Precalculus (6th Edition) Blitzer

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

Chapter 1 - Section 1.3 - More on Functions and Their Graphs - Exercise Set - Page 196: 31

Answer

The graph of the equation ${{y}^{4}}={{x}^{3}}+6$ is symmetric to the x-axis.

Work Step by Step

Consider the equation, ${{y}^{4}}={{x}^{3}}+6$ Check symmetry about the y-axis: An equation is symmetric about the y-axis if $-x$ is substituted in the function and the result is an equivalent equation then the graph of the equation is symmetric with respect to the y-axis. Substitute $x=-x$ in the equation, $\begin{align} & {{y}^{4}}={{\left( -x \right)}^{3}}+6 \\ & {{y}^{4}}=-{{x}^{3}}+6 \end{align}$ Therefore, the equation is not symmetric about the y-axis Now, check symmetry about the x-axis: An equation is symmetric about the x-axis, if $-y$ is substituted in the function and it leads to an equivalent equation than the graph is symmetric with respect to the x-axis. Substitute $y=-y$ in the equation ${{y}^{4}}={{x}^{3}}+6$. $\begin{align} & {{\left( -y \right)}^{4}}={{x}^{3}}+6 \\ & {{y}^{4}}={{x}^{3}}+6 \end{align}$ Therefore, the equation is symmetric about the x-axis. Now, check symmetry about the origin: An equation is symmetric about the origin if $x=-x,y=-y$ and it leads to an equivalent equation, this implies that the function is symmetric about the origin. Substitute $x=-x,y=-y$ in the equation ${{y}^{4}}={{x}^{3}}+6$. $\begin{align} & {{\left( -y \right)}^{4}}={{\left( -x \right)}^{3}}+6 \\ & {{y}^{4}}=-{{x}^{3}}+6 \end{align}$ Therefore, the equation is not symmetric about the origin. Hence, the equation is symmetric about the x-axis only.
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