Explanation The given function is y = x 2 − 4 x 2 + 4 . 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 2 − 4 x 2 + 4 = 0 Since the above equation is satisfied for value of x = − 2 , 2 the curve y = x 2 − 4 x 2 + 4 intercept on x -axis. Set x = 0 in the curve y = x 2 − 4 x 2 + 4 and obtain the y- intercept of the graph as follows: Since the curve y = x 2 − 4 x 2 + 4 is defined for x = 0 , the curve y = x 2 − 4 x 2 + 4 intercept on y -axis. Thus, the given curve has x- intercept and y- intercept. Obtain the symmetry of the given function as follows. Substitute, x = − x in the given equation y = x 2 − 4 x 2 + 4 and check the symmetry of the given function as follows: Here, the curve y = x 2 − 4 x 2 + 4 is same as y = ( − x ) 2 − 4 ( − x ) 2 + 4 . Hence, the curve y = x 2 − 4 x 2 + 4 is symmetric about y -axis. It is to be noted that, the curve y = x 2 − 4 x 2 + 4 is not same as − y = x 2 − 4 x 2 + 4 . Thus, the curve y = x 2 − 4 x 2 + 4 is not symmetric about x-axis. Obtain the asymptotes of the given function as follows. As x → ∞ or x → − ∞ , the curve y = x 2 − 4 x 2 + 4 tends to the graph of y → 1 . Thus, the horizontal asymptote of the graph is y → 0 . Since the function y = x 2 − 4 x 2 + 4 is defined for x = 0 , f ( x ) → − 1 so there are no the vertical asymptote of the graph. It is to be noted that, the function y = x 2 − 4 x 2 + 4 is defined for all real numbers. Thus, the domain of the function is ( − ∞ , ∞ ) . The function y = x 2 − 4 x 2 + 4 attains positive and negative values

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Chapter 24.6, Problem 18E

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Chapter 24 Solutions

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

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

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Chapter 24 Solutions