Graphing Trigonometric Functions
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1 LESSON Graphing Trigonometric Functions Graphing Sine and Cosine UNDERSTAND The table at the right shows - and f ()-values for the function f () 5 sin, where is an angle measure in radians. Look at the f ()-values. As the value of increases, the value of f () increases, then decreases, then increases, then decreases again. The values in the table form a repeating pattern, with the pattern starting over again at p. f () p p p p 5p p 7p UNDERSTAND The graph of 5 sin is shown on the coordinate plane below. The curve of the graph resembles waves, which is wh this is sometimes referred to as a sine wave. Several features are labeled, including the crest, or highest point of the graph, and trough, or lowest point of the graph. crest period trough sin Trigonometric functions like this are called periodic functions, because the repeat over and over from left to right. Here is the formal definition of a periodic function: A function f is periodic if there eists a number p such that f ( p) 5 f () for all. In that equation, p is the period of the function. It is the horizontal distance required for one complete ccle of the graph. The frequenc of a periodic function refers to the number of complete ccles in a given interval. For an interval of p, the frequenc of a function with period p is p p. In the sine function above, the period is p and the frequenc is. Duplicating this page is prohibited b law. Triumph Learning, LLC 78 Unit : Trigonometric Functions
2 Connect Make a table and graph for 5 cos. Create a table of values. p p p p p p 5p p Plot the points and graph the equation. cos Duplicating this page is prohibited b law. Triumph Learning, LLC DISCUSS Compare the shapes of the sine and cosine graphs. Lesson : Graphing Trigonometric Functions 79
3 Transforming Trigonometric Functions UNDERSTAND A general form of the sine function is f () 5 a sin [b( h)] k. As with other kinds of functions, the parameters a, b, h, and k have different effects on the graph of the function. The midline of a periodic function is the horizontal line that runs through the middle of the graph; half of the graph lies above the line, and the other half below it. In f () 5 a sin [b( h)] k, k represents the midline; it translates the function up or down. The graph of 5 sin 5 is shown at the right. In this case, k 5 5. The graph has a midline at 5 5; it is a translation 5 units up of 5 sin. midline The amplitude of a graph is the vertical distance from the midline to the highest (or lowest) point on the graph. In f () 5 a sin [b( h)] k, the parameter a affects the amplitude b verticall stretching or shrinking the function. If a,, the function is also reflected over the midline. In 5 sin 5, a 5. As shown on the graph, the amplitude is, which represents a vertical stretch of 5 sin b. 8 amplitude sin 5 UNDERSTAND Together, the parameters h and b affect the phase shift, or the amount that the function is translated horizontall from the parent function. The phase shift is equal to h b units. If the phase shift is negative, the function moves left. If the phase shift is positive, it moves right. The parameter b stretches or shrinks the function horizontall. When b,, the function is stretched horizontall from the vertical line determined b the phase shift, 5 h b. When b., the function is shrunk horizontall toward the line 5 h b. Eamine the graphs below. The graph of 5 sin ( p )is a horizontal translation of the parent function b p to the right. The graph of 5 sin is a horizontal stretch b. sin sin Duplicating this page is prohibited b law. Triumph Learning, LLC 8 Unit : Trigonometric Functions
4 Connect Graph the function f () 5 cos. Eamine the function. Compare the function to the general form f () 5 a cos [b( h)] k. In f () 5 cos, a 5, b 5, and both h and k equal. Since a 5, the graph will be stretched verticall b. Since b 5, the graph will be shrunk horizontall b. Perform the horizontal shrink. Take points on the graph of 5 cos, and multipl the -coordinates b. Perform the vertical stretch. Multipl each -value on the graph above in Step b. Duplicating this page is prohibited b law. Triumph Learning, LLC cos TRY Graph 5 sin. Lesson : Graphing Trigonometric Functions 8
5 EXAMPLE A Make a table and graph for 5 tan. Create a table of values. p p p undefined p p p undefined p p Determine how the undefined values affect the graph. The function is undefined at p, or.5p. Using our calculator, find values of 5 tan for values of that are close to.5p. Tr numbers to the left of.5p, such as.p,.5p, and.9p. For these -values, the -values get larger and larger, approaching. Tr numbers to the right of.5p, such as.p,.55p, and.5p. For these -values, the -values get smaller and smaller, approaching. Use the points and asmptotes to graph 5 tan. DISCUSS You can conclude that the undefined -values represent vertical asmptotes of the function. What are the domain and range of the function? Duplicating this page is prohibited b law. Triumph Learning, LLC 8 Unit : Trigonometric Functions
6 EXAMPLE B Graph the function f () 5 cos ( p). Eamine the function. Compare the function to the general form f () 5 a cos [b( h)] k. In the given function, a 5, b 5, h 5 p, and k 5. Since a 5, the graph will be stretched verticall b. Since h 5 p, the graph will be translated p units left. Since k 5, the graph will be translated unit up. Perform the vertical stretch. Multipl the -coordinates on the graph of 5 cos b. Perform the translations. Move ever point on the previous graph p units left and unit up. Duplicating this page is prohibited b law. Triumph Learning, LLC CHECK Test several values of, and check that the -values match what is shown on the graph. Lesson : Graphing Trigonometric Functions 8
7 EXAMPLE C Graph the function f () 5 tan ( p ). Identif its period, frequenc, and phase shift. Graph the function. The function represents a vertical shrink b, a horizontal stretch b, a horizontal translation of p units to the left, and a vertical translation of units down. Find the period and frequenc. The graph passes through the points (, ) and (p, ). The portion of the graph between those points appears to be a complete ccle. So, the period is the horizontal distance between those points, or the difference of their -coordinates. p 5 p To find the frequenc over an interval of p, divide p b the period. p p 5 The graph has a period of p and a frequenc of. On an interval of length p, the graph completes two-thirds Find the phase shift. Use h b, with h 5 p and b 5. p 5 p The phase shift is p, or p units to the left. TRY of a ccle. Look back at the graph of 5 cos ( p) in Eample B. Find the period, frequenc, and phase shift. Duplicating this page is prohibited b law. Triumph Learning, LLC 8 Unit : Trigonometric Functions
8 EXAMPLE D The graph shows a sine function. Identif ke features of the graph. Then, write its equation. 8 Identif the midline and amplitude. The highest point on the graph is 5. The lowest point is 5 5. The midline is halfwa between those two points, at 5. The amplitude is the absolute value of the difference of the crest and the midline. () 5 The midline of the graph is 5, and its amplitude is. Find the phase shift. The graph of 5 sin passes through the point (, ). The midline for this graph is 5. So, if this graph has no phase shift, it will pass through (, ). This graph includes the point ( p, ). So, this graph represents a translation of 5 sin b p units to the right. 8 Duplicating this page is prohibited b law. Triumph Learning, LLC Write the equation of the function. Since the amplitude is, a 5. Since the midline is 5, k 5. Since the phase shift is p to the right, h 5 p. Write an equation in the form f () 5 a sin [b( h)] k. The equation f () 5 sin ( p ) models the graph. DISCUSS The graph passes through the midline again p units to the right and p units to the left of ( p, ). This matches what the graph of 5 sin does, so there is no horizontal stretch or shrink. The phase shift is a translation of p units to the right of the graph of 5 sin. Could ou write a cosine function to model the graph? Lesson : Graphing Trigonometric Functions 85
9 Practice For each graph, identif the amplitude, midline, period, and frequenc over an interval of p... 8 amplitude: midline: period: frequenc: amplitude: midline: period: frequenc: Graph each function.. f () 5 sin. f () 5 cos Duplicating this page is prohibited b law. Triumph Learning, LLC 8 Unit : Trigonometric Functions
10 Graph each function. 5. f () 5 cos ( p). f () 5 tan 7. f () 5 sin 8. f () 5 sin Duplicating this page is prohibited b law. Triumph Learning, LLC Lesson : Graphing Trigonometric Functions 87
11 For questions 9 and, use the given graph. 9. Assume that the function shown is a sine function. Identif ke features of the graph. Then, write its equation. amplitude: midline: period: horizontal translation: equation:. Assume that the function shown is a cosine function. Identif ke features of the graph. Then, write its equation. amplitude: midline: Solve. period: horizontal translation: equation:. Consider the functions f () 5 5 sin and g() 5 5 sin ( p). Compare the functions and their graphs, and analze the phase shift in g() as compared to f (). Duplicating this page is prohibited b law. Triumph Learning, LLC 88 Unit : Trigonometric Functions
12 Solve.. CRITIQUE A student was asked to identif ke features of the graph of 5 sin. His answers are shown below. amplitude: midline: -ais period: p frequenc: Are the student s answers correct? Eplain.. APPLY A biccle wheel has a radius of inches. A point on the tire begins on the ground. The tire makes a complete revolution ever second. The function f (t) 5 cos (pt) gives the height of the point, in inches, after t seconds. Fill in the table. Then, graph the function on the coordinate plane. t f (t) Duplicating this page is prohibited b law. Triumph Learning, LLC Will the function ever reach a value of f (t) 5? Eplain. Lesson : Graphing Trigonometric Functions 89
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