Su'aalo tusaale ah oo ka hadlaya Astaamaha Hawlaha Laba-jibbaaran

Su'aalo Tusaale ah oo Ka Hadlaya Astaamaha Hawlaha Laba-jibbaaran

Shaqada labajibbaaran waa mowduuc muhiim ah oo ku jira xisaabta oo inta badan lagala kulmo manhajka dugsiga sare. Qaabka guud ee shaqada labajibbaaran waa \( f(x) = ax^2 + bx + c \), halkaas oo \( a \), \( b \), iyo \( c \) ay yihiin joogtooyin leh \( a \neq 0 \). Doodda astaamaha shaqooyinka labajibbaaran waxaa ka mid ah dhinacyo kala duwan sida dhidibka isku-dhafka, leexashada, qiimaha ugu badan ama ugu yar, iyo jihada parabola. Maqaalkani wuxuu ka hadli doonaa dhowr dhibaato oo tusaale ah iyo xalalkooda si loo fahmo astaamaha shaqooyinka labajibbaaran.

1. Su'aal: Go'aaminta dhidibka Isku-dhafka iyo Lafaha

Tusaale ahaan dhibaatooyinka:
Marka la eego shaqo labajibbaaran \( f(x) = 2x^2 – 4x + 1 \). Go'aami dhidibka isku-dheelitirka iyo geeska shaqada.

Dood:
Si loo go'aamiyo dhidibka isku-dheelitirka shaqada labajibbaaran \( ax^2 + bx + c \), waxaan isticmaalnaa qaacidada:
\[ x = -\frac{b}{2a} \]

Shaqada la bixiyay \( f(x) = 2x^2 – 4x + 1 \), qiimayaasha \( a = 2 \) iyo \( b = -4 \). Qiimahan ku beddel qaacidada:
\[ x = -\frac{-4}{2 \cdot 2} \]
\[ x = \frac{4}{4} \]
\[ x = 1 \]

Markaa, dhidibka isku-dheelitirka shaqada waa \( x = 1 \).

Si aan u helno geeska, waxaan qiimaha dhidibka isku-dheelitirka ku beddeleynaa shaqada:
\[ f(1) = 2(1)^2 – 4(1) + 1 \]
\[ f(1) = 2 – 4 + 1 \]
\[ f(1) = -1 \]

Markaa, geeska shaqadu waa \( (1, -1) \).

2. Su'aal: Go'aaminta Jihada Parabola

Tusaale ahaan dhibaatooyinka:
Go'aami jihada parabola ee shaqada labajibbaaran \( f(x) = -3x^2 + 6x – 2 \).

Dood:
Jihada parabola ee shaqada afargeeslaha ah waxaa lagu go'aamiyaa qiimaha isku-dhafka \( a \).

– Haddii \( a > 0 \), parabola-du waxay kor u furmeysaa.
– Haddii \( a < 0 \), parabola-du waxay hoos u furmeysaa. Shaqada la bixiyay \( f(x) = -3x^2 + 6x - 2 \), qiimaha \( a = -3 \). Tan iyo \( a < 0 \), parabola-du waxay hoos u furmeysaa. 3. Dhibaato: Helitaanka Xididdada Shaqada Afar-geesoodka ah Tusaale ahaan Dhibaato: Soo hel xididdada shaqada afar-geesoodka ah \( f(x) = x^2 - 5x + 6 \). Xalka: Xididdada shaqada afar-geesoodka ah waxaa laga heli karaa iyadoo la isku darayo ama la isticmaalayo qaacidada afar-geesoodka ah. Waxaan ku dari doonnaa: \[ x^2 - 5x + 6 = 0 \] Soo hel laba tiro oo isku dhufanaya si aad u siiso 6 oo ku dar bixinta -5. Tirooyinkani waa -2 iyo -3. \[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \] Sidaa darteed, xididdadu waa: \[ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 \] \[ x = 2 \quad \text{or} \quad x = 3 \] 4. Su'aal: Qiimaha ugu badan ama ugu yar Tusaale Su'aal: Go'aami qiimaha ugu yar ee shaqada labajibbaaran \( f(x) = 2x^2 - 4x + 5 \).

Dood: Si loo go'aamiyo qiimaha ugu yar ee shaqada labajibbaaran \( ax^2 + bx + c \), waxaan u baahanahay inaan hubinno in parabola uu kor u furmayo ama hoos u furmayo. Haddii \( a > 0 \), parabola uu kor u furmayo wuxuuna leeyahay qiime ugu yar; haddii \( a < 0 \), parabola uu hoos u furmayo wuxuuna leeyahay qiime ugu badan. Shaqada la bixiyay \( f(x) = 2x^2 - 4x + 5 \), qiimaha \( a = 2 \), sidaa darteed parabola wuxuu kor u furmayaa wuxuuna leeyahay qiime ugu yar. Qiimaha ugu yar wuxuu ka dhacaa geeska. Horey ayaan u naqaanay dhidibka isku-dhafka \( x = -\frac{b}{2a} \). Shaqadan: \[ x = -\frac{-4}{2 \cdot 2} \] \[ x = \frac{4}{4} \] \[ x = 1 \] Ku beddel \( x = 1 \) shaqada si aad u hesho qiimaha ugu yar: \[ f(1) = 2(1)^2 - 4(1) + 5 \] \[ f(1) = 2 - 4 + 5 \] \[ f(1) = 3 \] Sidaa darteed, qiimaha ugu yar ee shaqada waa 3. 5. Su'aal: Garaafka Hawlaha Labajibbaaran Su'aal Tusaale ah: Samee garaaf shaqada labajibbaaran \( f(x) = -x^2 + 4x - 3 \). Dood: Si loo sawiro shaqada labajibbaaran, waxaan u baahanahay inaan helno dhowr astaamood oo muhiim ah sida dhidibka isku-dhafka, geeska, iyo xididdada shaqada, iyo sidoo kale jihada parabola. 1. Dhidibka Isku-dhafka: \[x = -\frac{b}{2a} \] Shaqada \( f(x) = -x^2 + 4x - 3 \), qiimayaasha \( a = -1 \) iyo \( b = 4 \). \[ x = -\frac{4}{2(-1)} \] \[ x = -\frac{4}{-2} \] \[ x = 2 \] 2. Dooxada: Ku beddel \( x = 2 \) shaqada si aad u hesho geeska: \[ f(2) = -(2)^2 + 4(2) - 3 \] \[ f(2) = -4 + 8 - 3 \] \[ f(2) = 1 \] Dooxada waa \( (2, 1) \). 3. Jihada Parabola: Maadaama \( a = -1 \), parabola-du waxay hoos u furmeysaa. 4. Xididdada Shaqooyinka Laba-jibbaaran: \[ -x^2 + 4x - 3 = 0 \] Waxaan isticmaali karnaa qaacidada laba-jibbaaran: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Wixii \( a = -1 \), \( b = 4 \), iyo \( c = -3 \): \[ x = \frac{-4 \pm \sqrt{4^2 - 4(-1)(-3)}}{2(-1)} \] \[ x = \frac{-4 \pm \sqrt{16 - 12}}{-2} \] \[ x = \frac{-4 \pm \sqrt{4}}{-2} \] \[ x = \frac{-4 \pm 2}{-2} \] Laba xal: \[ x = \frac{-4 + 2}{-2} = 1 \] \[ x = \frac{-4 - 2}{-2} = 3 \] Xididdada waa \( x = 1 \) iyo \( x = 3 \). Macluumaadkan oo dhan, waxaan ku sawiri karnaa shaqada labajibbaaran. Parabola-gan wuxuu leeyahay leexasho at \( (2, 1) \), wuxuu u furmayaa hoos, wuxuuna leeyahay xidido at \( x = 1 \) iyo \( x = 3 \). Gunaanad Tusaalooyinka laga wada hadlay, waxaan si fiican u fahmi karnaa astaamaha kala duwan ee shaqada labajibbaaran. Aqoonta sida loo go'aamiyo dhidibka isku-dhafka, leexashada, qiimaha ugu badan ama ugu yar, jihada parabola, iyo xididdada shaqada labajibbaaran waa lama huraan si loo qeexo qaabka iyo sifooyinka parabola. Faham wanaagsan oo ku saabsan fikradahan ayaa siin doona aasaas adag ardayda si ay u sahamiyaan mowduucyo horumarsan oo ku saabsan xisaabta.