Question

what is vertical stretch and translation

Answer 1

for eg ?

Answer 2

a vertical stretch multiplies all the y-coordinates by some value (and changes the shape of the curve without changing the y-intercepts)

ex: y = 6sin (t) changes the amplitude of the sin curve from 1 to 6

a vertical translation keeps the shape of the sin curve, but adds a constant value to all the y-coordinates

ex: y = sin (t) + 6 shifts the entire curve up by 6 units

a horizontal stretch moves all the points farther away from the y-axis (at the same y-values)-- changing the shape

ex: y = sin (t/2) moves all the points twice as far from the y-axis (and the x-intercepts will be twice as far apart as before)
y = sin (2t) is a horizontal shrink (moving every point closer to the y-axis, and halving the distance between x-intercepts)
period is changed

a horizontal shift moves the entire curve right (or left)-- but maintains the distance between x-intercepts (period is unchanged)
ex: y = sin (t + 2) moves it 2 radians to the left)
y = sin (t - 3) moves it 3 radians to the right