Question

Graph each function over a one-period interval.y = ½ sec x

Accepted Answer

Identify the basic function: The function given is $$ y = \frac{1}{2} \sec x . $$The secant function, $$ \sec x $$, is the reciprocal of the cosine function, $$ \cos x . $$Determine the period of the function: The period of $$ \sec x  is $$the same as $$ \cos x $$, which is $$ 2\pi . $$Therefore, the period of $$ y = \frac{1}{2} \sec x  is $$also $$ 2\pi . $$Identify the vertical asymptotes: The secant function has vertical asymptotes where the cosine function is zero. For $$ \cos x = 0 $$, $$ x = \frac{\pi}{2} + k\pi $$, where $$ k  is an $$integer. Within one period $$ [0, 2\pi] $$, the vertical asymptotes are at $$ x = \frac{\pi}{2} $$ and $$ x = \frac{3\pi}{2} . $$Determine the amplitude and transformations: The coefficient $$ \frac{1}{2} $$ affects the vertical stretch of the secant function. This means the graph of $$ y = \frac{1}{2} \sec x $$ will be half as tall as the standard $$ \sec x $$ graph. Sketch the graph: Plot the vertical asymptotes at $$ x = \frac{\pi}{2} $$ and $$ x = \frac{3\pi}{2} . $$Between these asymptotes, sketch the secant curve, which will have a minimum at $$ x = 0 $$ and a maximum at $$ x = \pi $$, with the curve approaching the asymptotes as $$ x $$ approaches $$ \frac{\pi}{2} $$ and $$ \frac{3\pi}{2} .$$

Graph each function over a one-period interval.
y = ½ sec x

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Secant Function

Graphing Trigonometric Functions

Amplitude and Vertical Stretch