MAT 115: Precalculus Mathematics Constructing Graphs of Trigonometric Functions Involving Transformations by Hand Overview Below are the guidelines for constructing a graph of a trigonometric function involving the function transformations discussed in this course (translations, scale changes, and reflections). Each function possesses its own subtleties, so some of the work involved to create the desired graph may be tedious, but the general concepts apply to all trigonometric graphs. Before analyzing the details of constructing graphs of trigonometric functions, it is important to note the standard requirements for all such graphs in this course. Unless specifically stated otherwise, here is a list of those guidelines: 1. Graphs of trigonometric functions should span at least two periods. In most cases, graphing one period to the left and one period to the right of the phase shift is sufficient, although most graphs provide more meaningful insight if they cover both positive and negative x-values, so either graphing additional periods to show part of the figure on each side of the y-axis or centering the two periods graphed at an x-value other than the phase shift may be more appropriate. 2. Axes must be properly labeled using an appropriate scale. 3. Arrowheads must be properly placed on the ends of curves as necessary to indicate the corresponding portions of the graph continue indefinitely. 4. Graphs must be properly labeled by writing the function adjacent to the curve itself, including a title at the top of the graph, and/or using a legend as appropriate. Information to Be Known Ahead of Time Certain fundamental information regarding each of the six basic trigonometric functions must be known before beginning the procedure of graphing more complicated trigonometric functions involving transformations by hand: 1. The general shape of the graph of each basic trigonometric function 2. The domain, range, period, and amplitude of each basic trigonometric function 3. The value of each basic trigonometric function when evaluated at each of the quadrantal angles The graphs of each of the six basic trigonometric functions have been previously discussed, so they are not included in this document. However, the other necessary values are provided on the next page.
Domain, Range, Period, and Amplitude of the Trigonometric Functions Function Domain Range Period Amplitude sin x x (, ) y [ 1, 1] 2π 1 cos x x (, ) y [ 1, 1] 2π 1 tan x x (2n + 1) π 2 y (, ) π None csc x x nπ y (, 1] [1, ) 2π sec x x (2n + 1) π 2 y (, 1] [1, ) 2π 1 (or None) 1 (or None) cot x x nπ y (, ) π None Values of Trigonometric Functions Evaluated at Quadrantal Angles f(x) f(0) f ( π 2 ) f(π) f ( 3π 2 ) f(2π) sin x 0 1 0 1 0 cos x 1 0 1 0 1 tan x 0 Undefined 0 Undefined 0 csc x Undefined 1 Undefined 1 Undefined sec x 1 Undefined 1 Undefined 1 cot x Undefined 0 Undefined 0 Undefined
Graphing Procedure Constructing the graph of a trigonometric function involving transformations is a matter of manipulating known values of the associated parent function summarized on the previous page, plotting the key points and asymptotes obtained on a set of coordinate axes using an appropriate scale, and properly connecting those key points with appropriate curvature based on the general shape of the parent function. The first step is to determine the domain, range, period, amplitude, and phase shift of the given function through analysis of the sequence of transformations applied to the associated parent function, which is accomplished as follows: Domain: Range: Period: Amplitude: Phase Shift: Manipulate the domain of the parent function by applying the horizontal transformations, in order, if any exist. If none are present, the domain is the same as that of the parent function. Manipulate the range of the parent function by applying the vertical transformations, in order, if any exist. If none are present, the range is the same as that of the parent function. Manipulate the period of the parent function by applying the horizontal scale change, if any. If none is present, the period is the same as that of the parent function. Manipulate the amplitude of the parent function by applying the vertical scale change, if any. If none is present, the amplitude is the same as that of the parent function. The phase shift is the value obtained by combining the mathematical quantities and operations associated with the horizontal transformations, in order. Once the preceding quantities are determined, an appropriate scale for the coordinate axes on which the graph is to be drawn must be determined. The middle of the two periods to be graphed on the x-axis should be located at the value of the phase shift, and each tick mark on the x-axis should represent onefourth of the period if it is sine, cosine, cosecant, secant (since the period of those parent functions is 2π, which spans four quadrants, thus requiring each period to be divided into fourths), and one-half the period if it is tangent or cotangent (since the period of those parent functions is π, which spans two quadrants, thus requiring each period to be divided into halves). The y-axis should simply cover the key values in the range using an appropriate scale. Using the properly labeled coordinate axes, the function values corresponding to each tick mark along the x-axis must be plotted. The proper values can be obtained through manipulation of the values of the associated parent function at each of the quadrantal angles provided in the table on the preceding page. The y-value of the point located at the x-value corresponding to the phase shift is found in the f(0) column in that table. Each tick mark along the x-axis corresponds to a quadrantal angle for the parent function, so the y-value at each tick mark is calculated by applying the vertical transformations, in order, to the value found in the corresponding column of the table for the parent function, noting that any values given as Undefined represent vertical asymptotes. Once the y-values are obtained for each tick mark (including vertical asymptotes if appropriate), the graph can be roughly sketched by mimicking the shape of the parent function based on the key points previously plotted.
Example 1 f(x) = 2 3 cos (6x 5π 4 ) Parent function: y = cos x Domain: x (, ) Range: y [ 1, 1] Period: 2π Amplitude: 1 Transformations applied (in order): 1. Horizontal translation 5π 4 units to the right (mathematical equivalent: x + 5π 4 ) 2. Horizontal compression by a factor of 6 (mathematical equivalent: x 6) 3. Vertical stretch by a factor of 3 (mathematical equivalent: y 3) 4. Vertical reflection; reflection over the x-axis (mathematical equivalent: ) 5. Vertical translation 2 units up (mathematical equivalent: y + 2) Quantities associated with given function: Domain: x (, ) x+ 5π 4 x (, ) x 6 x (, ) Range: y [ 1, 1] y 3 y [, 3] y [, 3] y [ 1, 5] Period: 2π 6 = 2π 6 = π 3 Amplitude: 1 3 = 3 Phase Shift: 5π 6 = 5π 1 = 5π 4 4 6 24 Distance between tick marks along x-axis: π 3 4 = π 3 1 4 = π 12 Tick marks along x-axis will be at x = π 8, π 24, π, π, 5π, 7π, 3π, 11π, 13π 24 8 24 24 8 24 24
Function values (y-values) corresponding to tick marks on x-axis: Function value at middle of two periods to be graphed: f ( 5π 24 ): f(0) = 1 y 3 3 1 ( 5π 24, 1) Function values at tick marks spanning one period to right of middle of two periods to be graphed: f ( 7π 24 ): f (π 2 ) = 0 y 3 2 ( 7π 24, 2) f ( 3π ): f(π) = 1 y 3 8 3 5 ( 3π 8, 5) f ( 11π 24 ): f (3π 2 ) = 0 y 3 2 ( 11π 24, 2) f ( 13π 24 ): f(2π) = 1 y 3 3 1 ( 13π 24, 1) Function values at tick marks spanning one period to left of middle of two periods to be graphed: f ( π 8 ): f (3π 2 ) = 0 y 3 2 ( π 8, 2) f ( π ): f(π) = 1 y 3 24 3 5 ( π 24, 5) f ( π 24 ): f (π 2 ) = 0 y 3 2 ( π 24, 2) f ( π 8 ): f(0) = 1 y 3 3 1 ( π 8, 1) Final graph: f(x) = 2 3 cos (6x 5π 4 )
Example 2 g(x) = 1 + 1 3 cot ( 1 x + 2π) 4 Parent function: y = cot x Domain: x nπ Range: y (, ) Period: π Amplitude: None Transformations applied (in order): 1. Horizontal translation 2π units to the left (mathematical equivalent: x 2π ) 2. Horizontal stretch by a factor of 4 (mathematical equivalent: x 4) 3. Vertical compression by a factor of 3 (mathematical equivalent: y 3) 4. Vertical translation 1 unit down (mathematical equivalent: y 1) Quantities associated with given function: Domain: x nπ x 2π x nπ 2π x 4 x 4(nπ 2π) = 4nπ 8π Range: y (, ) y 3 y (, ) y (, ) Period: Amplitude: Phase Shift: π 4 = 4π None 2π 4 = 8π Distance between tick marks along x-axis: 4π 2 = 2π Tick marks along x-axis will be at x = 12π, 10π, 8π, 6π, 4π Since tick marks are all negative (i.e., to left of y-axis), consider extending additional periods to right to obtain at least one period to right of y-axis: Tick marks at x = 12π, 10π, 8π, 6π, 4π, 2π, 0, 2π, 4π
Function values (y-values) corresponding to tick marks on x-axis: Function value at middle of two periods to be graphed: f( 8π): f(0) is Undefined Vertical asymptote at x = 8π Function values at tick marks spanning one period to right of middle of two periods to be graphed: f( 6π): f ( π 2 ) = 0 y 3 1 ( 6π, 1) f( 4π): f(π) is Undefined Vertical asymptote at x = 4π Function values at tick marks spanning one period to left of middle of two periods to be graphed: f( 10π): f ( 3π 2 ) = 0 y 3 1 ( 10π, 1) f( 12π): f(π) is Undefined Vertical asymptote at x = 12π Function values at additional tick marks to obtain one period to right of y-axis: f( 2π): f ( π 2 ) = 0 y 3 1 ( 2π, 1) f(0): f(0) is Undefined Vertical asymptote at x = 0 f(2π): f ( 3π 2 ) = 0 y 3 1 (2π, 1) f(4π): f(π) is Undefined Vertical asymptote at x = 4π Final graph: g(x) = 1 + 1 3 cot ( 1 x + 2π) 4