Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.2 Mathematical Models: A Catalog of Essential Functions - 1.2 Exercises - Page 33: 7

Answer

(a) $y = 2x + b$ (b) $y = mx + 1 - 2m$ (c) $y = 2x-3$ The graph sketches are shown in the following image:
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Work Step by Step

(a) A linear equation has the form $y = mx + b$. Thus, the family of linear equations with slope 2 would have the form $y = 2x + b$. The sketches show arbitrary values of b for said equation. (b) A linear equation can also have the form $y-y_1 = m(x-x_1)$. Thus, the family of linear equations such that $f(2) = 1$ would have the form: $y-1 = m(x-2)$ $y-1 = mx - 2m$ $y = mx + 1 - 2m$ The sketches show arbitrary values of m for said equation. (c) Input $m=2$, the characteristic of the family of linear equations in (a), to the equation in (b) to find the equation that belongs to both families. This results in: $y = 2x + 1 - 4$ $y = 2x -3$
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