- WE WANT TO GRAPH THE GIVEN SECANT FUNCTION. WE WANT TO FIND THE PERIOD, FIND THE HORIZONTAL SHIFT, AND THEN GRAPH THE FUNCTION. SO BEFORE WE BEGIN, LET'S REVIEW THE KEY COMPONENTS OF THE COSECANT AND SECANT FUNCTIONS IN THIS FORM HERE. ALSO NOTICE HOW THIS PIECE OF THE FUNCTION IS IN FACTORED FORM. THE FIRST THING TO RECOGNIZE IS THAT THE ABSOLUTE VALUE OF "A" IS NOT THE AMPLITUDE, BECAUSE COSECANT AND SECANT HAVE NO AMPLITUDE. EVEN THOUGH THE VALUE OF "A" WILL VERTICALLY STRETCH OR VERTICALLY COMPRESS THE FUNCTION, THESE TWO FUNCTIONS DO NOT HAVE AN AMPLITUDE. NEXT, THE PERIOD = 2PI DIVIDED BY B, AGAIN, WHERE THIS IS IN FACTORED FORM. NEXT, C, THIS CONSTANT HERE, WILL VERTICALLY SHIFT THE GRAPH UP OR DOWN. IF C IS POSITIVE, THE FUNCTION IS SHIFTED UP C UNITS. IF C IS NEGATIVE, THE FUNCTION IS SHIFTED DOWN THE ABSOLUTE VALUE OF C UNITS. AND THEN FINALLY D, THIS VALUE HERE, HORIZONTALLY SHIFTS THE GRAPH, OR TRANSLATES THE GRAPH, WHERE IF D > 0, SO THIS WOULD BE IN THE FORM OF X - D, THE FUNCTION IS SHIFTED RIGHT D UNITS. BUT IF D IS NEGATIVE, MEANING THIS WOULD BE IN THE FORM OF X + D, THE FUNCTION IS SHIFTED THE ABSOLUTE VALUE OF D UNITS TO THE LEFT. SO GOING BACK TO OUR EXAMPLE, THE FIRST STEP IS GOING TO BE TO FACTOR 4X - 4. SO LET'S GO AHEAD AND WRITE THIS AS Y = 2 x SECANT OF-- FACTORING OUT THE 4 WOULD GIVE US 4 x THE QUANTITY X - 1. NOTICE IN THIS FORM B IS +4, WHICH WILL HELP US FIND THE PERIOD. AND HERE WE HAVE X - 1, WHICH WILL HELP US FIND THE HORIZONTAL SHIFT. SO THE PERIOD IS GOING TO BE EQUAL TO 2PI DIVIDED BY B, WHICH WOULD BE 2PI DIVIDED BY 4 OR PI/2 RADIANS, AND THIS IS GOING TO BE APPROXIMATELY 1.57. AND THEN THE HORIZONTAL SHIFT, BECAUSE WE HAVE X - 1, IT WILL ACTUALLY SHIFT THE GRAPH TO THE RIGHT 1 UNIT. REMEMBER, IF IT WAS X + 1, IT WOULD SHIFT THE GRAPH TO THE LEFT 1 UNIT. NOW THAT WE HAVE THE PERIOD AND THE HORIZONTAL SHIFT, LET'S FOCUS ON GRAPHING THIS FUNCTION. BEFORE WE GRAPH THE TRANSFORMATION OF THE SECANT FUNCTION, LET'S FOCUS ON SOME OF THE KEY COMPONENTS OF THE BASIC SECANT FUNCTION GRAPHED HERE IN BLUE, AS WELL AS THE BASIC COSINE FUNCTION GRAPHED HERE IN RED. REMEMBER, THE SECANT FUNCTION AND COSINE FUNCTION ARE RECIPROCALS OF ONE ANOTHER. AND OFTEN IT'S HELPFUL TO GRAPH THE CORRESPONDING COSINE FUNCTION TO GRAPH A GIVEN SECANT FUNCTION. NOTICE HERE IN RED WHERE THE COSINE FUNCTION IS EQUAL TO ZERO, THE SECANT FUNCTION HAS VERTICAL ASYMPTOTES HERE, HERE, HERE, AND HERE. ALSO NOTICE WHERE THE BASIC COSINE FUNCTION IS EQUAL TO 1 SO IS THE SECANT FUNCTION, BECAUSE THEY'RE RECIPROCALS OF ONE ANOTHER. SO HERE, HERE AND HERE BOTH FUNCTIONS ARE EQUAL TO 1. AND WHERE THE BASIC COSINE FUNCTION EQUALS -1, SO DOES THE BASIC SECANT FUNCTION. SO WE'RE GOING TO USE KEY POINTS ON THE CORRESPONDING COSINE FUNCTION TO GRAPH THE GIVEN TRANSFORMATION OF THE SECANT FUNCTION. MEANING WE'RE GOING TO START BY GRAPHING Y = 2 x COSINE OF 4 x THE QUANTITY X - 1. THIS GRAPH WILL HELP US GRAPH THE CORRESPONDING SECANT FUNCTION. SO TO GRAPH THIS COSINE FUNCTION WE'RE GOING TO FOCUS ON THIS PIECE OF THE BASIC COSINE FUNCTION, FROM 0 TO 2PI RADIANS HERE. NOTICE AT X = 0 THE COSINE FUNCTION IS 1, THEN IT GOES TO 0, -1, 0, AND 1. SO WE'LL USE THIS PATTERN TO GRAPH THIS TRANSFORMATION OF THE COSINE FUNCTION. TO BEGIN, NOTICE HOW THE GRAPH IS SHIFTED RIGHT ONE UNIT, SO WE WON'T START AT THE Y AXIS. WE'LL SHIFT THE GRAPH ONE UNIT TO THE RIGHT. SO WE'RE GOING TO START OUR GRAPH HERE, IF THIS IS ONE UNIT, AND THEN BECAUSE THE PERIOD IS PI/2 RADIANS, 1 + PI/2 RADIANS WOULD GIVE US ONE COMPLETE GRAPH OF THIS COSINE FUNCTION. SO 1 + PI/2 RADIANS WOULD BE APPROXIMATELY 1 + 1.57. LET'S GO AHEAD AND LABEL THIS 2.57 HERE. AND THEN JUST AS ALWAYS WE'LL DIVIDE THIS PERIOD INTO FOUR EQUAL PIECES. SO WE'LL CUT THIS IN HALF, AND THEN IN HALF AGAIN, AND NOW WE'LL GRAPH THESE KEY FIVE POINTS OF THE COSINE FUNCTION. BUT REMEMBER, WE DO HAVE A COEFFICIENT HERE OF 2, SO WHEN X = 1 INSTEAD OF PLOTTING THE COSINE FUNCTION VALUE OF 1, IT WOULD BE 2 x 1 OR 2. SO IF THIS IS 1 AND THIS IS 2, WE WOULD START THIS PIECE OF THE COSINE FUNCTION HERE, AND THEN IF WE MOVE 1/4 OF THE PERIOD TO THE RIGHT, IT'S GOING TO DROP DOWN TO THE X AXIS HERE. WE MOVE ONE MORE FOURTH OF THE PERIOD TO THE RIGHT WE'LL BE DOWN TO THE MINIMUM, WHICH ISN'T GOING TO BE -1. IT'S GOING TO BE 2 x -1 OR -2, WHICH WOULD BE HERE. AND THEN THE NEXT FOURTH WOULD BE BACK AT THE X AXIS. THE NEXT FOURTH WOULD BE BACK AT A MAXIMUM. SO HERE IS ONE PERIOD OF THE CORRESPONDING COSINE FUNCTION. IT WOULD LOOK LIKE THIS. LET'S GO AHEAD AND REPEAT THIS A COUPLE MORE TIMES. IF WE MOVE TO THE LEFT ANOTHER 1.57 UNITS WE CAN GRAPH ANOTHER PERIOD OF THIS COSINE FUNCTION. IF WE MOVE TO THE LEFT 1.57 UNITS WOULD BE APPROXIMATELY HERE AT APPROXIMATELY -.57. SO IF WE DIVIDE THIS INTERVAL INTO FOUR EQUAL PIECES WE WOULD BE HERE, HERE, AND HERE. FOLLOWING THE SAME PATTERN WE'LL START HERE AT OUR MAXIMUM, BACK TO THE MIDLINE, DOWN TO A MINIMUM, MIDLINE, AND MAXIMUM. SO IT WOULD LOOK LIKE THIS. AND LET'S GO AHEAD AND DO THAT ONE MORE TIME. IF WE GO LEFT ANOTHER 1.57 UNITS WE'D BE BACK AT APPROXIMATELY - 2.14, WHICH WOULD BE SOMEWHERE IN HERE. WE'LL DIVIDE THIS INTERVAL INTO FOUR EQUAL PIECES HERE, HERE, AND HERE AND, AGAIN, FOLLOW THE SAME PATTERN. ON THE LEFT OF THIS INTERVAL WE'LL HAVE A MAXIMUM, THEN MIDLINE, MINIMUM, MIDLINE, MAXIMUM. AND HERE WE GO. LET'S GO AHEAD AND STOP HERE. AND NOW BECAUSE THIS IS THE CORRESPONDING COSINE FUNCTION, WE SHOULD NOW BE ABLE TO EASILY GRAPH THE GIVEN SECANT FUNCTION. REMEMBER, WHERE THIS COSINE FUNCTION IS EQUAL TO ZERO, OUR SECANT FUNCTION WILL HAVE VERTICAL ASYMPTOTES. SO WE'D HAVE A VERTICAL ASYMPTOTE HERE, HERE, HERE, HERE, HERE, AND HERE. AND NOW THE CORRESPONDING SECANT FUNCTION WILL SHARE THE SAME MAXIMUM AND MINIMUM VALUES OF THE CORRESPONDING COSINE FUNCTION. SO STARTING AT THIS PIECE HERE, IT'S GOING TO PASS THROUGH AT THIS POINT AND APPROACH THE VERTICAL ASYMPTOTE HERE. AND LOOKING AT THIS PIECE OF THE COSINE FUNCTION, THE CORRESPONDING SECANT FUNCTION WOULD LOOK LIKE THIS. LOOKING AT THIS PIECE, THE CORRESPONDING SECANT FUNCTION WOULD LOOK LIKE THIS. THIS PIECE WOULD LOOK LIKE THIS. YOU CAN SEE HOW GRAPHING THIS COSINE FUNCTION, WHILE IT TOOK SOME TIME, IS VERY HELPFUL FOR GRAPHING THE CORRESPONDING SECANT FUNCTION. OF COURSE, WE'D GRAPH MORE IF WE NEEDED, BUT HOPEFULLY YOU GET THE IDEA. I HOPE YOU HAVE FOUND THIS HELPFUL.