Mehlala ea Lipotso tse Buisanang ka Mesebetsi ea Quadratic
Mesebetsi ea quadratic ke sehlooho sa bohlokoa se tšohloang lipalo, haholo-holo lipalo tsa bobeli. Mosebetsi ona o na le sebopeho se akaretsang \( f(x) = ax^2 + bx + c \), moo \(a\), \(b\), le \(c\) e leng li-constant tse nang le \(a \neq 0\). Sengoloa sena se tla tšohla mehlala e 'maloa ea mathata a amanang le mesebetsi ea quadratic hammoho le litlhaloso tse qaqileng ho thusa baithuti ho utloisisa mohopolo ona hamolemo.
1. Ho Fumana Metso ea Mosebetsi oa Quadratic
Potso ea 1: Fumana metso ea mosebetsi o latelang oa quadratic:
\[ f(x) = 2x^2 – 3x – 5 \]
Puisano:
Ho fumana metso ea mosebetsi oa quadratic, re ka sebelisa foromo ea quadratic, e leng:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Mosebetsing oa quadratic \( f(x) = 2x^2 – 3x – 5 \), re ka khetholla boleng ba \(a\), \(b\), le \(c\):
– \( a = 2 \)
– \( b = -3 \)
– \( c = -5 \)
Mehato ke e latelang:
1. Fumana khetho (\( \Delta \)):
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-3)^2 – 4(2)(-5) \]
\[ \Delta = 9 + 40 \]
\[ \Delta = 49 \]
2. Sebelisa foromo ea quadratic ho fumana metso:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 2} \]
\[ x = \frac{3 \pm 7}{4} \]
Kahoo re fumana litharollo tse peli:
\[ x_1 = \frac{3 + 7}{4} = \frac{10}{4} = 2.5 \]
\[ x_2 = \frac{3 – 7}{4} = \frac{-4}{4} = -1 \]
Kahoo metso ea mosebetsi ke \( x = 2.5 \) le \( x = -1 \).
2. Ho Fumana Dintlha tsa Mosebetsi wa Quadratic
Potso ea 2: Fumana ntlha ea khutlo ea mosebetsi o latelang oa quadratic:
\[ g(x) = -x^2 + 4x – 3 \]
Puisano:
Sehlohlolo sa mosebetsi wa quadratic se ka fumaneha ka ho sebedisa foromo:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
Mosebetsing wa quadratic \( g(x) = -x^2 + 4x – 3 \), re ka hlwaya boleng ba \(a\), \(b\), le \(c\):
– \( a = -1 \)
– \( b = 4 \)
– \( c = -3 \)
Mehato ke e latelang:
1. Fumana boleng ba vertex:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-4}{2(-1)} \]
\[ x_{\text{vertex}} = \frac{-4}{-2} \]
\[ x_{\text{vertex}} = 2 \]
2. Fumana boleng ba \( y \) ka ho kenya \( x_{\text{vertex}} \) sebakeng sa mosebetsi:
\[ y_{\text{vertex}} = g(2) \]
\[ y_{\text{vertex}} = – (2)^2 + 4(2) – 3 \]
\[ y_{\text{vertex}} = -4 + 8 – 3 \]
\[ y_{\text{vertex}} = 1 \]
Kahoo ntlha ya tshebetso ke \( (2, 1) \).
3. Ho taka Grafo ea Mesebetsi ea Quadratic
Potso ea 3: Thala kerafo ea mosebetsi o latelang oa quadratic:
\[ h(x) = x^2 – 2x – 3 \]
Puisano:
Pele re taka kerafo ea mosebetsi oa quadratic, re hloka ho tseba lintlha tse 'maloa tsa bohlokoa, tse kang metso, vertex, le tataiso ea parabola.
Ho Fumana Metso
Re ka sebelisa foromo ea quadratic ho fumana metso ea \( h(x) = x^2 – 2x – 3 \):
\[a = 1 \]
\[b = -2 \]
\[ c = -3 \]
1. Bala karohano:
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-2)^2 – 4(1)(-3) \]
\[ \Delta = 4 + 12 \]
\[ \Delta = 16 \]
2. Bala metso:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-2) \pm \sqrt{16}}{2(1)} \]
\[ x = \frac{2 \pm 4}{2} \]
Kahoo re fumana litharollo tse peli:
\[ x_1 = \frac{2 + 4}{2} = \frac{6}{2} = 3 \]
\[ x_2 = \frac{2 – 4}{2} = \frac{-2}{2} = -1 \]
Ho Fumana Ntlha ea Tlhōrō
3. Sebelisa foromo ea vertex:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-(-2)}{2(1)} \]
\[ x_{\text{vertex}} = 1 \]
4. Bala boleng ba \( y \):
\[ y_{\text{vertex}} = h(1) \]
\[ y_{\text{vertex}} = (1)^2 – 2(1) – 3 \]
\[ y_{\text{vertex}} = 1 – 2 – 3 \]
\[ y_{\text{vertex}} = -4 \]
Kahoo ntlha e ka hodimo ke \( (1, -4) \).
Likerafo tsa ho taka
– Metso e ho \( x = 3 \) le \( x = -1 \).
– Karolo e ka holimo e ho \( (1, -4) \).
– Kaha \( a > 0 \), parabola e buleha hodimo.
Rala lintlha tsena tsa bohlokoa ka kerafong 'me u thale parabola e fetang ho tsona.
Ka ho utloisisa metso, ntlha e ka holimo, le tataiso ea parabola, re ka taka kerafo e phethahetseng haholoanyane ea mosebetsi oa quadratic.
Qetello
Mosebetsi oa quadratic ke mohopolo oa motheo lipalo tse nang le lits'ebetso tse pharaletseng. Ho utloisisa mesebetsi ea quadratic ho re thusa ho tebisa kutloisiso ea rona ea likhopolo tse ling lipalo le mahlale a sebelisitsoeng. Ka ho ikoetlisa ka mehlala le ho utloisisa mehato ea ho e rarolla, ho tšeptjoa hore kutloisiso ea rona ea mesebetsi ea quadratic e tla tebisa le ho sebetsa haholoanyane.
