Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 1 - Section 1.3 - New Functions from Old Functions - 1.3 Exercises: 7


Translating the function by $4$ to the left, then reflecting it with respect to the $y$ axis and finally translating it downwards by $1$ we get $$y=-\sqrt{-x^2-5x-4}-1$$

Work Step by Step

First, note that the "width" of the graph remains the same because $3-0=-1-(-4)$ so there is no horizontal stretching, there is only leftward translation by $4$. Also, the "height" of the graph is unchanged because $1.5-0=-1-(-2.5)$ but the function is reflected with respect to the $y$ axis and then translated downwards by $1$. Applying these transformations one after another (the order here does matter!) we get: $$f(x)\to f(x+4)\to -f(x+4) \to -f(x+4)\to-f(x+4)-1.$$ This gives $$y=-\sqrt{3(x+4)-(x+4)^2}-1=\\-\sqrt{3x+12-x^2-8x-16}-1=\\-\sqrt{-x^2-5x-4}-1$$
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