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Name ________________________________________ Date __________________ Class__________________ LESSON 4-7 Reading Strategies Read a Table Exponential and logarithmic functions as well as linear, quadratic, and polynomial functions can all undergo the same types of transformations. Parent function: f (x) Vertical translation f (x ) + k k > 0 up k < 0 down Horizontal translation f (x − h) h > 0 right h < 0 left Reflection Vertical stretch or compression −f (x) across x-axis f (−x) across y-axis af (x) a > 1 stretch 0 < a < 1 compression Horizontal stretch or compression ⎛1 ⎞ f ⎜ x⎟ ⎝b ⎠ b > 1 stretch 0 < b < 1 compression Use the chart to answer the questions below. 1. How does the graph of f (x) = ln x compare to the graph of g(x) = ln x − 4? _________________________________________________________________________________________ 2. How does the graph of f (x) = ex compare to the graph of g(x) = e(x − 5)? _________________________________________________________________________________________ 3. Describe the transformation of g(x) = 8 ex from the parent function. _________________________________________________________________________________________ 4. How does the graph of g(x) = −ln x compare to the graph of f (x) = ln x? _________________________________________________________________________________________ 5. Write the transformed function, g(x), if f (x) = ln x is translated 7 units left. 6. Write the transformed function, g(x), if f (x) = ex is reflected over the y-axis. 7. Write the transformed function, g(x), if f (x) = log x 1 is horizontally compressed by a factor of . 6 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 4-58 Holt McDougal Algebra 2 3. g(x) = −7 ⋅ 9x − 1 4. x = 1 ⎛2 ⎞ 4. g(x)= −3ln ⎜ x + 8 ⎟ + 7 ⎝3 ⎠ 5. f(x)= −log(3x + 11) − 2 6. f(x) = −4 ⋅ 7x − 1 + 5 7. Translate 2 units up and 2 units right and then reflect across the x-axis 8. $27,647.16 Reteach Challenge 1. 3 1. They are identical. The graphs are reflections of f(x) = log x over the x-axis. 1 2. log = log x −1 = −1 x log x = − log x x 3. They are identical. The function is vertically compressed by a factor of 0.5 and then translated 2 units up. ( ) ( ) 4. log 100 x = log100 + log x 1 = log 102 + log x 2 = 2log10 + 2. 4; y = 0 1 log x log x = 2 + 2 2 5. The three functions are identical. The function is horizontally compressed by a factor of 2 and then shifted 1.5 units right. 4x 4x 2 x − 1.5 ) 6. 22 x − 3 = 2 ( = 4 x − 1.5 = 1.5 = 8 4 Problem Solving 1. a. About 3 ft tall b. About 0.45 ft; that is the initial height of the plant. 2. a. g(t) = ln t 3. x = 0 b. The parent function is translated 1.25 units left and stretched vertically by a factor of 2. c. B 3. A 4. H Reading Strategies 1. Possible answer: The shapes of the curves are the same, but the curve for g(x) is shifted 4 units down from the curve for f(x). Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A51 Holt McDougal Algebra 2 2. Possible answer: The shapes of the curves are the same, but the curve for g(x) is shifted 5 units right of the curve for f(x). 3. Possible answer: The curve for g(x) is a vertical stretch of the parent function by a factor of 8. 4. Possible answer: The curve for g(x) is the curve for f(x) reflected across the x-axis. 5. g(x) = ln (x + 7) 6. g(x) = e−x 7. g(x) = log 6x 3. Yes; 3 2 2 3 5. f(x) = 4(2.3)x 6. f(x) = 3035(0.35)x 7. f(x) = −4 + 0.5 ln x 8. f(x) = 1.97 − 0.7 ln x 9. a. 350(1.18)x b. 11.9 min c. 7,194,299 Reteach 4-8 CURVE FITTING WITH 1. 18, 72 Practice A 2. 8, 24, 72, 216 4.5 18 72 = 4, = 4, = 4; data set is 1.125 4.5 18 exponential with a constant ratio of 4. EXPONENTIAL AND LOGARITHMIC MODELS 1. a. 0.25, 1, 4,16, 64 64 16 4 1 = 4, = 4, = 4, =4 b. 16 4 1 0.25 24 72 216 = 3, = 3, = 3; data set is 8 24 72 exponential with a constant ratio of 3. c. Yes, 4 3. a. B(t) ≈ 251.9(1.4)t d. Yes 2. Yes; 5 b. About 5.8 min 3. No Challenge 4. a. a = 24, b = 0.5 1. Independent variable, years; dependent variable, number of subscribers; exponential model b. f(x) = 24(0.5x) 5. a. a = 3.3, b = 2 b. f(x) = 3.2(2x) 2. y ≈ 0.67(1.39)x; yes, because r2 = 0.9767 Practice B 1. No 3. Predicted value is about 67,341,000, which is close to the estimated number. 2. Yes; 6 3. Yes; 3 4. No x 5. f(x) = 29(0.49) 3.8(1.15)x 4. Possible answer: Predicted value is about 130,110,000, which is slightly higher than the estimated number. 6. f(x) = 7. a. f(x) = 0.97(1.16)x 5. Possible answer: Predicted value is 485,703,000, which is more than twice the estimated number. b. 15 cm c. $367.36 8. a. f(x) = 1.14 + 8.42 ln x 6. Looking at the curve, as you move to the right, the graph of the model rises very sharply and grossly overestimates the number of subscribers. b. 9.4 s c. 35.6 m/s Practice C 1. Yes; 1.25 4. Yes; Problem Solving 2. No 1. a. W(t) = 47.34(1.13)t b. 2016 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A52 Holt McDougal Algebra 2