1.04A - Rational Intro and Basic Function

University:
High School
Course:
High School Physics
Academic year:

2024
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1.04 Rational Functions Also known as: Inverse Variation 1.04A Intro/ Basic Function 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 2637 ÷ 4 659 4)2637 24 23 20 37 𝟏 36 2637 ÷ 4 = 𝟔𝟓𝟗 + 𝟒 1 Long Division 𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 2𝑥 2 + 3𝑥 − 7 ÷ 𝑥 + 4 2𝑥 −5 𝑥 + 4)2𝑥 2 + 3𝑥 − 7 2𝑥 2 + 8𝑥 −5𝑥 −7 −5𝑥 − 20 13 2𝑥 2 Ex 2 𝟏𝟑 + 3𝑥 − 7 ÷ 𝑥 + 4 = 𝟐𝒙 − 𝟓 + 𝒙+𝟒 3𝑥 + 2 ÷ 𝑥 − 1 3 𝑥 − 1) 3𝑥 + 2 3𝑥 − 3 5 𝟓 3𝑥 + 2 ÷ 𝑥 − 1 = 𝟑 + 𝒙−𝟏 Ex 3 • Defined by the quotient of two polynomials 𝑃 𝑓 𝑥 = 𝑄 • Where 𝑄 is at least a first-degree polynomial whose coefficients are not zero. • Our course: only first degree polynomials Rational function: definition • Determine the two polynomials 𝑃 and 𝑄 defining the rational function. 3𝑥 + 2 𝑔 𝑥 = 𝑥−1 1 ℎ 𝑥 = 4𝑥 2 𝑘 𝑥 = +5 𝑥 𝑃 = 3𝑥 + 2 𝑄 =𝑥−1 𝑃=1 𝑄 = 4𝑥 𝑃 = 2 + 5𝑥 𝑄=𝑥 Examples • A boat rental costs $20 total. How much would this cost per person? 𝒙 𝒇(𝒙) 0 1 3 1 2 1 ∉ℝ 2 10 4 5 5 4 10 2 60 40 Graph this function 20 20 40 20 𝑓 𝑥 = 𝑥 1 1 2 The graph has two asymptotes: X=0 (y-axis) Y=0 (x-axis) Boat Rental continued • An asymptote is: • a line which a curve approaches but never reaches (restrictions can be represented by an asymptote) • They can be vertical, horizontal or oblique (slanted) lines. Asymptote 1 𝑥 • Rule: 𝑓 𝑥 = ; 𝑤ℎ𝑒𝑟𝑒 𝑥 ≠ 0 The Basics Domain Range Intercepts Variation Extremes 𝑑𝑜𝑚 𝑓 = 𝑥 ∈ ℝ 𝑥 ≠ 0 = ℝ∗ 𝑟𝑎𝑛 𝑓 = 𝑦 ∈ ℝ 𝑦 ≠ 0 = ℝ∗ N/A 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ℝ∗ min 𝑓 = 𝐷𝑁𝐸 max 𝑓 = 𝐷𝑁𝐸 Sign 𝑓 > 0 over x > 0 𝑓 < 0 over x > 0 Vertex 0, 0 The intersection point of the asymptotes Visions 1 P67: 1a,c,e,g,i Homework