Cov Lus Nug Piv Txwv Sib Tham Txog Kev Tsim Cov Kev Ua Haujlwm Quadratic
Kev tsim cov quadratic functions yog ib qho tseem ceeb hauv algebra uas feem ntau tshwm sim hauv cov kev kawm lej theem nrab thiab theem siab. Kev nkag siab txog quadratic functions yog qhov tseem ceeb vim tias lawv feem ntau siv rau hauv ntau yam ntsiab lus, xws li kev tshuaj xyuas cov ntaub ntawv, kev ua qauv physics, thiab kev lag luam. Hauv tsab xov xwm no, peb yuav tham txog ntau yam teeb meem piv txwv thiab yuav daws lawv li cas los tsim cov quadratic functions.
Kev Nkag Siab Txog Cov Kev Ua Haujlwm Quadratic
Ib qho quadratic function yog ib qho second-degree polynomial function uas muaj daim ntawv dav dav:
\[ f(x) = ax^2 + bx + c \]
qhov twg \(a\), \(b\), thiab \(c\) yog cov tsis hloov pauv, thiab \(a \neq 0\).
Daim duab ntawm ib qho quadratic function yog ib qho nkhaus hu ua parabola. Parabolas muaj symmetry thiab ib daim duab uas nyob ntawm lub cim ntawm qhov tsis hloov pauv \(a\). Yog tias \(a > 0\), lub parabola qhib rau saum toj. Ntawm qhov tod tes, yog tias \(a < 0\), lub parabola qhib rau hauv qab. Cov Ntsiab Lus Tseem Ceeb ntawm Quadratic Functions - Cov hauv paus ntawm ib qho quadratic equation: Cov nqi ntawm \(x\) uas \(f(x) = 0\), uas tuaj yeem pom siv cov mis quadratic \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). - Vertex: Qhov siab tshaj plaws lossis qis tshaj plaws ntawm parabola, uas pom siv cov mis \((x, y)\) qhov twg \(x = -\frac{b}{2a}\) thiab \(y = f(-\frac{b}{2a})\). - Axis ntawm symmetry: Txoj kab ntsug uas faib cov parabola ua ob ntu symmetrical, uas yog ntawm \(x = -\frac{b}{2a}\).
Piv Txwv Lo Lus Nug 1: Tsim Ib Lub Quadratic Function Los Ntawm Peb Lub Ntsiab Lus Lo Lus Nug: Txheeb xyuas tus qauv rau lub quadratic function uas hla dhau cov ntsiab lus (1, 2), (2, 5), thiab (3, 10). Kev daws teeb meem: 1. Peb pib nrog daim ntawv dav dav ntawm quadratic function: \[ f(x) = ax^2 + bx + c \] 2. Hloov lub ntsiab lus (1, 2) rau hauv kab zauv: \[ a(1)^2 + b(1) + c = 2 \] \[ a + b + c = 2 \] (Equation 1) 3. Hloov lub ntsiab lus (2, 5) rau hauv kab zauv: \[ a(2)^2 + b(2) + c = 5 \] \[ 4a + 2b + c = 5 \] (Equation 2) 4. Hloov lub ntsiab lus (3, 10) rau hauv kab zauv: \[ a(3)^2 + b(3) + c = 10 \] \[ 9a + 3b + c = 10 \] (Equation 3) 5. Tam sim no peb muaj peb lub systems ntawm kab zauv linear: \[ \begin{cases} a + b + c = 2 \\ 4a + 2b + c = 5 \\ 9a + 3b + c = 10 \\ \end{cases} \] 6. Txhawm rau daws qhov teeb meem, peb rho tawm cov kab zauv thib ob thiab thawj zaug: \[ (4a + 2b + c) - (a + b + c) = 5 - 2 \] \[ 3a + b = 3 \] (Kab zauv 4) 7. Rho cov kab zauv thib peb thiab thib ob: \[ (9a + 3b + c) - (4a + 2b + c) = 10 - 5 \] \[ 5a + b = 5 \] (Equation 5) 8. Rho cov kab zauv 5 thiab kab zauv 4: \[ (5a + b) - (3a + b) = 5 - 3 \] \[ 2a = 2 \] \[ a = 1 \] 9. Hloov \(a = 1\) rau hauv kab zauv 4: \[ 3(1) + b = 3 \] \[ 3 + b = 3 \] \[ b = 0 \] 10. Hloov \(a = 1\) thiab \(b = 0\) rau hauv kab zauv 1: \[ 1 + 0 + c = 2 \] \[ c = 1 \] 11. Yog li, qhov kev ua haujlwm quadratic yog: \[ f(x) = 1x^2 + 0x + 1 \] \[ f(x) = x^2 + 1 \] Piv txwv Lo lus nug 2: Txheeb xyuas Quadratic Function los ntawm Vertex thiab Ib Qho Point Lo lus nug: Txheeb xyuas cov mis rau quadratic function uas muaj vertex ntawm (-1, 4) thiab hla dhau qhov point (1, 0). Kev daws teeb meem: 1. Daim ntawv txheem ntawm ib qho quadratic function nrog vertex \((h, k)\) yog: \[ f(x) = a(x - h)^2 + k \] 2. Hloov qhov vertex (-1, 4) rau hauv daim ntawv txheem: \[ f(x) = a(x + 1)^2 + 4 \] 3. Hloov qhov taw tes (1, 0) rau hauv qhov equation kom nrhiav tau \(a\): \[ 0 = a(1 + 1)^2 + 4 \] \[ 0 = a(2)^2 + 4 \] \[ 0 = 4a + 4 \] \[ 4a = -4 \] \[ a = -1 \] 4. Yog li, qhov kev ua haujlwm quadratic yog: \[ f(x) = -1(x + 1)^2 + 4 \] \[ f(x) = - (x + 1)^2 + 4 \] 5. Kev faib tawm rau daim ntawv txheem: \[ f(x) = - (x^2 + 2x + 1) + 4 \] \[ f(x) = -x^2 - 2x - 1 + 4 \] \[ f(x) = -x^2 - 2x + 3 \] Piv txwv Lo lus nug 3: Hloov Vertex Form mus rau Daim Ntawv Txheem Lo lus nug: Hloov quadratic function \( f(x) = 2(x - 3)^2 + 5 \) mus rau daim ntawv txheem \( ax^2 + bx + c \). Kev daws teeb meem: 1. Ua ntej, peb yuav tsum nthuav dav: \[ f(x) = 2(x - 3)^2 + 5 \] 2. Nthuav qhov binomial: \[ (x - 3)^2 = x^2 - 6x + 9 \] 3. Hloov rov qab rau hauv qhov kev ua haujlwm: \[ f(x) = 2(x^2 - 6x + 9) + 5 \] 4. Faib 2 rau txhua qhov ntawm binomial: \[ f(x) = 2x^2 - 12x + 18 + 5 \] 5. Muab txhua qhov sib xyaw ua ke: \[ f(x) = 2x^2 - 12x + 23 \] Yog li, daim ntawv txheem ntawm qhov kev ua haujlwm quadratic yog: \[ f(x) = 2x^2 - 12x + 23 \] Xaus Lus Kev tsim cov kev ua haujlwm quadratic los ntawm ntau yam ntaub ntawv yog ib qho kev txawj tseem ceeb hauv kev lej. Los ntawm kev xyaum ua tas li nrog ntau hom teeb meem, peb tuaj yeem txhim kho peb txoj kev nkag siab thiab kev txawj ntse hauv kev daws cov kab zauv quadratic. Cov ntsiab lus tseem ceeb uas yuav tsum nco ntsoov suav nrog kev nrhiav thiab kev tswj hwm cov txheej txheem rau kev rho tawm cov ntaub ntawv los ntawm daim ntawv vertex, hloov ntawm vertex thiab daim ntawv txheem, thiab tsim cov haujlwm los ntawm cov ntsiab lus muab. Nrog kev nkag siab zoo txog cov ncauj lus no, peb tuaj yeem daws cov teeb meem lej nyuaj dua yav tom ntej.