1 Mathematical Methods Units 1 and 2

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1 Mathematical Methods Units and Further trigonometric graphs In this section, we will discuss graphs of the form = a sin ( + c) + d and = a cos ( + c) + d. Consider the graph of = sin ( ). The following tale of values ma e constructed. 5 5 ( ) 5 = sin ( ) These points (, ) ma then e plotted on a set of aes and joined together a smooth curve (see graph). Verif this graph using a graphics calculator. This curve is identical to the asic curve = sin, ut translated units to the right. The period is. The amplitude is. = sin ( ) 5 = sin Now consider the graph of = cos ( + ) +. The following tale of values ma e completed: + ( + ) 5 5 = cos ( + ) +

2 Chapter 5 Etension Further trigonometric graphs These points (, ) are then plotted on a set of aes and joined together a smooth curve (see graph at right). Verif this graph using a graphics calculator. This curve is the same shape as the asic curve = cos ut translated units left and unit up. The amplitude is. The period is. In general, if >, c > and d >, the following rules appl. + = cos ( + ) + = cos Function Basic shape Amplitude Period Horizontal Vertical = a sin ( + c) + d = a sin a c left d up = a cos ( + c) + d = a cos a c left d up = a sin ( c) d = a sin a c right d down = a cos ( c) d = a cos a c right d down WORKED Eample Sketch the graph of each of the following for [, ]. a = sin ( + ) = cos ( ) THINK WRITE a Compare the function with a = sin + is of the form = a sin ( + c) + d. = a sin ( + c) + d where a =, =, c =, d = State the amplitude using the value of a. Amplitude = Find the period sustituting = Period = into. = Identif the asic function ( = a sin ). Basic function is = sin. 5 State the horizontal (c units left). Horizontal is units left. Continued over page

3 Mathematical Methods Units and THINK 7 9 State the vertical (d units up). Since d =, there is no vertical. Draw the -ais using [, ] and multiples of since the horizontal is a multiple of. Draw the -ais using [, ]. Draw, in pencil, the new -ais ( ) through = (the horizontal ). Sketch in the asic curve = sin over the domain, using the new aes as a guide. Use a calculator to check that end points are correctl shown on the graph. WRITE No vertical. When =, = sin = sin = ' = sin ( + ) When =, = sin + 5 = sin = + Compare the function with = cos is of the form = a cos ( + c) + d. = a cos ( + c) + d where a =, =, c =, d = State the amplitude using a. Amplitude = 5 Find the period sustituting = into. Identif the asic function ( = a cos ). State the horizontal. Since c is negative, the is to the right. Period = Basic function is = cos. Horizontal is units right.

4 Chapter Etension logarithmic graphs THINK 7 9 State the vertical. Since d is negative, the is downwards. Draw the -ais using [, ] and multiples of since the horizontal is a multiple of. Draw the -ais using [, ]. Draw, in pencil, the new -ais ( ) through = (the vertical ). Draw, in pencil, the new -ais ( ) through = (the horizontal ). Sketch the asic curve = cos over the domain, using the new aes as a guide. Note: the graph of = cos is the upside down version of = cos. That is, the graph is reflected in the -ais. Check the end points of the curve using a calculator. WRITE Vertical is unit down. When =, = cos ( ) =. When =, = cos ( ) =. Verif that the two graphs are correct using a graphics calculator. ' = cos ( ) ' rememer rememer The graph of = a sin ( + c) + d (or = a cos ( + c) + d) has the same asic shape as = a sin (or = a cos ). For positive, c and d:. Amplitude is a.. Period is.. Horizontal is c units left.. Vertical is d units up. If c or d is negative, then the opposite direction applies for the appropriate.

5 Mathematical Methods Units and 5. Further trigonometric graphs Complete each of the following statements. a The graph of = sin + is the same as = sin translated units.

5 5 Mathematical Methods Units and 5. Further trigonometric graphs Complete each of the following statements. a The graph of = sin + is the same as = sin translated units. The graph of = cos is the same as translated units down. c The graph of = sin ( + ) is the same as = sin translated units. d The graph of = cos ( ) is the same as translated units. e The graph of = sin ( ) + is the same as translated units and units. f The graph of = cos ( + ) 5 is the same as translated units left and units. Cop and complete the tale elow. Function a = sin ( + ) = sin ( ) c = cos ( ) d = cos + e = sin ( ) + f = cos ( + ) g = sin ( + ) h = cos ( ) + Basic function Horizontal Vertical i = sin ( + ) WORKED Eample Sketch the graphs of each of the following for [, ]. a = cos + = sin c = sin ( ) d = cos ( + ) e = sin ( + ) f = cos ( ) + g = sin ( ) h = cos ( + ) i = sin ( ) + j = cos ( + ) k = sin ( + ) +

6 CHAPTER 5 Circular functions Eercise 5. Further trigonometric graphs a up = cos c, left d = cos,, right e = sin, left,, up f = cos,, 5, down e g f Answers answers Function Basic function Horizontal Vertical a = sin ( + ) = sin left none = sin ( ) = sin right none c = cos ( ) = cos right none d = cos + = cos none up h 5 e = sin ( ) + = sin right up f = cos ( + ) = cos left down g = sin ( + ) = sin left down h = cos ( - ) + = cos right up i i = sin ( + ) = sin left down a c d 5 j k

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs

Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2