Transform sine vertically, stretch and translate together

Graph 1: Vertical Stretch, Vertical Translation

Interact with the applet to see how the values a and d transform the curve y=sin(x°).

[br][br]Notice that when d=0, the graph rises from the x axis as much as it falls. The rise and fall from the horizontal center (sometimes known as the sinusoidal axis) is known as the [url=https://www.mathsisfun.com/algebra/amplitude-period-frequency-phase-shift.html]amplitude[/url].[br][br]Notice how the y value of the local maximum and of the local minimum can be calculated by the values of a and d.

1. Order of operations

Suppose you wish to transform [math]y=\text{sin}(x°)[/math] to [math]y=3\text{sin}(x°)+5[/math]. Does it make mathematical sense to [br][br]1. stretch (multiply y values by 3) then to translate (add 5 to all new y values)?[br][br]or to [br][br]2. translate (add 5 to all y values) then to stretch (multiply new y values by 3)?[br][br](Hint - take the point [math](90,1)[/math] as an example - does the y value become [math]1\times3+5[/math] or [math]\left(1+5\right)\times3[/math]?)

Graph 2.

2(a)

Graph 2 has equation [math]y=f(x)[/math] where [math]f(x)=a\text{sin}(x°)+d[/math]. Which one of the following is a correct equation for this function?

[math]f\left(x\right)=5\text{sin}\left(x°\right)+1[/math] [math]f\left(x\right)=3\text{sin}\left(x°\right)+2[/math] [math]f\left(x\right)=2\text{sin}\left(x°\right)+3[/math] [math]f\left(x\right)=-2\text{sin}\left(x°\right)+3[/math]

2(b)

What is the range of the function [math]f(x)[/math], shown in graph 2?

Graph 3.

3(a)

The graph of the function [math]y=g(x)[/math] where [math]g(x)=a\text{sin}(x°)+d[/math] is shown in graph 3. What is the equation of the function [math]g(x)[/math]?

3(b)

What is the range of the function [math]g(x)[/math]?

4

The function [math]h(x)[/math] is defined [math]h(x)=7\text{sin}(x°)+10[/math]. What is the range of [math]h\left(x\right)[/math]?