1.04B - Rational Transformed and Graphing

1.04 Rational Functions 1.04B Transformed Rule Forms, Parameters, Properties and Graphing •𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) Ex: 𝑓 𝑥 = +𝑘 500 2𝑥−14 < 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑜𝑟𝑚 > +6 500 𝑓 𝑥 = +6 2 𝑥−7 250 𝑓 𝑥 = +6 𝑥−7 “𝑏 is incorporated in 𝑎” < 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑜𝑟𝑚 > Rational Function Ex 2: 𝑓 𝑥 = 66 4𝑥−12 +1 33 𝑓 𝑥 = +1 2(𝑥 − 3) Ex 3: 𝑓 𝑥 = 𝑓 𝑥 = Rational Function 17 2𝑥+1 17 1 2 𝑥+ 2 −3 −3 •𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) •𝑓 𝑥 = 𝑎 (𝑥−ℎ) •𝑓 𝑥 = 𝑎1 𝑥+𝑏1 𝑎2 𝑥+𝑏2 +𝑘 +𝑘 < 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑜𝑟𝑚 > “𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑” → General form/ homographicform the quotient between two linear Rational Function SWITCHING FORMS STANDARD • 𝑓 𝑥 = GENERAL 𝑎 𝑏(𝑥−ℎ) → +𝑘 𝑓 𝑥 = 𝑎1 𝑥+𝑏1 𝑎2 𝑥+𝑏2 Find the common denominator!! • 𝑓 𝑥 = −3 (𝑥−2) −2 ⟹ ⟹ 𝑓 𝑥 = −3 (𝑥−2) 𝑓 𝑥 = −3−2(𝑥−2) (𝑥−2) ⟹ 𝑓 𝑥 = ⟹ 𝑓 𝑥 = − 2(𝑥−2) (𝑥−2) −3−2𝑥+4 𝑥−2 −2𝑥+1 ,𝑥 𝑥−2 Converting the Rule ≠2 • 𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) → +𝑘 𝑓 𝑥 = 𝑎1 𝑥+𝑏1 𝑎2 𝑥+𝑏2 • Find the common denominator!! • 𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) +𝑘 ⟹ ⟹ ⟹ • Rearrange so x’s first 𝑾𝒉𝒆𝒓𝒆: 𝑎1 = 𝑘𝑏 𝑎2 = 𝑏 𝑏1 = 𝑎 − 𝑘𝑏ℎ 𝑏2 = −𝑏ℎ 𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) 𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) 𝑓 𝑥 = 𝑎+𝑘𝑏𝑥−𝑘𝑏ℎ 𝑏(𝑥−ℎ) ⟹ 𝑓 𝑥 = ⟹ 𝑓 𝑥 = + 𝑘𝑏 𝑥−ℎ 𝑏(𝑥−ℎ) + 𝑘𝑏𝑥−𝑘𝑏ℎ 𝑏 𝑥−ℎ 𝑘𝑏𝑥+𝑎−𝑘𝑏ℎ 𝑏𝑥−𝑏ℎ (𝑘𝑏)𝑥+(𝑎−𝑘𝑏ℎ) (𝑏)𝑥−(𝑏ℎ) Converting the Rule – NO # • 𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) +𝑘 → 𝑓 𝑥 = 𝑎1 𝑥+𝑏1 𝑎2 𝑥+𝑏2 𝑵𝑶𝑻𝑬: +𝑘 𝑎 ⟹ 1 𝑓 𝑥 =𝑘𝑏 + ⟹ 𝑓 𝑥 = =𝑘 = 𝑎⟹2 𝑓 𝑥 =𝑏 • Find the common denominator!! • 𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) 𝑾𝒉𝒆𝒓𝒆: 𝑎1 = 𝑘𝑏 𝑎2 = 𝑏 𝑏1 = 𝑎 − 𝑘𝑏ℎ 𝑏2 = −𝑏ℎ ⟹ 𝑓 𝑥 = 𝑎 𝑘𝑏 𝑥−ℎ 𝑏(𝑥−ℎ) 𝑏(𝑥−ℎ) 𝑎+𝑘𝑏(𝑥−ℎ) 𝑏(𝑥−ℎ) 𝑎+𝑘𝑏𝑥−𝑘𝑏ℎ 𝑏𝑥−𝑏ℎ 𝑘𝑏𝑥+𝑎−𝑘𝑏ℎ 𝑏𝑥−𝑏ℎ (𝑘𝑏)𝑥+(𝑎−𝑘𝑏ℎ) (𝑏)𝑥−(𝑏ℎ) 𝑏⟹2 𝑓 𝑥 −𝑏ℎ = = −ℎ = 𝑎2 𝑏 Vertex = intersection point of asymptotes = (ℎ, 𝑘) • General • Standard 𝑎 𝑓 𝑥 = +𝑘 𝑏 𝑥−ℎ 𝑎1 𝑥 + 𝑏1 𝑓 𝑥 = 𝑎2 𝑥 + 𝑏2 𝑾𝒉𝒆𝒓𝒆: ℎ= 𝑏2 − 𝑎2 𝑘= 𝑎1 𝑎2 Transformed Rational STANDARD GENERAL • 𝑓 𝑥 = 𝑎1 𝑥+𝑏1 𝑎2 𝑥+𝑏2 → 𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) +𝑘 • Long Division!! • 𝑓 𝑥 = 4𝑥+2 2𝑥−2 2𝑥 − 2 2 )4𝑥 + 2 4𝑥 − 4 6 3 𝑓 𝑥 = +2 𝑥−1 4𝑥 + 2 6 = 2+ 2𝑥 − 2 2𝑥 − 2 6 = 2 + 2 𝑥−1 Converting the Rule PARAMETERS • Vertical Scale Change • 𝒇 𝒙 = 𝟏 𝒙 𝒈 𝒙 = 𝟏𝟓 𝒙 Role of “a” • 𝒇 𝒙 = 𝟏 𝒙 𝒈 𝒙 = −𝟏 𝒙 • Scale Change • 𝒇 𝒙 = 𝟏 𝒙 𝒈 𝒙 = 𝟏 𝟓𝒙 Role of “ b” • 𝒇 𝒙 = 𝟏 𝒙 𝒈 𝒙 = 𝟏 −𝒙 • Vertical Scale Change • 𝒇 𝒙 = 𝟏 𝟐 𝒙 “a” vs “ b” • 𝒈 𝒙 = 𝟏 𝟐𝒙 • Vertical Scale Change • 𝒇 𝒙 = −𝟏 𝒙 “a” vs “ b” • 𝒈 𝒙 = 𝟏 −𝒙 𝑎 𝑏 𝑓 𝑥 = + 𝑘, 𝑥 ≠ ℎ (𝑥 − ℎ) Rule • Horizontal Translation • 𝒇 𝒙 = 𝟏 𝒙 𝒈 𝒙 = 𝟏 𝒙−𝟑 Role of “ h” • 𝒇 𝒙 = 𝟏 𝒙 𝒈 𝒙 = 𝟏 𝒙+𝟑 • Vertical Translation • 𝒇 𝒙 = 𝟏 𝒙 𝟏 𝒙 𝒈 𝒙 = +𝟑 Role of “ k” • 𝒇 𝒙 = 𝟏 𝒙 𝟏 𝒙 𝒈 𝒙 = −𝟑 • Domain • 𝑑𝑜𝑚𝑓 = ℝ\{ℎ} • 𝑑𝑜𝑚𝑓 = ℝ\ −𝑏2 𝑎2 • 𝑑𝑜𝑚𝑓 = {𝑥 ∈ ℝ|𝑥 ≠ ℎ} • Range • 𝑟𝑎𝑛𝑓 = ℝ\{𝑘} • 𝑟𝑎𝑛𝑓 = ℝ\ 𝑎1 𝑎2 • 𝑟𝑎𝑛𝑓 = {𝑦 ∈ ℝ|𝑦 ≠ 𝑘} Properties Domain Range Variation Extremes Sign Vertex B.O.P 𝑑𝑜𝑚 𝑓 = 𝑥 ∈ ℝ 𝑥 ≠ ℎ = ℝ\{𝒉} 𝑟𝑎𝑛 𝑓 = 𝑦 ∈ ℝ 𝑦 ≠ 𝑘 = ℝ\{𝒌} 𝑎 and 𝑏 same sign 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 𝑑𝑜𝑚𝑎𝑖𝑛 𝑎 and 𝑏 different signs 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 𝑑𝑜𝑚𝑎𝑖𝑛 𝑁𝑜𝑛𝑒 Changes at the zero and the vertical asymptote ℎ, 𝑘 The intersection point of the asymptotes 1 1 1 1 −2, − , −1, −1 , − , −2 , , 2 , 1, 1 , 2, 2 2 2 2 1. Factor 𝑏 2. Draw the asymptotes • Vertical 𝑥 = ℎ (𝑑𝑜𝑡𝑡𝑒𝑑) • Horizontal 𝑦 = 𝑘 𝑑𝑜𝑡𝑡𝑒𝑑 3. Decide if graph is in: • Quadrants I and III 𝑎𝑏 > 0 • Quadrants II and IV 𝑎𝑏 < 0 4. Use Order Pair Formula to find new points for −2, −12 , −1, −1 , −12, −2 , 12, 2 , 1,1 , (2, 12) How to Graph • Graph 𝑓 𝑥 = 6 −2𝑥+4 −2 6 𝑓 𝑥 = −2 −2(𝑥 − 2) −3 𝑓 𝑥 = − 2, 𝑥 ≠ 2 𝑥−2 𝑎 = −3, ℎ = 2, 𝑘 = −2 Example • Graph 𝑓 𝑥 = −3 𝑥−2 • 𝑥 + 2, −3𝑦 − 2 • −2, − • −1, −1 → 1, 1 • 1 3 − , −2 → , 4 2 2 1 5 , 2 → , −8 2 2 • 1 2 → 0, − • 1, 1 → 3, −5 • 2, 1 2 → 4, − 7 2 Example − 2, 𝑥 ≠ 2 ⇒ 1 2 𝑎 = −3, ℎ = 2, 𝑘 = −2 (𝑏 = 1) Find the six ordered pairs used to produce this graph, write on this slide Graph 𝑓 𝑥 = 2 𝑥+2 − 3, 𝑥 ≠ −2 Find the six ordered pairs used to produce this graph, write on this slide Graph 𝑓 𝑥 = −4 𝑥−2 + 5, 𝑥 ≠ 2 Find the six ordered pairs used to produce this graph, write on this slide Graph 𝑓 𝑥 = −6 −3𝑥+3 − 3, 𝑥 ≠ 1 • 𝑓 𝑥 = 𝑎 𝑏 𝑥−ℎ +𝑘 • Asymptotes: 𝑥 = ℎ and 𝑦 = 𝑘 • Properties: 𝑑𝑜𝑚 𝑓 = ℝ\{ℎ} 𝑟𝑎𝑛 𝑓 = ℝ\{𝑘} 𝑓 ↗ if 𝑎𝑏 < 0 𝑓 ↘ if 𝑎𝑏 > 0 𝑉 ℎ, 𝑘 • 𝑓 𝑥 = 𝑎1 𝑥+𝑏1 𝑎2 𝑥+𝑏2 ℎ=− 𝑏2 𝑎2 𝑘= 𝑎1 𝑎2 • Switching Forms: Standard to General = Common Denom. General to Standard = Long Division • Basic Ordered Pairs: −2, − 1 1 1 1 , −1, −1 , − , −2 , , 2 , 1, 1 , 2, 2 2 2 2 Summary of Learning • Visions 1 • P68: #4bdf Homework