Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 1 - Functions and Their Graphs - Section 1.5 Graphing Techniques: Transformations - 1.5 Assess Your Understanding - Page 98: 22


$y= x^3-4$

Work Step by Step

In order to find the answer, we will have to recall the following some point about the graph of $y=f(x)$. a) The graph of the function $y=-f(x)$ involves a reflection about the $x$-axis of the original function $f(x)$. b) The graph of the function $y=f(x)+a$ defines a vertical shift of $|a|$ units upward when $a \gt 0$, and downward side when $a\lt 0$ of the original function $f(x)$. c) The graph of $y=f(x-p)$ defines a horizontal shift of $|p|$ units to the right when $p \gt 0$, and to the left when $p \lt 0$ of the original function $f(x)$. (d) The graph of $y=k\cdot f(x)$ can be obtained a vertical stretch when $k\gt 1$ or compression when $0\lt k \lt1$) of the original function $f(x)$. As mentioned in point $(b)$, the resulting graph would attain a $4$-unit shift downward of the original function $f(x)$ and $a=-4$. So, we have new function as: $y=f(x)+a \\ y= x^3-4$
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