- WELCOME TO THE FIRST OF TWO
VIDEOS ON THE CONIC SECTION OF PARABOLA.
AND THE GOAL OF THIS VIDEO IS TO IDENTIFY THE KEY COMPONENTS
TO A PARABOLA, AND ALSO TO GRAPH OUR PARABOLA IN STANDARD FORM.
A PARABOLA IS A SET OF ALL POINTS XY
THAT ARE EQUAL DISTANCE FROM A FIXED LINE
CALLED THE DIRECTRIX AND A FIXED POINT CALLED THE FOCUS.
SO THIS BLUE LINE WOULD BE OUR DIRECTRIX
AND THIS RED POINT WOULD BE THE FOCUS.
THE PARABOLA CONSIST OF ALL THE POINTS
THAT HAVE THE SAME DISTANCE FROM THE POINT AS THEY ARE
TO THIS HORIZONTAL LINE.
SO THE LENGTH OF THIS SEGMENT
HERE WOULD EQUAL THE LENGTH OF THE SEGMENT HERE.
THE LENGTH OF THIS SEGMENT HERE
WOULD EQUAL THE LENGTH OF THIS SEGMENT HERE,
AND THE SAME FOR THIS SEGMENT AND THIS SEGMENT.
AND THE SAME WOULD BE TRUE FOR ANY POINT ON THIS PARABOLA.
BEFORE WE TAKE A LOOK AT THE STANDARD FORM
OF AN EQUATION OF PARABOLA LET'S GO AND TAKE A LOOK
AT THE GRAPHS OF PARABOLAS.
SOME PARABOLAS HAVE A VERTICAL AXIS AS WE SEE HERE ON THE LEFT.
SOMETIMES THIS IS CALLED THE AXIS OF SYMMETRY,
AND ON THE RIGHT WE HAVE A HORIZONTAL AXIS
OR HORIZONTAL LINE OF SYMMETRY.
REMEMBER THIS IS THE LINE WHERE
IF YOU WERE TO FOLD THE PARABOLA ACROSS THAT,
IT WOULD MATCH UP PERFECTLY WITH THE OTHER HALF.
OR WE CAN SAY IT'S SYMMETRICAL ACROSS THAT LINE.
AS WE SAW IN THE PREVIOUS SCREEN,
EVERY PARABOLA WILL HAVE A DIRECTRIX, AS WELL AS A FOCUS.
AND THEN LASTLY, EVERY PARABOLA ALSO HAS A VERTEX.
IT'S THE ONLY POINT THAT IS ON THE AXIS AND ON THE PARABOLA.
LET'S TAKE A LOOK AT THE STANDARD FORM OF AN EQUATION
OF A PARABOLA NOW.
IF THE PARABOLA HAS A VERTICAL AXIS,
AS WE SEE HERE ON THE LEFT, IT WILL BE IN THIS FORM.
NOTICE THE X PART OF THE EQUATION IS BEING SQUARED.
SO WHENEVER THE X PART OF THE EQUATION
IS BEING SQUARED WE WILL HAVE A VERTICAL AXIS AS WE SEE HERE.
THIS SHOULDN'T BE TOO HARD TO REMEMBER
BECAUSE WE HAVE GRAPHED PARABOLAS
IN THE FORM OF Y = X SQUARED BEFORE,
AND THEY ALWAYS EITHER OPENED UP OR OPENED DOWN.
THE COORDINATE OF THE VERTEX OF THE PARABOLA WILL BE (H,K),
SO WE CAN DETERMINE THE VERTEX BY INSPECTING THE EQUATION.
NEXT WE SHOULD BE ABLE TO LOOK AT THE EQUATION AND DETERMINE
THE VALUE OF A.
AND THAT'S GOING TO BE EXTREMELY IMPORTANT BECAUSE A WILL TELL US
THE DISTANCE FROM THE VERTEX TO THE FOCUS.
AND IT WILL ALSO TELL US THE DISTANCE FROM THE VERTEX
TO THE DIRECTRIX.
SO IF THIS WAS OUR GRAPH,
WE COULD DETERMINE THE COORDINATES OF THE FOCUS
BY ADDING A TO THE Y COORDINATE OF THE VERTEX.
AND WE COULD FIND THE EQUATION OF THE DIRECTRIX
BY SUBTRACTING A FROM THE Y COORDINATE OF THE VERTEX.
NOW, IF THE Y PART OF THE EQUATION IS BEING SQUARED,
THIS INDICATES THAT WE WILL HAVE A HORIZONTAL AXIS,
AS WE SEE HERE ON THE RIGHT.
BUT THE VERTEX WILL STILL BE (H,K) AND A WILL STILL TELL US
THE DISTANCE FROM THE VERTEX TO THE FOCUS,
AS WELL AS THE DISTANCE FROM THE VERTEX TO THE DIRECTRIX.
BUT IN THIS CASE, IF THIS WAS OUR GRAPH,
WE WOULD ADD A TO THE X COORDINATE OF THE VERTEX
TO FIND THE FOCUS, AND SUBTRACT A FROM THE X COORDINATE
OF THE VERTEX TO DETERMINE THE EQUATION OF THE DIRECTRIX.
SO BASED UPON WHETHER WE HAVE A VERTICAL OR HORIZONTAL AXIS,
THAT'LL AFFECT WHETHER WE ADD OR SUBTRACT A TO THE X
OR Y COORDINATE OF THE VERTEX.
THERE'S ONE MORE THING I WANT TO MENTION
BEFORE WE TRY TO GRAPH SOME OF OUR OWN.
NOW, THESE NOTES HERE JUST FORMALIZE WHAT WE JUST SAID
ABOUT HOW TO DETERMINE THE COORDINATES OF THE FOCUS,
AS WELL AS THE EQUATION OF THE DIRECTRIX.
SO THE LAST THING WE WANT TO TALK ABOUT IS THE LATUS RECTUM,
AND THIS IS THE CORD THROUGH A FOCUS PARALLEL TO THE DIRECTRIX,
AS WE SEE HERE IN GREEN
FOR A VERTICAL AXIS AND HORIZONTAL AXIS.
AND WHAT'S NICE ABOUT THAT IS IT HAS A LENGTH OF 4A.
AND WE CAN DETERMINE THE VALUE
OF 4A BY LOOKING AT THE EQUATION IN STANDARD FORM.
SO THIS WILL TELL US HOW WIDE OR HOW NARROW OUR PARABOLA WILL BE.
LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE EXAMPLES OF OUR OWN.
THE FIRST THING WE SHOULD NOTICE ABOUT THIS EQUATION
IS THE X PART IS BEING SQUARED,
THEREFORE WE'RE GOING TO HAVE A VERTICAL AXIS.
NEXT WE SHOULD BE ABLE TO DETERMINE THE COORDINATES
OF THE VERTEX BY LOOKING AT THIS EQUATION.
REMEMBER THE VERTEX HAS THE COORDINATES (H,K),
SO IF WE SEE X + 2 THE X COORDINATE WOULD BE -2.
AND IF WE SEE Y - 1 THE Y COORDINATE WOULD BE 1.
NEXT, IF WE CAN DETERMINE THE VALUE OF A,
WE SHOULD BE ABLE TO DETERMINE THE COORDINATES OF THE FOCUS,
AS WELL AS THE EQUATION OF THE DIRECTRIX.
WELL, 4A = 4, THEREFORE A = 1.
LET'S GO AHEAD AND GRAPH SOME OF THIS NOW.
WE'RE GOING TO PLOT THE VERTEX AT (-2,1).
NEXT, WE KNOW WE HAVE A VERTICAL AXIS
AND IT PASSES THROUGH THE VERTEX,
SO LET'S GO AHEAD AND SKETCH THAT.
AND THIS LINE WOULD HAVE THE EQUATION X = -2.
ONE THING WE DIDN'T MENTION BEFORE WAS IF A IS POSITIVE
THIS GRAPH IS GOING TO OPEN UP,
IF A IS NEGATIVE IT'S GOING TO OPEN DOWN.
SO THIS GRAPH IS GOING TO OPEN UP.
LET'S GO AHEAD AND WRITE THAT DOWN.
SO IF THIS PARABOLA OPENS UP
WE KNOW THE FOCUS WILL BE INSIDE THE PARABOLA,
AND IT WILL BE ONE UNIT ABOVE THE COORDINATES OF THE VERTEX.
SO WE'LL ADD ONE TO THE Y COORDINATE TO FIND THE FOCUS.
SO THIS WOULD BE THE POINT (-2,2).
NOW, THE DIRECTRIX WILL BE A HORIZONTAL LINE ONE UNIT
BELOW THE VERTEX IN THIS CASE.
SO IF WE GO ONE UNIT BELOW THE VERTEX THIS POINT HERE
WOULD BE ONE POINT ON THE DIRECTRIX,
SO THIS EQUATION HERE--
SO THIS LINE HERE WOULD BE THE EQUATION OF THE DIRECTRIX.
AND THIS WOULD BE Y = 0.
NOW, WE KNOW THE PARABOLA OPENS UP,
SO IT WILL LOOK SOMETHING LIKE THIS,
BUT WE'RE NOT REALLY SURE HOW WIDE IT WILL BE.
BUT NOW IF WE SKETCH THE ENDPOINTS
OF THE LATUS RECTUM WE'LL HAVE TWO MORE ADDITIONAL POINTS
ON THAT PARABOLA.
THE LATUS RECTUM WOULD HAVE LENGTH 4A,
SO WE'RE GOING TO SKETCH A SEGMENT 4 UNITS LONG
THROUGH THIS GREEN POINT PARALLEL TO THE DIRECTRIX
AND THAT WOULD LOOK LIKE THIS.
SO THIS ENDPOINT AND THIS ENDPOINT ARE ON THAT PARABOLA,
AND THAT SHOULD BE ENOUGH INFORMATION
TO MAKE A NICE SKETCH, WOULD LOOK SOMETHING LIKE THIS.
AGAIN, WE HAVE OUR DIRECTRIX, WE HAVE OUR FOCUS,
AND WE HAVE OUR PARABOLA.
NOW, YOU NOTICE I DID INCLUDE SOME NOTES DOWN HERE,
BUT I THINK IF WE CAN LOGICALLY DETERMINE WHERE THE FOCUS
AND DIRECTRIX SHOULD BE,
IT'S GOING TO BE A LOT EASIER THAN TRYING TO MEMORIZE
THESE NOTES BASED UPON WHETHER IT HAS A VERTICAL,
HORIZONTAL AXIS.
LET'S GO AND TAKE A LOOK AT ONE MORE.
NOTICE NOW THE Y PART OF THE EQUATION IS BEING SQUARED,
WHICH INDICATES WE'RE GOING TO HAVE A HORIZONTAL AXIS.
OUR VERTEX WILL STILL HAVE THE COORDINATES (H,K),
SO THE X COORDINATE WILL BE 1 AND THE Y COORDINATE WILL BE -2.
LET'S GO AHEAD AND FIND THE VALUE OF A.
WE KNOW THAT 4A = -8, SO THAT MEANS A WILL = -2.
NOTICE NOW WE HAVE A NEGATIVE VALUE OF A,
AND SO WHAT THAT MEANS IS THAT OUR PARABOLA,
INSTEAD OF OPENING RIGHT IN A POSITIVE DIRECTION,
WILL OPEN LEFT IN THE NEGATIVE DIRECTION.
LET'S GO AHEAD AND PLOT OUR VERTEX (1,-2).
NOW WE'RE GOING TO TAKE THE ABSOLUTE VALUE
OF A NOW TO DETERMINE THE DISTANCE FROM THE VERTEX
TO THE FOCUS.
SO THE FOCUS WILL BE TWO UNITS FROM THE VERTEX,
AND WILL BE TO THE LEFT BECAUSE THE PARABOLA OPENS TO THE LEFT.
SO WE'LL MOVE TWO UNITS TO THE LEFT.
THIS WILL BE OUR FOCUS (-1,-2).
AND THE DIRECTRIX WILL BE TWO UNITS
TO THE RIGHT OF THE VERTEX,
SO IT'LL PASS THROUGH THIS POINT AND IT WILL BE A VERTICAL LINE.
AND THIS IS THE EQUATION X = 3 AND THIS IS OUR DIRECTRIX.
SO RIGHT NOW WE ONLY HAVE ONE POINT ON THIS PARABOLA
THAT WE KNOW OPENS TO THE LEFT,
BUT IF WE CAN SKETCH THE LATUS RECTUM
WE CAN DETERMINE HOW WIDE OR HOW NARROW THIS PARABOLA WILL BE.
THE LATUS RECTUM WILL HAVE A LENGTH OF EIGHT UNITS.
SO WE'RE GOING TO SKETCH THIS SEGMENT
THAT STARTS FOUR UNITS ABOVE THE FOCUS
AND STOPS FOUR UNITS BELOW THE FOCUS.
SO IT WOULD LOOK LIKE THIS.
AND THE ENDPOINTS OF THIS SEGMENT WOULD BE TWO POINTS
ON THE PARABOLA AS WE SEE HERE.
SO OUR PARABOLA WILL PASS THROUGH THESE THREE POINTS.
IT'LL LOOK SOMETHING LIKE THIS AND SOMETHING LIKE THIS.
OKAY. THAT'LL DO IT FOR THIS VIDEO. IN THE NEXT VIDEO
WE'LL TALK ABOUT HOW TO GRAPH PARABOLAS IN GENERAL FORM,
AND WE'LL ALSO REVIEW THE MAJOR COMPONENTS OF A PARABOLA.
THANK YOU FOR WATCHING.