Answer :
Answer:
Step-by-step explanation:
Alright, lets get started.
The given equation is:
[tex]y= csc [3(x+4)]+6[/tex]
[tex]y=csc(3x+12) +6[/tex]
Period = [tex]\frac{2\pi }{b}=\frac{2\pi }{3}[/tex]
hence the period is = [tex]\frac{2\pi }{3}[/tex]
Phase shift = [tex]\frac{-c}{b}=\frac{-12}{3}=-4[/tex]
hence phase shift 4 units to the left
ertical shift is 6 units up
So, the answer is C, period 2pi/3, phase shift 4 units left and vertical shift 6 units up : Answer
Hope it will help :)
Shifts are position change. The shifts and period in given function is given by: Option C: period: 2π/3 ; phase shift: 4 units left; vertical shift: 6 units up
What is period of a function?
Suppose that a function f(x) is such that:
[tex]f(x) = f(x + T); \: \forall \: x \in D(f)[/tex]
where D(f) is domain of function f, then we say that function is periodic and its period is of 'T' length.
It means function is generating same values after T units travel on x axis(input axis).
What is vertical and horizontal shift(phase shift)?
- Phase shift: When a point is shifted horizontally on the coordinate plane, then it is called to be shifted horizontally. If it shifted, say p units, then its phase shift is of p units.
- Vertical shift: When a point is shifted vertically on the coordinate plane, then it is called to be shifted vertically. If it shifted, say q units, then its vertical shift is of q units.
For functions, usually output of functions are taken as y coordinate (vertical height) and input is taken as horizontal distance from origin(x coordinate).
Thus, as the function is [tex]y = csc[3(x+4)] + 6[/tex]
we see that its input value variable x is added with 4, so its phase shift is of 4 units, and that is actually left shift (since the values which used to come after 4 units of original input, are now going to come 4 units early(so left as left side has smaller values than right side in coordinate plane) ).
The output is added with 6 at the end, so its vertical shift is of 6 units. Unlike above logic, the heights are now up instead of down as there is no past future of the graph on input, its just that all heights are increased with 6 units and height increment means upward motion in coordinate plane.
We have a fact that:
[tex]csc(x) = csc(2\pi + x)[/tex]
Its original period is of 2π
The given function, thus, has:
[tex]y = csc[3(x+4)] + 6 = csc[3(x+4) + 2\pi] + 6 = csc[3(x + 2\pi/3 + 4)] + 6[/tex]
Thus, its period is [tex]2\pi/3[/tex] as increasing input by [tex]2\pi/3[/tex] makes the function land on same output.
The shifts and period in given function is given by: Option C: period: 2π/3 ; phase shift: 4 units left; vertical shift: 6 units up
Its plot, with comparison to plot of [tex]y = csc3x[/tex] is given below.
Learn more about phase shift and vertical shifts here:
https://brainly.com/question/10390652
