Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 10 - Section 10.5 - The Binomial Theorum - Exercise Set - Page 1093: 69

Answer

See below:
1571103654

Work Step by Step

Let us consider the following functions: $\begin{align} & {{f}_{1}}\left( x \right)={{\left( x+1 \right)}^{4}} \\ & {{f}_{2}}\left( x \right)={{x}^{4}} \\ & {{f}_{3}}\left( x \right)={{x}^{4}}+4{{x}^{3}} \\ & {{f}_{4}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}} \\ \end{align}$ $\begin{align} & {{f}_{5}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}}+4x \\ & {{f}_{6}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}}+4x+1 \\ \end{align}$ The graph of the function ${{f}_{2}}\left( x \right)={{x}^{4}}$ has the same shape as the graph of the function ${{f}_{1}}\left( x \right)={{\left( x+1 \right)}^{4}}$, but is shifted 1 unit toward the left. The graph of the function ${{f}_{3}}\left( x \right)={{x}^{4}}+4{{x}^{3}}$ is closer to the graph of the function ${{f}_{1}}\left( x \right)={{\left( x+1 \right)}^{4}}$. The graph of the function ${{f}_{4}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}}$ is approaching closer to the graph of the function ${{f}_{1}}\left( x \right)={{\left( x+1 \right)}^{4}}$ in comparison to the graphs of the functions ${{f}_{2}}\left( x \right)={{x}^{4}}$ and ${{f}_{3}}\left( x \right)={{x}^{4}}+4{{x}^{3}}$. The graph of the function ${{f}_{5}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}}+4x $ is closest to the graph of the function ${{f}_{1}}\left( x \right)={{\left( x+1 \right)}^{4}}$ in comparison to the graphs of the other three functions. And the graph of the function ${{f}_{6}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}}+4x+1$ coincides with the graph of the function ${{f}_{1}}\left( x \right)={{\left( x+1 \right)}^{4}}$ . That is, the graphs of the two functions are the same. So, the graphs of the functions ${{f}_{2}}\left( x \right)={{x}^{4}}$, ${{f}_{3}}\left( x \right)={{x}^{4}}+4{{x}^{3}}$, ${{f}_{4}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}}$, and ${{f}_{5}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}}+4x $ are approaching to the graph of the function ${{f}_{1}}\left( x \right)={{\left( x+1 \right)}^{4}}$, while the graph of the function ${{f}_{6}}\left( x \right)={{x}^{4}}+4{{x}^{3}}+6{{x}^{2}}+4x+1$ is the same as the graph of the function ${{f}_{1}}\left( x \right)={{\left( x+1 \right)}^{4}}$.
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