Chapter 2 Graphs of Trig Functions,Graphs of Basic Parent Trigonometric Functions. It is instructive to view the parent trigonometric functions on the same axes as their reciprocals. Identifying patterns between the two functions can be helpful in graphing them. Looking at the sine and cosecant functions,we see that they intersect at their maximum. and minimum values i e when 1 The,vertical asymptotes not shown of the. cosecant function occur when the sine,function is zero. Looking at the cosine and secant functions,we see that they intersect at their maximum. and minimum values i e when 1 The,vertical asymptotes not shown of the secant. function occur when the cosine function is,Looking at the tangent and cotangent. functions we see that they intersect when,sin cos i e at an. integer The vertical asymptotes not,shown of the each function occur when the. other function is zero,Version 2 0 Page 16 of 109 January 1 2016. Chapter 2 Graphs of Trig Functions,Characteristics of Trigonometric Function Graphs. All trigonometric functions are periodic meaning that they repeat the pattern of the curve called a. cycle on a regular basis The key characteristics of each curve along with knowledge of the parent. curves are sufficient to graph many trigonometric functions Let s consider the general function. where A B C and D are constants and is any of the six trigonometric functions sine cosine. tangent cotangent secant cosecant, Amplitude is the measure of the distance of peaks and troughs. from the midline i e center of a sine or cosine function. amplitude is always positive The other four functions do not. have peaks and troughs so they do not have amplitudes For. the general function defined above amplitude A, Period is the horizontal width of a single cycle or wave i e the distance it travels before it repeats. Every trigonometric function has a period The periods of the parent functions are as follows for. sine cosine secant and cosecant period 2 for tangent and cotangent period. For the general function defined above, Frequency is most useful when used with the sine and. cosine functions It is the reciprocal of the period i e. Frequency is typically discussed in relation to the sine and cosine functions when considering. harmonic motion or waves In Physics frequency is typically measured in Hertz i e cycles per. second 1 Hz 1 cycle per second, For the general sine or cosine function defined above frequency. Version 2 0 Page 17 of 109 January 1 2016,Chapter 2 Graphs of Trig Functions. Phase Shift, Phase shift is how far has the function been shifted horizontally. left or right from its parent function For the general function. defined above,phase shift, A positive phase shift indicates a shift to the right relative to the. graph of the parent function a negative phase shift indicates a shift. to the left relative to the graph of the parent function. A trick for calculating the phase shift is to set the argument of the trigonometric function equal to. zero B C 0 and solve for The resulting value of is the phase shift of the function. Vertical Shift, Vertical shift is the vertical distance that the midline of a curve lies. above or below the midline of its parent function i e the axis. For the general function defined above vertical shift D. The value of D may be positive indicating a shift upward or. negative indicating a shift downward relative to the graph of the. parent function,Putting it All Together, The illustration below shows how all of the items described above combine in a single graph. Version 2 0 Page 18 of 109 January 1 2016,Chapter 2 Graphs of Trig Functions. Summary of Characteristics and Key Points Trigonometric Function Graphs. Function Sine Cosine Tangent Cotangent Secant Cosecant. Parent Function sin cos tan cot sec csc,Domain except except except except. where is an Integer where is an Integer,where is odd where is odd. Vertical Asymptotes none none,where is an where is odd where is an. where is odd,Integer Integer,Range 1 1 1 1 1 1 1 1. Period 2 2 2 2,midway between midway between, intercepts where is an Integer where is odd none none. asymptotes asymptotes, Odd or Even Function Odd Function Even Function Odd Function Odd Function Even Function Odd Function. General Form sin cos tan cot sec csc,Amplitude Stretch Period 2 2 2 2. Phase Shift Vertical Shift,when vertical asymptote vertical asymptote. when vertical asymptote,when vertical asymptote vertical asymptote. when vertical asymptote,when vertical asymptote vertical asymptote. 1 An odd function is symmetric about the origin i e An even function is symmetric about the axis i e. 2 All Phase Shifts are defined to occur relative to a starting point of the axis i e the vertical line 0. Version 2 0 Page 19 of 109 January 1 2016,Chapter 2 Graphs of Trig Functions. Graph of a General Sine Function,General Form,The general form of a sine function is. In this equation we find several parameters of the function which will help us graph it In particular. Amplitude The amplitude is the magnitude of the stretch or compression of the. function from its parent function sin, Period The period of a trigonometric function is the horizontal distance over which. the curve travels before it begins to repeat itself i e begins a new cycle For a sine or cosine. function this is the length of one complete wave it can be measured from peak to peak or. from trough to trough Note that 2 is the period of sin. Phase Shift The phase shift is the distance of the horizontal translation of the. function Note that the value of in the general form has a minus sign in front of it just like. does in the vertex form of a quadratic equation So. o A minus sign in front of the implies a translation to the right and. o A plus sign in front of the implies a implies a translation to the left. Vertical Shift This is the distance of the vertical translation of the function This is. equivalent to in the vertex form of a quadratic equation. Example 2 1, The midline has the equation y D In this example the midline. is y 3 One wave shifted to the right is shown in orange below For this example. Phase Shift,Vertical Shift,Version 2 0 Page 20 of 109 January 1 2016. Chapter 2 Graphs of Trig Functions,Graphing a Sine Function with No Vertical Shift. A wave cycle of the sine function has three zero points points on the x axis Example. at the beginning of the period at the end of the period and halfway in between. Step 1 Phase Shift,The first wave begins at the,point units to the right of The point is. the Origin,Step 2 Period The first,The first wave ends at the wave ends at the point. point units to the right of,where the wave begins,Step 3 The third zero point The point is. is located halfway between,the first two,Step 4 The value of the The point is. point halfway between the,left and center zero points is. Step 5 The value of the The point is,point halfway between the. center and right zero points,Step 6 Draw a smooth, curve through the five key This will produce the graph. points of one wave of the function,Step 7 Duplicate the wave. to the left and right as Note If 0 all points,desired on the curve are shifted. vertically by units,Version 2 0 Page 21 of 109 January 1 2016. Chapter 2 Graphs of Trig Functions,Graph of a General Cosine Function. General Form,The general form of a cosine function is. In this equation we find several parameters of the function which will help us graph it In particular. Amplitude The amplitude is the magnitude of the stretch or compression of the. function from its parent function cos, Period The period of a trigonometric function is the horizontal distance over which. the curve travels before it begins to repeat itself i e begins a new cycle For a sine or cosine. function this is the length of one complete wave it can be measured from peak to peak or. from trough to trough Note that 2 is the period of cos. Phase Shift The phase shift is the distance of the horizontal translation of the. function Note that the value of in the general form has a minus sign in front of it just like. does in the vertex form of a quadratic equation So. o A minus sign in front of the implies a translation to the right and. o A plus sign in front of the implies a implies a translation to the left. Vertical Shift This is the distance of the vertical translation of the function This is. equivalent to in the vertex form of a quadratic equation. Example 2 2, The midline has the equation y D In this example the midline. is y 3 One wave shifted to the right is shown in orange below For this example. Phase Shift,Vertical Shift,Version 2 0 Page 22 of 109 January 1 2016. Chapter 2 Graphs of Trig Functions,Graphing a Cosine Function with No Vertical Shift. A wave cycle of the cosine function has two maxima or minima if 0. one at the beginning of the period and one at the end of the period and a Example. minimum or maximum if 0 halfway in between,Step 1 Phase Shift. The first wave begins at the,point units to the right of The point is. Step 2 Period The first,The first wave ends at the wave ends at the point. point units to the right of,where the wave begins,Step 3 The value of the The point is. point halfway between those,in the two steps above is. Step 4 The value of the The point is,point halfway between the. left and center extrema is,Step 5 The value of the The point is. point halfway between the,center and right extrema is. Step 6 Draw a smooth, curve through the five key This will produce the graph. points of one wave of the function,Step 7 Duplicate the wave. to the left and right as Note If 0 all points,desired on the curve are shifted. vertically by units,Version 2 0 Page 23 of 109 January 1 2016. Chapter 2 Graphs of Trig Functions,Graph of a General Tangent Function. General Form,The general form of a tangent function is. In this equation we find several parameters of the function which will help us graph it In particular. Scale factor The tangent function does not have amplitude is the magnitude of the. stretch or compression of the function from its parent function tan. Period The period of a trigonometric function is the horizontal distance over which. the curve travels before it begins to repeat itself i e begins a new cycle For a tangent or. cotangent function this is the horizontal distance between consecutive asymptotes it is also. the distance between intercepts Note that is the period of tan. Phase Shift The phase shift is the distance of the horizontal translation of the. function Note that the value of in the general form has a minus sign in front of it just like. does in the vertex form of a quadratic equation So. o A minus sign in front of the implies a translation to the right and. o A plus sign in front of the implies a implies a translation to the left. Vertical Shift This is the distance of the vertical translation of the function This is. equivalent to in the vertex form of a quadratic equation. Example 2 3, The midline has the equation y D In this example the midline. is y 3 One cycle shifted to the right is shown in orange below For this example. Note that for the,tangent curve we,typically graph half Scale Factor. of the principal,cycle at the point Period,of the phase shift. and then fill in the Phase Shift,other half of the. cycle to the left Vertical Shift,see next page,Version 2 0 Page 24 of 109 January 1 2016. Chapter 2 Graphs of Trig Functions, Graphing a Tangent Function with No Vertical Shift. A cycle of the tangent function has two asymptotes and a zero point halfway in Example. between It flows upward to the right if 0 and downward to the right if 0. Step 1 Phase Shift,The first cycle begins at the,zero point units to the The point is. right of the Origin,Step 2 Period, Place a vertical asymptote The right asymptote is at. units to the right of the,beginning of the cycle,Step 3 Place a vertical The left asymptote is at. asymptote units to the,left of the beginning of the. Step 4 The value of the The point is,point halfway between the. zero point and the right,asymptote is,Step 5 The value of the The point is. point halfway between the,left asymptote and the zero. Step 6 Draw a smooth, curve through the three key This will produce the graph. points approaching the of one cycle of the function. asymptotes on each side,Step 7 Duplicate the cycle. to the left and right as Note If 0 all points,desired on the curve are shifted. vertically by units,Version 2 0 Page 25 of 109 January 1 2016. Chapter 2 Graphs of Trig Functions,Graph of a General Cotangent Function. General Form,The general form of a cotangent function is. In this equation we find several parameters of the function which will help us graph it In particular. Scale factor The cotangent function does not have amplitude is the magnitude of. the stretch or compression of the function from its parent function cot. Period The period of a trigonometric function is the horizontal distance over which. the curve travels before it begins to repeat itself i e begins a new cycle For a tangent or. cotangent function this is the horizontal distance bet. Graphs of Basic Parent Trigonometric Functions Every trigonometric function has a period The periods of the parent functions are as follows for

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