Tambayoyi Misali Game da Ayyukan Quadratic
Ayyukan quadratic muhimmin batu ne da aka tattauna a fannin lissafi, musamman a fannin lissafi na sakandare. Wannan aikin yana da tsari na gaba ɗaya \( f(x) = ax^2 + bx + c \), inda \(a\), \(b\), da \(c\) suke da madaidaitan ma'auni tare da \(a \neq 0\). Wannan labarin zai tattauna misalai da dama na matsaloli da suka shafi ayyukan quadratic tare da cikakkun bayanai don taimakawa ɗalibai su fahimci wannan ra'ayi sosai.
1. Tantance Tushen Aikin Huɗu
Tambaya ta 1: Nemo tushen aikin quadratic mai zuwa:
\[ f(x) = 2x^2 – 3x – 5 \]
Tattaunawa:
Domin nemo tushen aikin quadratic, za mu iya amfani da dabarar quadratic, wato:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
A cikin aikin quadratic \( f(x) = 2x^2 – 3x – 5 \), za mu iya gano ƙimar \(a\), \(b\), da \(c\):
– \( a = 2 \)
– \( b = -3 \)
– \( c = -5 \)
Matakan sune kamar haka:
1. Nemo mai nuna bambanci (\( \Delta \)):
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-3)^2 – 4(2)(-5) \]
\[ \Delta = 9 + 40 \]
\[ \Delta = 49 \]
2. Yi amfani da dabarar quadratic don nemo tushen:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 2} \]
\[ x = \frac{3 \pm 7}{4} \]
Don haka muna samun mafita guda biyu:
\[ x_1 = \frac{3 + 7}{4} = \frac{10}{4} = 2.5 \]
\[ x_2 = \frac{3 – 7}{4} = \frac{-4}{4} = -1 \]
Don haka tushen aikin shine \( x = 2.5 \) da \( x = -1 \).
2. Tantance Zagaye na Aikin Huɗu
Tambaya ta 2: Kayyade gefen aikin quadratic mai zuwa:
\[ g(x) = -x^2 + 4x – 3 \]
Tattaunawa:
Ana iya tantance kusurwar aikin quadratic ta amfani da dabarar:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
A cikin aikin quadratic \( g(x) = -x^2 + 4x – 3 \), za mu iya gano ƙimar \(a\), \(b\), da \(c\):
– \( a = -1 \)
– \( b = 4 \)
– \( c = -3 \)
Matakan sune kamar haka:
1. Nemo ƙimar \( x \) na vertex ɗin:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-4}{2(-1)} \]
\[ x_{\text{vertex}} = \frac{-4}{-2} \]
\[ x_{\text{vertex}} = 2 \]
2. Nemo ƙimar \( y \) ta hanyar maye gurbin \( x_{\text{vertex}} \) a cikin aikin:
\[ y_{\text{vertex}} = g(2) \]
\[ y_{\text{vertex}} = – (2)^2 + 4(2) – 3 \]
\[ y_{\text{vertex}} = -4 + 8 – 3 \]
\[ y_{\text{vertex}} = 1 \]
Don haka gefen aikin shine \( (2, 1) \).
3. Zana Jadawalin Aiki na Quadratic
Tambaya ta 3: Zana jadawalin aikin kwata-kwata mai zuwa:
\[ h(x) = x^2 – 2x – 3 \]
Tattaunawa:
Kafin mu zana jadawalin aikin quadratic, muna buƙatar sanin muhimman abubuwa da dama, kamar tushen, gefen, da kuma alkiblar parabola.
Tantance Tushen
Za mu iya amfani da dabarar quadratic don nemo tushen \( h(x) = x^2 – 2x – 3 \):
\[ a = 1 \]
\[ b = -2 \]
\[ c = -3 \]
1. Lissafa mai nuna bambanci:
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-2)^2 – 4(1)(-3) \]
\[ \Delta = 4 + 12 \]
\[ \Delta = 16 \]
2. Lissafa tushen:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-2) \pm \sqrt{16}}{2(1)} \]
\[ x = \frac{2 \pm 4}{2} \]
Don haka muna samun mafita guda biyu:
\[ x_1 = \frac{2 + 4}{2} = \frac{6}{2} = 3 \]
\[ x_2 = \frac{2 – 4}{2} = \frac{-2}{2} = -1 \]
Ƙayyade Matsakin Kololuwa
3. Yi amfani da dabarar vertex:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-(-2)}{2(1)} \]
\[ x_{\text{vertex}} = 1 \]
4. Lissafa ƙimar \( y \):
\[ y_{\text{vertex}} = h(1) \]
\[ y_{\text{vertex}} = (1)^2 – 2(1) – 3 \]
\[ y_{\text{vertex}} = 1 – 2 – 3 \]
\[ y_{\text{vertex}} = -4 \]
Don haka gefen shine \( (1, -4) \).
Zane-zanen Zane
– Tushen suna a \( x = 3 \) da \( x = -1 \).
– Ƙetaren yana a \( (1, -4) \).
– Tunda \( a > 0 \), parabola yana buɗewa sama.
Zana waɗannan muhimman abubuwan a kan jadawali sannan ka zana parabola da ke ratsa su.
Ta hanyar fahimtar tushen, gefen, da kuma alkiblar parabola, za mu iya zana cikakken jadawalin aikin quadratic.
Kammalawa
Aikin quadratic wani muhimmin ra'ayi ne a fannin lissafi wanda ke da amfani mai yawa. Fahimtar ayyukan quadratic yana taimaka mana mu zurfafa fahimtarmu game da wasu ra'ayoyi a fannin lissafi da kimiyyar da aka yi amfani da su. Ta hanyar yin aiki da misalai da fahimtar matakan da za a bi don magance su, ana fatan fahimtarmu game da ayyukan quadratic za ta zurfafa kuma ta zama mai amfani.
