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This is the second of our many quizzes on Graphing trig functions: sec, scs, cot. Click on this to access more such competitive quizzes on Trigonometry.

Graphing trig functions: sec, scs, cot Quiz 2

mathematics

 Which of the following is the graph of $y=\sec \left( 2x-1 \right)+3$.

 Apply the properties of $\sec x$ and find the period, vertical and horizontal shifts of given function and compare the options.

  The period $\sec x$ is $2\pi $ $ \\ $ A typical cycle of it is from $x=-\pi $ to $x=\pi $. $ \\ $ For $\sec \left( 2x-1 \right)$, a typical cycle is from $2x-1=-\pi $ to$2x-1=\pi $, which gives us from $x=\dfrac{-\pi +1}{2}$ to $x=\dfrac{\pi +1}{2}$. $ \\ $ Hence, the period is $\pi $. $ \\ $ The domain of $\sec x$ is all real except for $x=\left( \dfrac{\pi }{2} \right)\pm m\pi $, where m is an integer $ \\ $ The domain of $\sec \left( 2x-1 \right)$ is all real except for$2x-1=\left( \dfrac{\pi }{2} \right)\pm m\pi $, which gives us $x=\left( \dfrac{\pi }{4} \right)\pm \left( \dfrac{m\pi }{2} \right)+\dfrac{1}{2}$ $ \\ $ Therefore, the vertical shift is $3$ and the phase shift is $\dfrac{1}{2}$ to the right. $ \\ $ The range of $\sec \left( 2x-1 \right)$ is $\left( -\infty ,2 \right]\cup \left[ 4,\infty  \right),\left\{ y|y\le 2,y\ge 4 \right\}$ $ \\ $ The given equation represents graph number $(c)$.

 How many Y – intercepts do the graph of the function $y=5+\sec \left( 3x \right)$ have and also find its Y – intercept?

 Apply $x=0$ for finding the Y- intercept in the given function and find the value of $'y'$ that gives the required answer.

  The graph of given function is  $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/4931732a-8f0b-4113-b17f-02c05b322668729653538096162620.png"/> $ \\ $ Here, there is only one Y – intercept. $ \\ $ Given, $y=5+\sec \left( 3x \right)$ $ \\ $ To find Y-intercepts we must put $x=0$ and solve for it. $ \\ y=5+\sec \left( 3\left( 0 \right) \right) \\ y=5+\sec \left( 0 \right) \\ y=5+1 \\ y=6 \\ $ Therefore, there is $1$ Y-intercept in the given equation which is $\left( 0,6 \right)$.

 What is the vertical asymptote in the graph of $y=3\cot \left( x-\dfrac{\pi }{4} \right)+2$.

 Use the asymptotes of $\cot x$ and find the asymptotes of a given function by equating the inside value of the function to asymptotes of $\cot x$ which is the required answer.

Given, $y=3\cot \left( x-\dfrac{\pi }{4} \right)+2$ $ \\ $ For $y=\cot x$, vertical asymptotes will happen at $x=n\pi $, where $n$ is the integer. $ \\ $ Use the period of $y=\cot x$ which is $\left( 0,\pi  \right)$ to find the asymptotes of $y=3\cot \left( x-\dfrac{\pi }{4} \right)+2$ $ \\ $ Assuming the given equation as $y=a\cot \left( bx+c \right)+d$ and putting $\left( bx+c \right)$ equals to zero gives us, $ \\ \Rightarrow ~\left( x-\dfrac{\pi }{4} \right)=0 \\ \Rightarrow x=\dfrac{\pi }{4} \\ $ Now, putting $\left( bx+c \right)$  as $\pi $, gives us $ \\ \Rightarrow \left( x-\dfrac{\pi }{4} \right)=\pi  \\ \Rightarrow x=\dfrac{5\pi }{4} \\ $ Therefore, period for  $y=3\cot \left( x-\dfrac{\pi }{4} \right)+2$ will always happen at $\left( \dfrac{\pi }{4},\dfrac{5\pi }{4} \right)$, where $\dfrac{\pi }{4}$ and $\dfrac{5\pi }{4}$ are vertical asymptotes. $ \\ $ It will occur at $\dfrac{\pi }{4}$, $\dfrac{5\pi }{4}$ and every $x=\dfrac{\pi }{4}+n\pi $, where n is an integer. $ \\ $ The graph of given function is  $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/8a013c94-339f-493e-af9b-cc610eaf65831202826095876378764.png"/>

 How many complete repetitions of the function will be there in the graph of $y=\cot \left( \dfrac{x}{2} \right)-4$ in the interval $[0,8\pi ]$.

 Use the properties of cotangent trigonometry curve and find the number of points of discontinuity of a given function in $\left( 0,8\pi  \right)$ which gives the required answer as the function repeats at every discontinuity point.

 Given, $y=\cot \left( \dfrac{x}{2} \right)-4$ $ \\ \cot \dfrac{x}{2}=\dfrac{1}{\tan \dfrac{x}{2}}=\dfrac{\cos \dfrac{x}{2}}{\sin \dfrac{x}{2}} \\ $ Considering $\sin \dfrac{x}{2}=0$, the discontinuity will happen when,
$x=0,2\pi ,4\pi ,6\pi ,8\pi $ in the interval of $\left[ 0,8\pi  \right]$ $ \\ $ In the interval of $\left[ 0,2\pi  \right]$ we get a full complete cycle for the given equation.
Therefore, we get a total of 4 complete repetitions of the function in the interval of $\left[ 0,8\pi  \right]$. $ \\ $ The graph of given function is $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/c9e3bfcb-fb01-44ef-90a5-599496a3a7254336880923691533485.png"/>

 What will be the equation of the graph given below? $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/d7c4172a-2b08-4443-b4d1-8145d426f4156532277423844743184.png"/>

 Check the options for which the period, vertical shift and horizontal shift of given graph matches by comparing with $\csc x$ as all options are cosecant functions.

 We know that $\sin x$ has a period of $2\pi $, so does $\operatorname{cosec}x$.  $ \\ $ A typical cycle of it is from $x=0$ to $x=2\pi $. $ \\ $ For $\operatorname{cosec}\left( \dfrac{2x}{3}+\dfrac{\pi }{3} \right)$, a typical cycle is from $\dfrac{2x}{3}+\dfrac{\pi }{3}=0$ to$\dfrac{2x}{3}+\dfrac{\pi }{3}=2\pi $, which gives us from $x=-\dfrac{\pi }{2}$ to $x=\dfrac{5\pi }{2}$. $ \\ $ Hence, the period is $3\pi $. $ \\ $ The domain of $\operatorname{cosec}x$ is all real except for $x=m\pi $, where m is an integer $ \\ $ The domain of $\operatorname{cosec}\left( \dfrac{2x}{3}+\dfrac{\pi }{3} \right)$ is all real except for $\dfrac{2x}{3}+\dfrac{\pi }{3}=m\pi $, which gives us $x=\dfrac{3m\pi }{2}-\dfrac{\pi }{2}$ $ \\ $ Therefore, vertical shift is $-2$ (downward) and the phase shift is $-\dfrac{\pi }{2}$ (leftwards). $ \\ $ The range of $\operatorname{cosec}\left( \dfrac{2x}{3}+\dfrac{\pi }{3} \right)$ is $(-\infty \:,\:-4]\cup \:[4,\:\infty \:),\left\{ y|y\le -4,y\ge 4 \right\}$ $ \\ $ Therefore, equation $y=4~\csc \left( \dfrac{2}{3}x+\dfrac{\pi }{3} \right)-2$ represents the given graph.