Tambayoyi Misali Game da Halayen Ayyukan Quadratic
Aikin quadratic muhimmin batu ne a fannin lissafi wanda galibi ake fuskanta a manhajar makarantar sakandare. Tsarin aikin quadratic gabaɗaya shine \( f(x) = ax^2 + bx + c \), inda \( a \), \( b \), da \( c \) sune madaidaitan abubuwa tare da \( a \neq 0 \). Tattaunawar halayen ayyukan quadratic ta haɗa da fannoni daban-daban kamar axis of symmetry, vertex, maximum ko minimum values, da kuma alkiblar parabola. Wannan labarin zai tattauna misalai da dama na matsaloli da mafita don fahimtar halayen ayyukan quadratic.
1. Tambaya: Tantance Axis na Similar da Vertex
Misalin matsalar:
An ba da aikin kwata-kwata \( f(x) = 2x^2 – 4x + 1 \). Kayyade axis na daidaito da kuma gefen aikin.
Tattaunawa:
Domin tantance axis na daidaito na aikin quadratic \( ax^2 + bx + c \), muna amfani da dabarar:
\[ x = -\frac{b}{2a} \]
A cikin aikin da aka bayar \( f(x) = 2x^2 – 4x + 1 \), ƙimar \( a = 2 \) da \( b = -4 \). Sauya waɗannan ƙimar zuwa dabarar:
\[ x = -\frac{-4}{2 \cdot 2} \]
\[ x = \frac{4}{4} \]
\[x = 1 \]
Don haka, axis na daidaiton aikin shine \( x = 1 \).
Domin nemo gefen, muna maye gurbin ƙimar axis na daidaitawa zuwa aikin:
\[ f(1) = 2(1)^2 – 4(1) + 1 \]
\[ f(1) = 2 – 4 + 1 \]
\[ f(1) = -1 \]
Don haka, gefen aikin shine \( (1, -1) \).
2. Tambaya: Tantance Alkiblar Parabola
Misalin matsalar:
Kayyade alkiblar parabola na aikin quadratic \( f(x) = -3x^2 + 6x – 2 \).
Tattaunawa:
Alkiblar parabola na aikin quadratic ana tantance ta ne ta hanyar ƙimar coefficient \( a \).
– Idan \( a > 0 \), parabola yana buɗewa sama.
– Idan \( a < 0 \), parabola zai buɗe ƙasa.
A cikin aikin da aka bayar \( f(x) = -3x^2 + 6x - 2 \), ƙimar \( a = -3 \). Tunda \( a < 0 \), parabola yana buɗewa ƙasa.
3. Tambaya: Nemo Tushen Aikin Quadratic Misali Tambaya: Nemo tushen aikin quadratic \( f(x) = x^2 - 5x + 6 \).
Tattaunawa: Ana iya gano tushen aikin quadratic ta hanyar yin factoring ko amfani da dabarar quadratic. Za mu yi lissafinsa: \[ x^2 - 5x + 6 = 0 \] Nemo lambobi biyu waɗanda idan aka ninka su daidai da 6 kuma idan aka ƙara su daidai da -5. Lambobin sune -2 da -3.
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \] Saboda haka, tushen sune: \[ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 \] \[ x = 2 \quad \text{or} \quad x = 3 \] 4. Tambaya: Matsakaicin ko Mafi ƙarancin ƙima Misali Tambaya: Ƙayyade mafi ƙarancin ƙimar aikin quadratic \( f(x) = 2x^2 - 4x + 5 \).
Tattaunawa: Domin tantance mafi ƙarancin ƙimar aikin quadratic \( ax^2 + bx + c \), muna buƙatar duba ko parabola ɗin yana buɗewa sama ko ƙasa. Idan \( a > 0 \), parabola yana buɗewa sama kuma yana da mafi ƙarancin ƙima; idan \( a < 0 \), parabola yana buɗewa ƙasa kuma yana da matsakaicin ƙima.
A cikin aikin da aka bayar \( f(x) = 2x^2 - 4x + 5 \), ƙimar \( a = 2 \), don haka parabola yana buɗewa sama kuma yana da ƙimar mafi ƙaranci.
Mafi ƙarancin ƙima yana faruwa ne a lokacin kololuwar. Mun riga mun san axis na daidaito \( x = -\frac{b}{2a} \). Don wannan aikin: \[x = -\frac{-4}{2 \cdot 2} \] \[x = \frac{4}{4} \] \[x = 1 \] Sauya \(x = 1 \) cikin aikin don nemo mafi ƙarancin ƙima: \[f(1) = 2(1)^2 - 4(1) + 5 \] \[f(1) = 2 - 4 + 5 \] \[f(1) = 3 \] Saboda haka, mafi ƙarancin ƙimar aikin shine 3.
5. Tambaya: Jadawalin Aikin Quadratic Misali Tambaya: Ƙirƙiri jadawali na aikin quadratic \( f(x) = -x^2 + 4x - 3 \).
Tattaunawa: Domin zana jadawalin aikin quadratic, muna buƙatar nemo wasu muhimman halaye kamar axis of symmetry, vertex, da tushen aikin, da kuma alkiblar parabola.
1. Axis na Daidaitawa: \[x = -\frac{b}{2a} \] A cikin aikin \( f(x) = -x^2 + 4x - 3 \), ƙimar \( a = -1 \) da \( b = 4 \).
\[ x = -\frac{4}{2(-1)} \] \[ x = -\frac{4}{-2} \] \[ x = 2 \] 2. Vertex: Sauya \(x = 2 \) cikin aikin don nemo vertex: \[f(2) = -(2)^2 + 4(2) - 3 \] \[f(2) = -4 + 8 - 3 \] \[f(2) = 1 \] Vertex shine \( (2, 1) \).
3. Alkiblar Parabola: Tunda \( a = -1 \), parabola yana buɗewa ƙasa.
4. Tushen Aikin Quadratic: \[ -x^2 + 4x - 3 = 0 \] Za mu iya amfani da dabarar quadratic: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Ga \( a = -1 \), \( b = 4 \), da \( c = -3 \): \[ x = \frac{-4 \pm \sqrt{4^2 - 4(-1)(-3)}}{2(-1)} \] \[ x = \frac{-4 \pm \sqrt{16 - 12}}{-2} \] \[ x = \frac{-4 \pm \sqrt{4}}{-2} \] \[ x = \frac{-4 \pm 2}{-2} \] Magani guda biyu: \[ x = \frac{-4 + 2}{-2} = 1 \] \[ x = \frac{-4 - 2}{-2} = 3 \] Tushen su ne \( x = 1 \) da \( x = 3 \).
Da duk wannan bayanin, za mu iya zana aikin kwata-kwata. Wannan parabola yana da kusurwa a \( (2, 1) \), yana buɗewa ƙasa, tare da tushen a \( x = 1 \) da \( x = 3 \).
Kammalawa Ta hanyar misalan da aka tattauna, za mu iya fahimtar halaye daban-daban na ayyukan quadratic. Sanin yadda ake tantance axis na daidaitawa, vertex, matsakaicin ko mafi ƙarancin ƙima, alkiblar parabola, da kuma tushen aikin quadratic yana da matuƙar muhimmanci don bayyana siffa da halayen parabola. Fahimtar waɗannan ra'ayoyi sosai zai samar da tushe mai ƙarfi ga ɗalibai don bincika ƙarin batutuwa a fannin lissafi.
