1.04C - Rational Equations | High School

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High School
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High School Physics
Academic year:

2024
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1.04 Rational Functions 1.04C Rational Equations • Ex 2 • Ex 1 −6 𝑓 𝑥 = +4 2𝑥 − 1 1 𝑥−8 3 𝑔 𝑥 = 2𝑥 + 1 𝑓 8 =? 𝑔 15 =? 18 𝑓 8 = 5 3 𝑔 15 = − 31 Finding 𝑦 Given 𝑥 4 −3=0 𝑥−5 • Ex 1 4 =3 𝑥−5 4 𝑓 𝑥 = −3 𝑥−5 Find the zero 4(1) = 3(𝑥 − 5) 4 = 3𝑥 − 15 19 = 3𝑥 19 =𝑥 3 Finding 𝑥 Given 𝑦 2𝑥 + 7 =0 1 𝑥+1 3 • Ex 2 2𝑥 + 7 𝑓 𝑥 = 1 𝑥+1 3 (2𝑥 + 7)(1) = 0 Find the zero 2𝑥 + 7 = 0 2𝑥 = −7 7 𝑥=− 2 Finding 𝑥 Given 𝑦 1 𝑥+1 3 𝑎1 𝑥 + 𝑏1 𝑓 𝑥 = ⇒ 𝑎2 𝑥 + 𝑏2 𝑎1 𝑥 + 𝑏1 0= 𝑎2 𝑥 + 𝑏2 0 𝑎2 𝑥 + 𝑏2 = 1 𝑎1 𝑥 + 𝑏1 0 = 𝑎1 𝑥 + 𝑏1 𝑏1 𝑧=− 𝑎1 −𝑏1 = 𝑎1 𝑥 𝑏1 − =𝑥 𝑎1 Zeros Formula – General Form 𝑘 𝑎1 𝑥 + 𝑏1 𝑓 𝑥 = ℎ 𝑎2 𝑥 + 𝑏2 𝑧 General Form Shorts • Standard: 𝑎 𝑧=− +ℎ 𝑘𝑏 • General 𝑏1 𝑧=− 𝑎1 Two Formulas for Zeros SOLVING EQUATIONS −2 𝑓 𝑥 = +4 3𝑥 − 7 𝑔 𝑥 = 13 −2 + 4 = 13 3𝑥 − 7 −2 =9 3𝑥 − 7 −2(1) = 9(3𝑥 − 7) Find 𝑓 = 𝑔 −2 = 27𝑥 − 63 61 = 27𝑥 Example 1 61 =𝑥 27 2𝑥 + 1 𝑓 𝑥 = 5𝑥 − 2 𝑔 𝑥 =2 Find 𝑓 = 𝑔 2𝑥 + 1 =2 5𝑥 − 2 2𝑥 + 1 1 = 2 5𝑥 − 2 2𝑥 + 1 = 10𝑥 − 4 5 = 8𝑥 Example 2 5 =𝑥 8 2𝑥 + 6 𝑓 𝑥 = 𝑥+1 2𝑥 + 6 =𝑥 𝑥+1 2𝑥 + 6 1 = 𝑥 𝑥 + 1 𝑔 𝑥 =𝑥 2𝑥 + 6 = 𝑥 2 + 𝑥 Find 𝑓 = 𝑔 0 = 𝑥2 − 𝑥 − 6 0= 𝑥−3 𝑥+2 𝑥 = −2, 3 Example 3 213 𝑓 𝑥 = − 10 𝑥+7 213 − 10 = 2𝑥 − 62 𝑥+7 213 = 2𝑥 − 52 𝑥+7 𝑔 𝑥 = 2𝑥 − 62 213 = 2𝑥 2 − 38𝑥 − 364 Find 𝑓 = 𝑔 0 = 2𝑥 2 − 38𝑥 − 577 19 ± 1515 𝑥= 2 Example 4 𝑥 ≈ −9.9615, 28.9615 • Visions 1 • P67: 6acegi Homework