1.03B - Absolute Transformed and Graphing

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High School Physics
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2024
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Absolute Value Functions ABSOLUTE VALUE FUNCTIONS 1.03B – TRANSFORMED FUNCTION ABSOLUTE TRANSFORMED 𝑓 𝑥 =𝑎 𝑏 𝑥−ℎ +𝑘 • Thinking back to our properties… 𝑓 𝑥 =𝑎 𝑏 𝑥−ℎ +𝑘 =𝑎 𝑏 𝑥−ℎ +𝑘 = 𝑎𝑏 𝑥 − ℎ + 𝑘 𝑤ℎ𝑒𝑟𝑒 𝑏 > 0 • Product 𝑎𝑏 acts as a new “𝑎” … therefore: Symmetry in the Absolute 𝑓 𝑥 =𝑎 𝑥−ℎ +𝑘 𝑎≠0 The Transformed Rule PROPERTIES 𝑎 > 0, then 𝑎 < 0 𝑘=0 𝑘<0 𝑘>0 𝑎>0 • 𝑑𝑜𝑚 𝑓 = ℝ • 𝑟𝑎𝑛 𝑓 = [𝑘, +∞[ Properties – D, R, 𝑓 𝑥 =𝑎 𝑥−ℎ +𝑘 0=𝑎 𝑥−ℎ +𝑘 −𝑘 = 𝑎 𝑥 − ℎ 𝑘 − = 𝑥−ℎ 𝑎 • 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 ⇒ 𝑓(0) • 𝑧𝑒𝑟𝑜𝑠 𝑘 > 0 ⇒ 𝑛𝑜𝑛𝑒 𝑘=0⇒𝑥=ℎ If 𝑥 − ℎ ≥ 0 𝑘 𝑘 < 0 ⇒ 𝑥1 = ℎ − 𝑎 𝑘 𝑥2 = ℎ + 𝑎 then: 𝑥 − ℎ = − 𝑘 𝑎 𝑘 𝑎 𝑥 =− +ℎ If 𝑥 − ℎ ≤ 0 then: 𝑥 − ℎ = − − 𝑘 𝑎 𝑥 = +ℎ Properties - I 𝑘 𝑎 • 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ] − ∞, ℎ] • 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 [ℎ, +∞[ • min 𝑓 = 𝑘 • max 𝑓 = 𝐷𝑁𝐸 Properties – V, E 1. 𝑘 > 0 ⇒ 𝑓 is strictly positive over ℝ 2. 𝑘 = 0 ⇒ 𝑓 is positive over ℝ 𝑓 is negative at 𝑥 = {ℎ} 3. 𝑘 < 0 ⇒ 𝑓 is positive over ] − ∞, 𝑥1 ] ∪ [𝑥2 , +∞[ 𝑓 is negative over [𝑥1 , 𝑥2 ] Properties - S ABSOLUTE VALUE FUNCTION: PROPERTIES Domain Range 𝑑𝑜𝑚 𝑓 = ℝ Intercepts 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 ⇒ 𝑓(0) 𝑧𝑒𝑟𝑜𝑠 𝑟𝑎𝑛 𝑓 = [𝑘, +∞[ 𝑘 > 0 ⇒ 𝑛𝑜𝑛𝑒 𝑘=0⇒𝑥=ℎ 𝑘 𝑘 < 0 ⇒ 𝑥1 , 𝑥2 = ℎ ± 𝑎 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ] − ∞, ℎ] 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 [ℎ, +∞[ Variation min 𝑓 = 𝑘 max 𝑓 = 𝐷𝑁𝐸 Extremes Sign 1. 2. 3. 𝑘 > 0 ⇒ 𝑓 is strictly positive over ℝ 𝑘 = 0 ⇒ 𝑓 is positive over ℝ 𝑓 is negative at 𝑥 = {ℎ} 𝑘 < 0 ⇒ 𝑓 is positive over ] − ∞, 𝑥1 ] ∪ [𝑥2 , +∞[ 𝑓 is negative over [𝑥1 , 𝑥2 ] PROPERTIES If 𝑎 < 0 𝑘=0 𝑘<0 𝑘>0 𝑎<0 • 𝑑𝑜𝑚 𝑓 = ℝ • 𝑟𝑎𝑛 𝑓 =] − ∞, 𝑘] Properties – D, R, • 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 ⇒ 𝑓(0) • 𝑧𝑒𝑟𝑜𝑠 𝑘 𝑘 > 0 ⇒ 𝑥1 , 𝑥2 = ℎ ± 𝑎 𝑘=0⇒𝑥=ℎ 𝑘 < 0 ⇒ 𝑛𝑜𝑛𝑒 Properties - I • 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ] − ∞, ℎ] • 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 [ℎ, +∞[ • min 𝑓 = 𝐷𝑁𝐸 • max 𝑓 = 𝑘 Properties – V, E 1. 𝑘 > 0 ⇒ 𝑓 is positive over[𝑥1 , 𝑥2 ] 𝑓 is negative over] − ∞, 𝑥1 ] ∪ [𝑥2 , +∞[ 2. 𝑘 = 0 ⇒ 𝑓 is positive at 𝑥 = {ℎ} 𝑓 is negative over ℝ 3. 𝑘 < 0 ⇒ 𝑓 is strictly negative over ℝ Properties - S ABSOLUTE VALUE FUNCTION: PROPERTIES 𝑑𝑜𝑚 𝑓 = ℝ Domain Range Intercepts 𝑟𝑎𝑛 𝑓 =] − ∞, 𝑘] 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 ⇒ 𝑓(0) 𝑧𝑒𝑟𝑜𝑠 𝑘 𝑘 > 0 ⇒ 𝑥1 , 𝑥2 = ℎ ± 𝑎 𝑘=0⇒𝑥=ℎ 𝑘 < 0 ⇒ 𝑛𝑜𝑛𝑒 Variation 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 [ℎ, +∞[ 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ] − ∞, ℎ] Extremes min 𝑓 = 𝐷𝑁𝐸 max 𝑓 = 𝑘 Sign 1. 2. 3. 𝑘 > 0 ⇒ 𝑓 is positive over[𝑥1 , 𝑥2 ] 𝑓 is negative over] − ∞, 𝑥1 ] ∪ [𝑥2 , +∞[ 𝑘 = 0 ⇒ 𝑓 is positive at 𝑥 = {ℎ} 𝑓 is negative over ℝ 𝑘 < 0 ⇒ 𝑓 is strictly negative over ℝ ℎ, 𝑘 Vertex 𝑥=ℎ Axis of Symmetry GRAPHING Method 1 – Ordered Pair Formula • Ordered Pair Formula 𝑥 + ℎ, 𝑎𝑦 + 𝑘 𝑏 • Basic Ordered Pairs: −1,1 , 0,0 , (1, 1) Steps: 1. Apply a, h, k to the Ordered Pair Formula 2. Transform Pairs one at a time. To Graph 𝑓 𝑥 = |𝑥| 𝑓 𝑥 =2 𝑥−1 −3 Parameters GRAPHING Method 2 – Piecewise Method Steps: 1. 2. Identify the vertex ℎ, 𝑘 and plot it Identify Parameter 𝑎 and use it as a second point. 𝑅𝐼𝑆𝐸 𝑅𝑈𝑁 to find ALWAYS run RIGHT for a positive slope. 3. Consider symmetry to plot third point. Mirror point 2. To Graph 𝑓 𝑥 = |𝑥| 1 𝑓 𝑥 =2 − 3+ 5 − 𝑥 −𝑥 1+ 4 2 Parameters Try It: 𝑓 𝑥 = −2𝑥 + 6 𝑓 𝑥 = 2|𝑥 − 3| 𝑟𝑖𝑠𝑒 V 3,0 , 𝑎𝑑𝑑 𝑟𝑢𝑛 → (4, 2), mirror (4,2) → (2,2) *Note* The parameter “𝑎” is also the positive and negative slopes of the lines. a: vertical scale change and reflection h: horizontal translation k: vertical translation • P53: 1ace, 3ace • P54: 5ace • P55: 8ace Homework