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- WE WANT TO COMPLETE THE TABLE USING THE FUNCTION RULE
F OF X = 1/2X SQUARED, AND THEN AFTER COMPLETING
THE TABLE WE WANT TO GRAPH THE FUNCTION.
SO WE CAN SELECT ANY X VALUE THAT WE WANT,
BUT NOTICE HOW WHATEVER X VALUE WE SELECT
WE ARE GOING TO HAVE TO SQUARE IT, AND THEN MULTIPLY IT BY 1/2.
SO LET'S GO AHEAD AND PUT X = 0 IN THE MIDDLE
OF THE TABLE, AND THEN WE'LL SELECT VALUES
THAT ARE < 0 HERE AND > 0 HERE.
SO LET'S SELECT -2 AND -3, AND 2 AND 3.
SO WHEN X = -3 WE'LL SUBSTITUTE -3 FOR X TO DETERMINE
THE CORRESPONDING FUNCTION VALUE, OR Y VALUE.
SO WE'D HAVE 1/2 x -3 SQUARED.
WELL, -3 SQUARED IS 9, AND 9 x 1/2 WOULD BE 9/2,
WHICH TELLS US THE GRAPH OF THE FUNCTION WOULD CONTAIN
THE POINT WITH AN X COORDINATE OF -3 AND A Y COORDINATE OF 9/2.
BUT IF WE CONVERT THIS TO A DECIMAL IT'S GOING TO BE
EASIER TO PLOT, 9/2 = 4.5.
WHEN X = -2 THE FUNCTION VALUE,
OR Y VALUE, WOULD BE 1/2 x -2 SQUARED.
WELL, -2 SQUARED = 4, AND 1/2 x 4 = 2.
SO THE GRAPH OF THE FUNCTION WOULD ALSO CONTAIN THE POINT
WITH AN X COORDINATE OF -2 AND A Y COORDINATE OF 2.
WHEN X = 0 THE FUNCTION VALUE, OR Y VALUE,
WOULD BE 1/2 x 0 SQUARED, WHICH IS GOING TO BE 0.
SO THE GRAPH OF THE FUNCTION CONTAINS THE POINT 0,0.
AND THEN WHEN X = 2, Y WOULD BE 1/2 x 2 SQUARED.
WELL, 2 SQUARED IS THE SAME AS - 2 SQUARED,
SO WE'D HAVE 4 x 1/2, OR 2.
SO THE GRAPH OF THE FUNCTION CONTAINS THE POINT 2,2.
AND THEN WE HAVE X = 3, SO WE'D HAVE Y = 1/2 x 3 SQUARED,
WHICH IS 9/2 OR 4.5.
SO THE GRAPH OF THE FUNCTION ALSO CONTAINS THE POINT 3,4.5.
LET'S GO AHEAD AND PLOT THESE FIVE POINTS,
AND SEE IF IT GIVES US ENOUGH INFORMATION TO MAKE A NICE
GRAPH OF THE FUNCTION.
SO -3,4.5 WOULD BE HERE, -2,2 HERE, 0,0,
AND THEN WE HAVE 2,2 AND 3,4.5.
NOTICE HOW THESE POINTS DO HAVE SYMMETRY
ABOUT THE Y AXIS, MEANING THE RIGHT SIDE AND THE LEFT SIDE
ARE MIRROR IMAGES OF ONE ANOTHER.
AND THIS DOES LOOK LIKE IT'S ENOUGH INFORMATION
TO MAKE A NICE GRAPH OF OUR FUNCTION F OF X = 1/2X SQUARED.
THIS SHAPE IS ACTUALLY CALLED A PARABOLA.
IT LOOKS KIND OF LIKE A U-SHAPED GRAPH, SOMETHING LIKE THIS.
SO REGARDLESS OF WHAT KIND OF FUNCTION WE HAVE,
IF WE FIND ENOUGH POINTS USING A T TABLE, WE CAN ALWAYS MAKE
A NICE SKETCH OF THE GRAPH.
I HOPE THIS WAS HELPFUL.