Lesson How Functions Can Work

Lets start with the basic linear function.
f(x) = x

f(x) = x + 1 wouldn't you easily think that all points would increase one (so vertically shifts one unit)

f(x) = 2x + 1 you could easily understand that the slope is affected (the slope is steeper)

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Now, lets use a parabolic function.
f(x) = (x)^2

f(x) = (x - 1)^2 knowing: f(x) = a(x - h)^2 + k where (h,k) is the vertex, the x-value is shifted -h units, parabola shifts one unit to the right

f(x) = (x - 1)^2 + 2 whenever you add to a function like this, all the units shift vertically (graph shifts two units vertically)

f(x) = (2)(x - 1)^2 + 2 the value of determines the contractions of the parabola (the parabola bends in 2x)

f(x) = (2)(2x - 1)^2 + 2 further bending of the parabola, but distorts the vertex

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Now, lets use a trig function.
f(x) = sin(x)

f(x) = sin(x + 1) works the same, all values shifts -1 units horizontally

f(x) = sin(x + 1) + 1 just like the parabola, the 'wave' is vertically shifted one unit

f(x) = 2sin(x + 1) + 1 unlike the parabola, this value determines the height of the 'wave'

f(x) = 2sin(2x + 1) + 1 the coefficient determines the frequency (2/c) ~> where is the coefficient ~> (frequency = )