Lesson How Functions Can Work

Source code of 'How Functions Can Work'

This Lesson (How Functions Can Work) was created by by Nate(3500)

Lets start with the basic linear function.
f(x) = x
{{{graph(300,300,-5,5,-5,5,x)}}}
f(x) = x + 1 wouldn't you easily think that all points would increase one (so vertically shifts one unit)
{{{graph(300,300,-5,5,-5,5,x+1)}}}
f(x) = 2x + 1 you could easily understand that the slope is affected (the slope is steeper)
{{{graph(300,300,-5,5,-5,5,2x+1)}}}
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Now, lets use a parabolic function.
f(x) = (x)^2
{{{graph(300,300,-5,5,-5,5,(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
{{{graph(300,300,-5,5,-5,5,(x - 1)^2)}}}
f(x) = (x - 1)^2 + 2  whenever you add to a function like this, all the units shift vertically (graph shifts two units vertically)
{{{graph(300,300,-5,5,-5,5,(x - 1)^2 + 2)}}}
f(x) = (2)(x - 1)^2 + 2 the value of {{{a}}} determines the contractions of the parabola (the parabola bends in 2x)
{{{graph(300,300,-5,5,-5,5,2(x - 1)^2 + 2)}}}
f(x) = (2)(2x - 1)^2 + 2 further bending of the parabola, but distorts the vertex
{{{graph(300,300,-5,5,-5,5,2(2x - 1)^2 + 2)}}}
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Now, lets use a trig function.
f(x) = sin(x)
{{{graph(300,300,-5,5,-5,5,sin(x))}}}
f(x) = sin(x + 1)  works the same, all values shifts -1 units horizontally
{{{graph(300,300,-5,5,-5,5,sin(x+1))}}}
f(x) = sin(x + 1) + 1 just like the parabola, the 'wave' is vertically shifted one unit
{{{graph(300,300,-5,5,-5,5,sin(x + 1) + 1,1)}}}
f(x) = 2sin(x + 1) + 1 unlike the parabola, this value determines the height of the 'wave'
{{{graph(300,300,-5,5,-5,5,2*sin(x + 1) + 1,1)}}}
f(x) = 2sin(2x + 1) + 1 the coefficient determines the frequency (2{{{pi}}}/c) ~> where {{{c}}} is the coefficient ~> (frequency = {{{pi}}})
{{{graph(300,300,-5,5,-5,5,2*sin(2x + 1) + 1,1)}}}