University:
High School
Course:
High School Mathematics
Academic year:

2023
Views:

131

Pages:

4
Author:

imagmahoryf

6.4 Transforming Graphs of Exponential and Logarithmic Functions A. VOCABULARY & CONCEPTS Exponential Function Transformations Core Concept f(x) Notation Examples Horizontal Translation f(x-h) y = 4^(x-2) (3 units right) Graph shifts left or right f(x+h) y = 4^(x+2) (2 units left) Vertical Translation f(x)+k y = 4^x + 5 (5 units up) Graph shifts up or down f(x)-k y = 4^x - 1 (1 unit down) Reflection -f(x) y = -4^x (flips over x-axis) Graph flips over x-axis f(-x) y = 4^-x (flips over y-axis) Horizontal Stretch or Shrink f(ax) y = 4^(3x) (shrink by a factor of 1/3) Graph stretches away from or shrinks toward y-axis f(1/ax) y = 4^(x/2) (stretch by a factor of 2) Vertical Stretch or Shrink a*f(x) y = 3(4^x) (stretch by a factor of 3) Graph stretches away from or shrinks toward x-axis (1/a)*f(x) y = (1/4)(4^x) (shrink by a factor of 1/4) Logarithmic Function Transformations Core Concept f(x) Notation Examples Horizontal Translation f(x-h) g(x) = log(x-4) (4 units right) Graph shifts left or right f(x+h) g(x) = log(x+7) (7 units left) Vertical Translation f(x)+k f(x) = log x + 3 (3 units up) Graph shifts up or down f(x)-k g(x) = log x - 1 (1 unit down) Reflection -f(x) g(x) = -log x (flips over x-axis) Graph flips over x-axis f(-x) g(x) = log(-x) (flips over y-axis) Horizontal Stretch or Shrink f(ax) g(x) = log(4x) (shrink by a factor of 1/4) Graph stretches away from or shrinks toward y-axis f(1/ax) g(x) = log(x/2) (stretch by a factor of 2) Vertical Stretch or Shrink a*f(x) g(x) = 5log x (stretch by a factor of 5) Graph stretches away from or shrinks toward x-axis (1/a)*f(x) g(x) = (1/2)log x (shrink by a factor of 1/2) y = 2^x x y -2 1/4 -1 1/2 0 1 1 2 2 4 y = log2 x x y 1/4 -2 1/2 -1 1 0 2 1 4 2 What do you notice? The domain and range are switched. (x,y) → (y,x). The logarithmic function is a reflection over y=x. Thus, logarithms are the INVERSE of exponential functions. B. EXAMPLES Describe the transformation of f represented by g. Then graph each function. 1. f(x) = 2^x, g(x) = 2^(x-3) + 1 Horizontal translation right 3, vertical translation up 1. f(x) = 2^x x y -2 1/4 -1 1/2 0 1 1 2 2 4 2. f(x) = e^-x, g(x) = e^-x - 5 Vertical translation down 5. f(x) = e^-x x y -2 e^2 ≈ 7.4 -1 e^1 ≈ 2.7 0 e^0 ≈ 1 1 e^-1 ≈ 0.4 2 e^-2 ≈ 0.1 3. f(x) = log₅x, g(x) = ½log₅(x+7) Vertical Shrink by a factor of ½. Horizontal translation left 7. y = 5^x x y -2 1/25 = .04 → (-7, -2) -1 1/5 = .2 → (-6, -1) 0 1 → (-7, 0) 1 5 → (-6, 1) 2 25 → (-5, 2) 4. f(x) = 2^x, g(x) = log₁/₃x - 4/3 Vertical translation down 4/3. y = (⅓)^x x y -2 9 → (9, -2) -1 3 → (3, -1) 0 1 → (1, 0) 1 ⅓ → (⅓, 1) 2 1/9 → (1/9, 2) 8. f(x) = (¼)x, g(x) = (¼)x-8 + 4 6. Let the graph of g be a translation 2 units up followed by a vertical stretch by a factor of 2 of the graph of f(x) = log1/3 x. Write a rule for g. g(x) = 2(log1/3(x + 2)) g(x) = 2log1/3 x + 4 7. Let the graph of g be a reflection in the y-axis, followed by a translation 4 units to the left of the graph of f(x) = log x. Write a rule for g. g(x) = log(-x) reflection over y-axis g(x) = log(-(x+4)) g(x) = log(-x-4)