Answer: g(x) = 2sin(x)+3
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Explanation:
We start with y = sin(x) or f(x) = sin(x). Recall that y = f(x) since both are outputs.
Vertically stretching by a factor of 2 means we multiply each y value by 2.
This means,
y = sin(x)
2y = 2sin(x)
Then we shift 3 units up. So we add 3 to the y coordinate
2y+3 = 2sin(x)+3
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We can track an example point to see this movement in action. For instance, let's focus on the point (pi/6,1/2) which is on f(x). I'm in radian mode.
Stretching f(x) vertically by a factor of 2 means (pi/6,1/2) moves to (pi/6, 1) after the stretching occurs. I multiplied the y coordinate by 2, and kept the x coordinate the same.
Then we shift up 3 units to go from (pi/6, 1) to (pi/6, 4). You add 3 to the y coordinate and keep the x coordinate the same. Note that plugging x = pi/6 into g(x) = 2sin(x)+3 leads to 4.
