 \] Mohato oa 3: Thala axis of symmetry axis of symmetry ke mola o otlolohileng \( x = 1 \). Mohato oa 4: Thala lintlha 'me u thale parabola Thala lintlha (0,1), (1,-1), le (2,1). Thala curve ea parabola e lekanang ka lintlha tsena. 6. Qetello Ho grapha mosebetsi oa quadratic ke sesebelisoa sa bohlokoa lipalo tse nang le mefuta e mengata ea lits'ebetso tsa lefats'e la nnete, ho tloha fisiks ho ea moruong le boenjiniere. Kutloisiso e felletseng ea parabola, mokhoa oa ho e grapha, le thepa e tsamaeang le eona e fana ka motheo o tiileng oa tlhahlobo e eketsehileng. Ka ho latela mehato e tšohliloeng le ho utloisisa litšobotsi tsa parabola, mang kapa mang a ka taka le ho sekaseka kerafo ea mosebetsi oa quadratic habonolo.
