2.01 ExpLog - Exponential Base and Transformed Functions

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

2024
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30
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2.01 Exponential Functions • 𝑦=2 𝑥 𝑥 • 𝑦= 𝑦 𝑥 1 𝑥 2 𝑦 • 𝑦 = 2−𝑥 𝑥 −3 −3 −3 −2 −2 −2 −1 −1 −1 0 0 0 1 1 1 2 2 2 3 3 3 Explore 𝑦 If 𝑐 = 1 𝑓 𝑥 =𝑐 You end up with A basic constant function Example: 11 = 1 12 = 1 13 = 1 18 = 1 1100 000 000 = 1 𝑥 If 𝑐 < 0 You end up with undefined points 𝑓 𝑥 = where 𝑐 ≠ 1 and 𝑐 > 0 Rule 1 (−2)2 (exponent) (power) 𝑥 𝑦=𝑐 (base) Where: 𝑐 ∈ The Function ∗ ℝ+ 𝑐≠1 1. Find Δ𝑥 2. Find Δ𝑦 3. Find quotient Δ𝑦2 Δ𝑦1 • Test It: 𝑥 𝑦 𝑥 𝑦 −3 1 8 1 4 1 2 −8 1 256 1 32 1 4 0 1 1 2 1 2 4 16 2 4 7 128 3 8 10 1024 −2 −1 4. BASE= Δ𝑥 Δ𝑦2 Δ𝑦1 −5 −2 Finding c from table (no transformations) 1. Find Δ𝑥 2. Find Δ𝑦 3. Find quotient Δ𝑦2 Δ𝑦1 4. BASE, 𝑐 = Try It Δ𝑥 𝑥 𝑦 −9 1 262144 1 16 −2 Δ𝑦2 Δ𝑦1 5 12 1024 16777216 The independent variable • 𝑥 is an exponent • There is a pattern in the sequence for the 𝑦-values • If you divide two consecutive differences ∆𝑦 , it gives you the base of the exponential function!! Multiplying factor Exponential Function 𝑓 𝑥 = 3𝑥 Table of Values 𝑥 0 𝑦 1 3 1 1 3 2 9 3 27 4 81 −1 54 𝑏𝑎𝑠𝑒 = =3 +2/3 18 +2 18 𝑏𝑎𝑠𝑒 = =3 +6 6 +18 6 +54 𝑏𝑎𝑠𝑒 = = 3 2 Example 18 6 2 • 𝑓 𝑥 =3 (2, 9) 𝑥 (1, 3) (0, 1) −1, If 𝑐 > 1 1 3 • 𝑓 𝑥 = 1 𝑥 2 (−3, 8) −2,4 (−1, 2) (0, 1) 1, If 0 < 𝑐 < 1 1 2 (it’s a fraction) • Is a curve with an asymptote at 𝑦 = 0 (that’s the x-axis) • Is a curve passing through (0, 1) and (1, 𝑐). Properties of the Basic Function Domain Range Intercepts 𝑑𝑜𝑚 𝑓 = ℝ 𝑟𝑎𝑛 𝑓 = ℝ+ ∗ 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 = 1 𝑍𝑒𝑟𝑜𝑠 = 𝑛𝑜𝑛𝑒 Variations 𝑐 > 1 ⇒ 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ℝ 0 < 𝑐 < 1 ⇒ 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ℝ Extremes Sign Basic Ordered Pairs The Graph 𝑛𝑜𝑛𝑒 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑣𝑒𝑟 ℝ 1 −1, , 0,1 , (1, 𝑐) 𝑐 • Visions 1 • P164: 19, 20, 21, 22 Homework PARAMETERS/TRANSFORMED PROPERTIES Exponential Functions 𝑓 𝑥 = 𝑎𝑐 𝑏 𝑥−ℎ +𝑘 Where: 𝑎≠0 𝑏≠0 𝑐>0 𝑐≠1 The Rule – Standard Form 𝑓 𝑥 = 15 3 𝑓 𝑥 = 15 3 𝑎 = 15 𝑐=3 𝑏=2 i.e. 2𝑥−2 −5 2(𝑥−1) −5 ℎ=1 𝑘 = −5 Effect of “c” • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 =3 2 𝑥 Vertical stretch “a” (𝑎 > 1) • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 1 𝑓 𝑥 = 2 3 𝑥 Vertical Compression “a” (0 < 𝑎 < 1) • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 =− 2 𝑥 Reflection over x-axis (𝒂 < 𝟎) “a” “a” • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 = 2 2𝑥 Horizontal Compression “b” (𝑏 > 1) • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 = 2 1 2𝑥 Horizontal Stretch “b” (0 < 𝑏 < 1) • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 = 2 −𝑥 Reflection over y=axis (𝒃 < 𝟎) “b” “b” • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 = 2 𝑥−2 Horizontal Translation of 2 units to the right “h” • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 = 2 𝑥−2 𝑥 −1 0 1 2 “h” 𝑦 - basic 𝑦-new 1 1 2 8 1 1 4 1 2 2 4 1 3 8 2 4 16 4 5 32 8 • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 = 2 𝑥−2 Vertical Translation of 2 units down **NEW ASYMPTOTE** “k” • < Basic > 𝑓 𝑥 = 2𝑥 • < Transformed > 𝑓 𝑥 = 2 𝑥−2 𝑥 −1 0 “k” 𝑦 - basic 𝑦-trans 1 3 − 2 2 1 −1 1 2 0 2 4 2 3 8 6 4 16 14 Properties if 𝒄 > 𝟏 Domain Range Intercepts Variation Extremes Sign ℝ 𝑎 > 0 → ]𝑘, +∞[ 𝑎 < 0 → ] − ∞, 𝑘[ Initial Value 𝑓(0) Zeros 𝑎 and 𝑘 same sign → 𝑛𝑜𝑛𝑒 𝑎 and 𝑘 opposite signs → 𝑜𝑛𝑒 𝑎, 𝑏 𝑠𝑎𝑚𝑒 𝑠𝑖𝑔𝑛 → 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 a, b opposite sign → 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑁𝑂𝑁𝐸 𝑖𝑓 𝑎 > 0 and 𝑘 < 0 → Changes at the zero 𝑖𝑓 𝑎 > 0 and 𝑘 > 0 → strictly positive 𝑖𝑓 𝑎 < 0 and 𝑘 < 0 → strictly negative 𝑖𝑓 𝑎 < 0 and 𝑘 > 0 → Changes at the zero Properties Properties if 𝟎 < 𝒄 < 𝟏 Domain Range Intercepts Initial Value ℝ 𝑎 > 0 → ]𝑘, +∞[ 𝑎 < 0 → ] − ∞, 𝑘[ 𝑓(0) 𝑎 and 𝑘 same sign → 𝑛𝑜𝑛𝑒 𝑎 and 𝑘 opposite signs → 𝑜𝑛𝑒 𝑎, 𝑏 𝑠𝑎𝑚𝑒 𝑠𝑖𝑔𝑛 → 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑎, 𝑏 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑔𝑛 → 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑁𝑂𝑁𝐸 Zeros Variation Extremes Sign 𝑖𝑓 𝑎 > 0 and 𝑘 < 0 → Changes at the zero 𝑖𝑓 𝑎 > 0 and 𝑘 > 0 → strictly positive 𝑖𝑓 𝑎 < 0 and 𝑘 < 0 → strictly negative 𝑖𝑓 𝑎 < 0 and 𝑘 > 0 → Changes at the zero Properties