Graphing Sine & Cosine
What is the amplitude, period, range of sine and cosine?
//Parent funtion graph//
y = sin(x)
● amplitude = 1 ; period = 2π ; range: -1≤y≤1
y = cos(x)
● amplitude = 1 ; period = 2π ; range: -1≤y≤1
What is the graphing parent function formula? What is the tangents function period and the location of vertical asymptotes?
graphing of the parent function y=tan(x)
■ period = pi
■ function is undefined at x=pi/2 + npi (locations of vertical
asymptotes)
What is the general parent function formula for cotangent? What is the period? how do you find the vertical asymptotes?
The graph of parent function y=cot(x)
■ period = pi
■ function is undefined at x= npi (locations of vertical asymptotes)
The graph of parent function y=sec(x) and y=csc(x)
■ period = 2pi
For function y = arcsin x if and only if sin y=x
what is the domain and range?
domain: -1≤x≤1 range: -pi/2 ≤y≤ pi/2
what is the general formula?
How do you find the amplitude and period of sine and cos?
General formula: y = a sin(bx-c)+d and y= a cos(bx-c)+d
■ amplitude = |a|
● changing the amplitude will vertically stretch or
shrink the graph of the parent functions
○ |a|>1 ⇒ vertical stretch
○ |a|<1 ⇒ vertical shrink
■ Period = 2π/b (b is a positive number)
● changing the period will horizontally stretch or shrink
the graph of the parent function
○ b>1 ⇒ horizontal shrink
○ 0<b<1 ⇒ horizontal stretch
How do you find the two consecutive vertical asymptotes?
Location of first asymptote: bx-c=-pi/2 ⇒ x =
-pi/(2b)+c/b
Location of second asymptote: 1st asymptote + period
⇒ x = -pi/(2b)+c/b+pi/b = pi/(2b)+c/b
How do you find the 2 consecutive vertical asymptotes? the first & second asymptotes
Location of first asymptote: bx-c=0 ⇒ x = c/b
Location of second asymptote: bx-c=pi ⇒ x = pi/b+c/b
what is the formula of secant & cosecant to graph the functions?
y = a sec(bx-c)+d
y=a csc(bx-c)+d
For the function y = arccos x if and only if cos y=x
what is the domain and range?
domain: -1≤x≤1 range: 0≤y≤ pi
How do you find the phase shift and vertical translation shift of sine and cosine?
■ Phase Shift = c/b
● Horizontally shift the graph of the parent function
○ c > 0 ⇒ horizontal shift to the right
○ c < 0 ⇒ horizontal shift to the left
■ Vertical Translation (Shift) = d
● Vertically shift the graph of the parent function
○ d>0 ⇒ vertical shift d units up
○ d<0 ⇒ vertical shift d units down
How do you locate the middle between the two consecutive asymptotes?
x = (first asymptote + second asymptote)/2
Middle is an intercept with midline (x-intercept if d =0))
How do you find the middle between the two consecutive asymptotes?
x = (first asymptote + second asymptote)/2
Middle is an intercept with midline (x-intercept if d =0))
How do you sketch the reciprocal functions?
If secant function, first graph y = a cos(bx-c)+d
If cosecant function, first graph y = a sin(bx-c)+d
For the function y = arctan x if and only if tan y=x
what is the domain and range?
domain: -inf<x<inf range: -pi/2 ≤y≤ pi/2
How do you find the start, middle, and end of the cycle of sine & cosine?
How to locate the start of one cycle: x=c/b (phase shift)
● If sine curve, the start of the cycle is an intercept with midline
● If cosine curve, the start of the cycle is at a max (or min if a<0)
How to locate the end of one cycle: x=c/b + 2π/b (phase shift + period)
● If sine curve, the end of the cycle is an intercept with midline
● If cosine curve, the end of the cycle is at a max (or min if a<0)
How to locate the middle of the cycle: x = (start of cycle + end of cycle)/2
● If sine curve, the middle is an intercept with midline
● If cosine curve, the middle is at a min (or max if a<0).
How do you locate the first-quarter point between the two consecutive asymptotes?
x = (first asymptote + middle)/2
For y-coordinate, go down ‘a’ from the midline
How do you find the first-quarter point between the two consecutive asymptotes?
x = (first asymptote + middle)/2
For y-coordinate, go up ‘a’ from the midline
How do you locate and sketch the vertical asymptotes?
The asymptotes are located where the graph from step 1
intercepts the midline (x-intercepts if d = 0)
if -1≤x≤1 and -pi/2 ≤y≤ pi/2 then...
if -1≤x≤1 and 0 ≤y≤ pi then...
sin(arcsin x) = x and arcsin(sin y) = y
cos(arccos x) = x and arccos(cos y) = y
How do you find the first and third quarter for sine and cosine?
Locating the first quarter point of the cycle:
x = (start of cycle + middle of cycle)/2
● If sine curve, the first quarter point is a max (or min if a<0).
● If cosine curve, the first quarter point is intercept with midline
Locating the third quarter-point of the cycle.
x = (middle of cycle + end of cycle)/2
● If sine curve, that point is a min (or max if a<0).
● If cosine curve, that point is an intercept with midline
How do you find the third-quarter point between the two consecutive asymptotes?
x = (middle + second asymptote)/2
For y-coordinate, go up ‘a’ from the midline
How do you find the third-quarter point between the two consecutive asymptotes?
x = (middle + second asymptote)/2
For y-coordinate, go down ‘a’ from the midline
The max or min value of the graph from step 1
step 1 remainder :
If secant function, first graph y = a cos(bx-c)+d
If cosecant function, first graph y = a sin(bx-c)+d
if x is a real number and -pi/2 < y < pi/2 then...
tan(arctan x) = x and arctan(tan y) = y