Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.1 Parametric Equations - Exercises - Page 603: 38


The answer is $y=\cos(x-3)+4$.

Work Step by Step

The maximum of $y=\cos x$ occurs at $(x,y)=(0,1)$. The parametric equation of $y=\cos x$ is $c(t)=(t,\cos t)$. To translate the maximum of $c(t)$ at $(0,1)$ to $(3,5)$ is to move the parametric curve horizontally 3 units and vertically 4 units. So we replace $c(t)=(t,\cos t)$ by $c(t)=(3+t,4+\cos t)$. So, the coordinates of the translated curve are $x=t+3$ and $y=\cos t+4$. Solving for $t$ from $x$ we get $t=x-3$. Substituting $t$ into $y$ gives $y=\cos(x-3)+4$.
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