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- WELCOME TO A LESSON ON GRAPHING A QUADRATIC FUNCTION
IN GENERAL FORM USING SYMMETRIC POINTS.
TO GET AN IDEA OF HOW THIS METHOD WORKS
LET'S LOOK AT THE GRAPH OF A QUADRATIC FUNCTION.
THESE TWO BLACK POINTS ARE SYMMETRIC POINTS
ON THE QUADRATIC FUNCTION
BECAUSE THEY'RE MIRROR IMAGES OF ONE ANOTHER
ACROSS THE RED VERTICAL LINE WHICH IS THE AXIS OF SYMMETRY.
SO THE IDEA IS, IF WE CAN FIND TWO SYMMETRIC POINTS
ON THE PARABOLA
THEN WE KNOW THE AXIS OF SYMMETRY
WOULD HAVE TO PASS THRU DIRECTLY
IN BETWEEN THE TWO POINTS
ALLOWING US TO ALSO FIND THE X COORDINATE OF THE VERTEX
AND THEREFORE FINDING THE CORRESPONDING Y COORDINATE
AS WELL FROM THE FUNCTION.
SO LOOKING AT OUR GRAPH
NOTICE HOW THIS POINT HAS COORDINATES 0, -1,
AND THIS POINT HAS COORDINATES 4, -1.
SYMMETRIC POINTS ON A PARABOLA
WILL ALWAYS HAVE THE SAME Y COORDINATES
AND IF WE TAKE THE AVERAGE OF THE X COORDINATES,
WE CAN FIND THE EQUATION OF THE AXIS OF SYMMETRY.
HERE 0 + 4 DIVIDED BY 2 = 2.
SO THE EQUATION OF THE AXIS OF SYMMETRY IS X = 2
WHICH MEANS THE X COORDINATE OF THE VERTEX WILL ALSO BE 2
AND THEN WE CAN FIND F OF 2
TO THE FIND THE Y COORDINATE OF THE VERTEX.
SO BEFORE WE DEFINE THE FORMAL STEPS FOR THIS METHOD
LET'S TAKE A LOOK AT AN EXAMPLE.
SO WE CAN WRITE THIS AS Y = X/SQUARED - 4X + 3.
AND NOW FOR THIS METHOD WE'RE GOING TO FACTOR OUT
THE GREATEST COMMON FACTOR BETWEEN THESE TWO X TERMS HERE
WHICH IN THIS CASE THE GREATEST COMMON FACTOR IS X.
SO NOW WE'LL WRITE THIS AS Y = X x THE QUANTITY X - 4
THEN WE STILL HAVE THIS + 3.
SO TO FIND THOSE TWO SYMMETRIC POINTS
WE'RE GOING TO FIND THE X VALUES
THAT MAKE THIS PRODUCT HERE EQUAL TO 0.
SO LET'S MAKE A TABLE OF VALUES.
IF X IS EQUAL TO 0 THIS PRODUCT WOULD BE 0
AND IF X IS EQUAL TO 4, X - 4 WOULD BE 0
AND THEREFORE THIS PRODUCT WOULD BE 0.
SO WE'LL SELECT X = 4 FOR THE SECOND X COORDINATE.
NOW WE'LL FIND THE CORRESPONDING Y VALUES.
WHEN X IS 0 THIS PRODUCT IS 0, SO Y IS EQUAL TO 0 + 3 OR 3
AND WHEN X IS 4, THIS FACTOR'S EQUAL TO 0
THEREFORE THIS PRODUCT IS EQUAL TO 0.
SO Y IS STILL 0 + 3 OR 3.
AGAIN, NOTICE HOW THE Y COORDINATES ARE THE SAME
THEREFORE THESE WILL BE SYMMETRIC POINTS
ACROSS THE AXIS OF SYMMETRY.
SO WE HAVE THE POINT 0, 3, WHICH WOULD BE HERE.
WE HAVE THE POINT 4, 3, WHICH SHOULD BE HERE.
AND NOW WE KNOW THE AXIS OF SYMMETRY
MUST PASS IN THE MIDDLE BETWEEN THESE TWO POINTS
SO THE AXIS OF SYMMETRY WOULD BE THIS VERTICAL LINE HERE.
NOTICE HOW THE EQUATION OF THIS VERTICAL LINE IS X = 2
AND BECAUSE THE VERTEX IS ON THIS AXIS OF SYMMETRY
WE KNOW THE X COORDINATE OF THE VERTEX MUST BE 2,
SO THAT WILL BE OUR THIRD POINT
AND NOW WE'LL EVALUATE THE FUNCTION AT X = 2.
SO F OF 2, WE'LL GO AHEAD AND USE DIVISIONAL FUNCTION
SO WE'LL HAVE 2/SQUARED - 4 x 2 + 3.
SO F OF 2 IS EQUAL TO 4, THIS WOULD BE - 8 + 3.
SO 4 - 8 + 3 = -1.
SO OUR Y COORDINATE IS -1 FOR THE VERTEX.
SO WE'LL GO AHEAD AND PLOT THIS POINT, 2, -1, IS HERE
AND NOW NOTICE THIS IS MORE THAN ENOUGH INFORMATION
TO MAKE A NICE SKETCH OF OUR QUADRATIC FUNCTION.
IT WOULD LOOK SOMETHING LIKE THIS.
SO LET'S GO AHEAD AND FORMALIZE THESE STEPS.
WE'LL START BY FACTORING OUT THE GREATEST COMMON FACTOR
OF THE 2X TERMS
AND THEN WE'LL SELECT THE X VALUES
THAT MAKE THE FACTORED PART EQUAL TO 0.
THEN WE'LL FIND THE CORRESPONDING Y VALUES.
THIS WILL GIVE US THE TWO SYMMETRIC POINTS
SO WE'LL PLOT THESE TWO POINTS
AND THEN WE'LL DETERMINE THE X VALUE BETWEEN THE TWO POINTS
WHICH WOULD GIVE US THE EQUATION
OF THE AXIS OF SYMMETRY.
THIS WILL BE THE X COORDINATE OF THE VERTEX.
WE'LL FIND THE CORRESPONDING Y VALUE
AND THEN PLOT THE VERTEX.
AND THEN FROM HERE WE CAN GRAPH THE FUNCTION.
LET'S TAKE A LOOK AT TWO MORE EXAMPLES.
AGAIN LET'S START BY REPLACING F OF X WITH Y.
SO WE'LL HAVE Y = -2X/SQUARED + 6X - 1
AND NOW WE'LL FACTOR OUT THE GREATEST COMMON FACTOR
OF THESE TWO X TERMS,
THE GREATEST COMMON FACTOR HERE WOULD BE 2X.
LET'S GO AHEAD AND FACTOR OUT -2X.
SO WE'LL HAVE Y = -2X x THE QUANTITY X - 3
THE SIGNS GOING TO CHANGE HERE AND THEN WE STILL HAVE -1.
AND NOW, WE'LL SELECT THE VALUES
THAT MAKE THIS PRODUCT HERE EQUAL 0.
SO WHEN X IS 0 THIS FACTOR'S EQUAL TO 0.
AND WHEN X IS 3 THIS FACTOR IS EQUAL TO 0.
SO NOW WE'LL FIND THE CORRESPONDING Y VALUES.
WHEN X IS 0 THIS PRODUCT IS 0 SO Y IS 0 -1 OR -1.
WHEN X IS 3 THIS IS 0
THEREFORE THIS PRODUCT IS 0 AGAIN.
SO Y IS EQUAL TO 0 - 1 OR -1.
THESE TWO Y COORDINATES SHOULD ALWAYS BE THE SAME
IF WE SELECT THE CORRECT X VALUES.
LET'S GO AHEAD AND PLOT THESE.
WE HAVE 0, -1 AND 3, -1,
AND WE SHOULD RECOGNIZE THE AXIS OF SYMMETRY
WOULD BE THIS VERTICAL LINE HERE
HALF WAY BETWEEN THESE TWO POINTS
WHICH WOULD BE X = 1.5 OR X = 3/2.
NOTICE 3/2 IS THE AVERAGE OF THE TWO X COORDINATES HERE.
THIS ALSO TELLS US THE X COORDINATE OF THE VERTEX
IS 3/2.
SO NOW WE'LL FIND F OF 3/2
TO FIND THE Y COORDINATE OF THE VERTEX.
SO F OF 3/2 IS GOING TO BE
EQUAL TO -2 x 3/2/SQUARED + 6 x 3/2 - 1.
SO THIS IS GOING TO BE 9/4.
SO -2/1 x 9/4 + 6/1 x 3/2 - 1.
THIS SIMPLIFIES HERE.
THIS SIMPLIFIES TO 1, THIS SIMPLIFIES TO 2.
SO THIS IS GOING TO EQUAL -9/2 AND THIS SIMPLIFIES AS WELL.
THIS SIMPLIFIES TO 1, THIS SIMPLIFIES TO 3.
SO + 9 AND THEN - 1.
THIS IS THE SAME AS -4.5.
-4.5 + 9 IS +4.5 - 1 IS 3.5
WHICH AS A FRACTION WOULD BE 7/2.
AND AGAIN, IF IT'S HELPFUL TO PLOT THIS
WE COULD THINK OF THIS AS THE POINT 1.5, 3.5,
WHICH WOULD BE THIS POINT HERE.
WE SHOULD NOTICE THIS PARABOLA DOES OPEN DOWN
WHICH IS GOOD
BECAUSE THE LINEAR COEFFICIENT IS NEGATIVE.
SO OUR GRAPH LOOKS SOMETHING LIKE THIS.
AND NOW LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
FIRST STEP, LET'S REPLACE F OF X WITH Y
AND NOW WE'LL FACTOR OUT THE GREATEST COMMON FACTOR
FROM THESE TWO X TERMS
WHICH IN THIS CASE IS JUST GOING TO BE X.
SO WE'LL HAVE Y = X x QUANTITY 1/2X - 1 - 2.
NOW WE'LL MAKE OUR TABLE OF VALUES.
THIS PRODUCT HERE IS GOING TO BE 0 WHEN X IS 0,
BUT ALSO WHEN 1/2X - = 0.
SO WE'D ADD 1 TO BOTH SIDES OF THE EQUATION,
MULTIPLY BOTH SIDES BY 2.
THIS SIMPLIFIES TO X.
SO X = +2 FOR THE 2nd X COORDINATE.
NOW WE'RE GOING TO FIND THE CORRESPONDING Y VALUES.
WHEN X IS 0 THIS PRODUCT HERE IS 0.
SO WE HAVE Y = 0 - 2 OR -2.
AND WHEN X IS 2 THIS IS 0.
SO WE HAVE Y = 0 - 2 OR -2 AGAIN.
LET'S GO AHEAD AND PLOT THESE TWO POINTS,
0, -2, OUR Y INTERCEPT
AND 2, -2 HERE.
WE SHOULD RECOGNIZE THE AXIS OF SYMMETRY
IS GOING TO BE THIS VERTICAL LINE
RIGHT BETWEEN THE TWO POINTS WITH A EQUATION OF X = 1
WHICH MEANS THE VERTEX MUST HAVE A X COORDINATE OF 1.
SO NOW WE'LL FIND THE CORRESPONDING Y VALUE
OR THE Y COORDINATE OF THE VERTEX BY DETERMINING F OF 1
WHICH WOULD BE 1/2 x 1/SQUARED - 1 - 2.
IT'S GOING TO BE 1/2 - 1.
THAT'S -1/2 - 2.
THAT'S -2 1/2 OR -5/2.
WHICH IS THE SAME AS -2.5.
SO THE VERTEX HAS COORDINATES 1, -2.5 WHICH WOULD BE HERE.
NOW THESE THREE POINTS ARE PRETTY CLOSE TOGETHER
SO WE MAY WANT TO FIND ONE MORE POINT ON THIS PARABOLA.
LET'S EVALUATE THE FUNCTION OF X = 4.
4/SQUARED IS 16, 16 x 1/2 = 8, 8 - 4 - 2 IS +2.
SO IF F OF 4 IS EQUAL TO 2
THAT MEANS THE PARABOLA WOULD CONTAIN THE POINT 4, 2,
WHICH WOULD BE THIS POINT HERE.
AND THERE'S A SYMMETRIC POINT
ON THE OTHER SIDE OF THE AXIS OF SYMMETRY
WHICH WOULD BE THIS POINT HERE.
NOW WE CAN MAKE A NICE SKETCH OF OUR PARABOLA.
IT WOULD LOOK SOMETHING LIKE THIS.
OKAY. I HOPE YOU FIND THIS METHOD HELPFUL.
ONE THING NICE ABOUT THIS METHOD
IS YOU DON'T HAVE TO MEMORIZE ANY SPECIAL FORMULAS.