Lesson How Functions Can Work
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Algebra: Functions, Domain, NOT graphing
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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)}}} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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)}}} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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)}}}