Respuesta :

Answer:

ii),iv and vi

Step-by-step explanation:

Given is a graph which has 3 pieces and we have to find out the equation of the graph from the given options

i) OPtion a) [tex]y=\sec\ \frac{x}{3}+2[/tex]

This graph will be y(0) = sec0+2 =3

But y intercept is not 3.  Hence incorrect

ii) [tex]y=-\sec\ \frac{x}{3}+2[/tex]

When x=0 we have y intercept = -3+2 =-1 matches with graph

Also x intercepts when sec x/3 =2

Or when x = multiples of pi(since sec pi/3 =2 )

So ii is correct

iii) [tex]y=-\sec\ ( \frac{x}{3}+\pi)+2[/tex]

This graph is the transformation of the parent graph by shifting horizontally pi units to left

y(0) = -secpi+2 =1 .

When x =pi, y = -(-2)+2 is not zero. So x intercepts do not match.  Incorrect

iv) [tex]y=\sec\ ( \frac{x}{3}+\pi)+2[/tex]

this matches with y intercept as 1, and x intercepts as -pi and pi

Hence correct

v) [tex]y=-\csc\ \left(\frac{x}{3}+\frac{3\pi}{2}\right)+2[/tex]

This is incorrect since x intercepts 2pi and -2pi do not match with the given graph

vi) [tex]y=\csc\ \left(\frac{x}{3}+\frac{3\pi}{2}\right)+2[/tex]

This has x intercepts as pi and -pi and also y intercept as 1.

Hence correct

Answer:

The correct options are:

  • [tex]y=f(x)=-\sec (\dfrac{x}{3})+2[/tex]
  • [tex]y=f(x)=\sec(\dfrac{x}{3}+\pi)+2[/tex]
  • [tex]y=f(x)=\csc (\dfrac{x}{3}+\dfrac{3\pi}{2})+2)[/tex]

Step-by-step explanation:

a)

[tex]y=f(x)=-\sec (\dfrac{x}{3})+2[/tex]

When we plot the graph of this function we see that it matches the given graph.

Hence, option: a is correct.

b)

  [tex]y=f(x)=\sec (\dfrac{x}{3})+2[/tex]

       when x=0 then we hvae:

  y=3≠1

( since in the given graph at x=0 the graph is at 1)

Hence, option: b is incorrect.

c)

     [tex]y=f(x)=-\sec(\dfrac{x}{3}+\pi)+2[/tex]

We know that:

[tex]\sec(\pi+\theta)=-sec \theta[/tex]

Hence, we get the above expression as:

 [tex]y=\sec (\dfrac{x}{3})+2[/tex] which is same as the expression in option: b

Hence, option: c is incorrect.

d)

[tex]y=f(x)=\sec(\dfrac{x}{3}+\pi)+2[/tex]

We know that:

[tex]\sec(\pi+\theta)=-sec \theta[/tex]

Hence, we get the above expression as:

 [tex]y=-\sec (\dfrac{x}{3})+2[/tex] which is same as the expression in option: a

Hence, option: d is correct.

e)

[tex]y=f(x)=-\csc (\dfrac{x}{3}+\dfrac{3\pi}{2})+2--------(1)[/tex]

We know that:

[tex]\csc (\dfrac{3\pi}{2}+\theta)=-\sec \theta[/tex]

Hence, we get expression (1) as:

 [tex]y=\sec (\dfrac{x}{3})+2[/tex] which matches the expression as in option: b.

Hence, option: e is incorrect.

f)

[tex]y=f(x)=\csc (\dfrac{x}{3}+\dfrac{3\pi}{2})+2--------(2)[/tex]

We know that:

[tex]\csc (\dfrac{3\pi}{2}+\theta)=-\sec \theta[/tex]

Hence, we get expression (2) as:

 [tex]y=-\sec (\dfrac{x}{3})+2[/tex] which matches the expression as in option: a.

Hence, option: f is correct.

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