Explanation The given function is y = x + 4 x 2 . It is known that, the points for which the graph crosses each axis is called the intercepts of the graph. Set y = 0 and obtain the x- intercept of the graph as follows: x + 4 x 2 = 0 x = − 4 x 2 x 3 = − 4 x = − 1.587 Since the above equation is satisfied for the value of x , the curve y = x + 4 x 2 intercept on x -axis at x = − 1.587 . Set x = 0 in the curve y = x + 4 x 2 and obtain the y- intercept of the graph as follows: Since the curve y = x + 4 x 2 is not defined for x = 0 , the curve y = x + 4 x 2 does not intercept on y-axis. Thus, the given curve is intercept on x-axis and does not intercept on y-axis. Obtain the symmetry of the given function as follows. Substitute, x = − x in the given equation y = x + 4 x 2 and check the symmetry of the given function as follows: Here, the curve y = x + 4 x 2 is same as y = − x + 4 x 2 . Hence, the curve y = x + 4 x 2 is not symmetric about y -axis. It is to be noted that, the curve y = x + 4 x 2 is not same as − y = x + 4 x 2 . Thus, the curve y = x + 4 x 2 is not symmetric about x -axis. Obtain the asymptotes of the given function as follows. As x → 0 , 4 x 2 → ∞ , the curve y = x + 4 x 2 tends to the graph of y = ∞ . Thus, the horizontal asymptote of the graph is y = ∞ . Since the function y = x + 4 x 2 is undefined for x = 0 , the vertical asymptote of the graph is x = 0 . It is to be noted that, the function y = x + 4 x 2 is defined for all real numbers except x = 0 . Thus, the domain of the function is ( − ∞ , ∞ ) − { 0 } . The function y = x + 4 x 2 attains negative and positive values both because the denominator x 2 is always positive but the total function value may be negative also. Thus, the range of the function y = x + 4 x 2 is ( − ∞ , ∞ )

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ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 24.6, Problem 6E

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Chapter 24 Solutions

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The graph of the function y = x + 4 x 2 .

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Chapter 24 Solutions