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X
- FROM THE GRAPH DETERMINE THE OPEN INTERVAL
FOR WHICH THE FUNCTION IS INCREASING OR DECREASING,
AND CONCAVE UP OR DOWN.
ALSO INDICATE ANY RELATIVE EXTREMA OR POINTS OF INFLECTION.
LET'S BEGIN BY DETERMINING THE INTERVAL
FOR WHICH THE FUNCTION IS INCREASING OR DECREASING.
THE FUNCTION IS INCREASING IF AS X INCREASES Y INCREASES,
AND THE FUNCTION IS DECREASING IF AS X INCREASES Y DECREASES.
IF WE READ THE GRAPH FROM LEFT TO RIGHT,
IF WE'RE GOING UP-HILL THE GRAPH IS INCREASING,
IF WE'RE GOING DOWN-HILL THE GRAPH IS DECREASING.
LOOKING AT THE GRAPH FROM LEFT TO RIGHT,
NOTICE HOW THE FUNCTION IS INCREASING
OVER THIS INTERVAL UNTIL IT REACHES THIS HIGH POINT
AT X = -2.
ALSO NOTICE IF WE WERE TO SKETCH A TANGENT LINE AT ANY POINT
OVER THIS INTERVAL THE SLOPE OF THE TANGENT LINE
WOULD BE POSITIVE.
BUT THEN NOTICE FROM X = -2 HERE TO X = 2
THE FUNCTION IS GOING DOWN-HILL OR DECREASING UNTIL IT REACHES
2 HERE AND THEN IT SWITCHES
BACK TO INCREASING FROM 2 TO INFINITY.
SO NOTICE HOW IT CHANGES FROM INCREASING TO DECREASING
AT X = -2 AND IT CHANGES FROM DECREASING TO INCREASING
AT X = 2.
AND THEREFORE, THE FUNCTION IS INCREASING
OVER THE OPEN INTERVAL FROM NEGATIVE INFINITY TO -2,
BUT THEN IT INCREASES ALSO ON THE OPEN INTERVAL
FROM 2 TO INFINITY.
AND IT'S DECREASING OVER THE OPEN INTERVAL FROM -2 TO 2.
AND NOTICE WHERE THE FUNCTION CHANGES FROM INCREASING
TO DECREASING WE HAVE A HIGH POINT,
AND THEREFORE A RELATIVE MAXIMUM HERE.
AND WHERE THE FUNCTION CHANGES FROM DECREASING TO INCREASING
WE HAVE A LOW POINT OR RELATIVE MINIMUM.
SO THE RELATIVE MAXIMUM IN VALUES ARE THE Y VALUES.
AND THE LOCATIONS ARE THE X VALUES.
SO WE CAN SAY WE HAVE A RELATIVE MAXIMUM OF APPROXIMATELY
LET'S SAY 8.3 AT X = -2.
SOMETIMES YOU'LL SEE THE RELATIVE MAX AND MINS
GIVEN AS ORDERED PAIRS AS WELL.
AND THEN WE HAVE A RELATIVE MINIMUM OF LET'S SAY X = -2.3
AT X = 2.
AND NOW LETS TALK ABOUT THE CONCAVITY.
LOOKING AT THE LEFT SIDE OF THE GRAPH,
NOTICE HOW THIS PIECE IS CONCAVE DOWN AND IT SEEMED TO CHANGE
CONCAVITY RIGHT AT THE Y-AXIS.
AND ON THE RIGHT THE FUNCTION IS CONCAVE UP.
SO, AGAIN, ON THE LEFT THE FUNCTION IS CONCAVE DOWN,
AND ON THE RIGHT THE FUNCTION IS CONCAVE UP.
AND, AGAIN, WE'RE GOING TO SAY
IT CHANGES CONCAVITY RIGHT AT X = 0 OR THIS POINT HERE.
SO WE CAN SAY THE FUNCTION IS CONCAVE UP OVER THE OPEN
INTERVAL FROM 0 TO INFINITY,
AND THE FUNCTION IS CONCAVE DOWN OVER THE OPEN INTERVAL
FROM NEGATIVE INFINITY TO ZERO.
AND THEN FINALLY, THE POINT OF INFLECTION
OR THE POINT WHERE IT CHANGES CONCAVITY LOOKS LIKE IT WOULD BE
THE POINT (0,3) THE POINT RIGHT HERE ON THE Y-AXIS.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.