You can use the method for reflection across x axis which makes all the y coordinate -ve.
The parent function is [tex]y = -3sin(x/2) -2[/tex]
The graph of the reflected sine curve and the original sine curve is along the x axis is attached below.
How does reflection work along an axis?
When a graph is reflected along an axis, say x axis, then that leads the graph to go just in opposite side of the axis as if we're seeing it in a mirror.
If you study it more, you will find that its symmetric, thus each point is equidistant from the axis of reflection as that of the image of that point.
Thus, if you're reflecting a point (x,y) along x axis, then its x abscissa will stay same but y ordinates will negate. Thus (x,y) turns to (x, -y)
Similarly, if you're reflecting a point (x,y) along y axis, the resultant image of the point will be (-x,y)
The sine function [tex]y = asin(bx) + c[/tex] has amplitude 'a' and period of [tex]2\pi/ b[/tex]with midline y = c
Since amplitude is specified 3, thus, a = 3
The period is [tex]2\pi/b = 4\pi \implies b = 1/2[/tex]
c = 2 due to midline being y =2
Thus, function is [tex]y = asin(bx) + c = 3sin(x/2) + 2[/tex]
The coordinates of its points will be [tex](x,y) = (x, 3sin(x/2) + 2)[/tex]
When reflected on x axis, the y coordinate will get negated, thus,
points will be [tex](x,y) = (x, -3sin(x/2)-2)[/tex]
Thus, reflected function of the reflected function(thus, parent function) is [tex]y = -3sin(x/2) -2[/tex]
The graph of this parent function and reflected function is attached below.
Learn more about reflection of graphs across axes here:
brainly.com/question/2235973
