- DETERMINE WHICH LINEAR 
FUNCTION WOULD GIVE THE SLOPES
OF THE TANGENT LINES 
TO THE QUADRATIC FUNCTION.
WE CAN ALSO SAY 
WHICH LINEAR FUNCTION
WOULD BE THE DERIVATIVE FUNCTION 
OF F OF X.
LET'S BEGIN BY ANALYZING 
OUR FUNCTION F OF X.
FIRST THING WE MIGHT NOTICE IS 
THAT THE VERTEX IS AT X = 2,
WHICH WOULD BE THIS POINT HERE.
NOTICE IF WE SKETCH A TANGENT 
LINE AT THIS POINT
IT WOULD BE A HORIZONTAL TANGENT 
LINE,
MEANING THE SLOPE WOULD BE ZERO.
WHICH MEANS, 
THE LINEAR FUNCTION
GIVEN THESE SLOPES OF THE 
TANGENT LINES WOULD HAVE TO HAVE
A FUNCTION VALUE OF ZERO 
AT X = 2.
SO IF WE TAKE A LOOK AT OUR 
LINEAR FUNCTIONS AT X = 2,
WE HAVE A FUNCTION VALUE HERE 
AT 2, HERE AT 0, HERE AT -2,
AND HERE AT -4.
NOTICE HOW LINE C 
IS THE ONLY LINEAR FUNCTION
THAT HAS A FUNCTION VALUE 
OF ZERO AT X = 2.
WHICH MEANS, LINE C MUST 
BE THE LINEAR FUNCTION
THAT GIVES THESE THE SLOPES 
OF THE TANGENT LINES.
BUT LET'S ANALYZE OUR FUNCTION 
F OF X FURTHER.
NOTICE TO THE LEFT OF THE VERTEX
WE'RE ON THE OPEN INTERVAL 
FROM NEGATIVE INFINITY TO 2,
WHICH WOULD BE THIS PIECE 
OF THE GRAPH.
THE FUNCTION IS GOING UP-HILL 
FROM LEFT TO RIGHT,
SO WE SAY THE FUNCTION 
IS INCREASING.
WHICH MEANS, THE TANGENT LINES 
ON THIS INTERVAL
WOULD HAVE A POSITIVE SLOPE.
SO, FOR EXAMPLE, IF WE SKETCH A 
TANGENT LINE AT THIS POINT HERE,
NOTICE HOW IT HAS A POSITIVE 
SLOPE.
AS WELL AS IF WE SKETCH 
A TANGENT LINE HERE,
THE TANGENT LINE HAS A POSITIVE 
SLOPE.
WHICH MEANS, THE LINEAR FUNCTION 
WOULD HAVE TO HAVE
POSITIVE FUNCTION VALUES 
ON THE SAME INTERVAL
FROM NEGATIVE INFINITY TO 2.
IF WE LOOK AT THESE LINEAR 
FUNCTIONS
ON THE SAME INTERVAL FROM 
NEGATIVE INFINITY TO 2,
LINES A AND B WOULD BE INCORRECT
BECAUSE NOTICE HOW THEY DO DROP 
BELOW THE X-AXIS
ON THIS INTERVAL.
AND, THEREFORE, THEY HAVE 
NEGATIVE FUNCTION VALUES,
WHICH WOULD INDICATE 
THAT THE TANGENT LINES
HAVE NEGATIVE SLOPES, 
WHICH WE KNOW THEY DON'T.
SO LINES A AND B DEFINITELY 
DON'T WORK.
AND NOW IF WE CONSIDER 
TO THE RIGHT OF THE VERTEX
OR ON THE OPEN INTERVAL 
FROM 2 TO INFINITY,
NOTICE HOW THE FUNCTION IS GOING 
DOWN-HILL FROM LEFT TO RIGHT.
SO WE SAY THE FUNCTION 
IS DECREASING,
WHICH MEANS ALL THE TANGENT 
LINES IN THIS INTERVAL
WOULD HAVE A NEGATIVE SLOPE.
FOR EXAMPLE, IF WE SKETCH A 
TANGENT LINE AT THIS POINT HERE,
NOTICE HOW IT DOES HAVE 
A NEGATIVE SLOPE.
WHICH MEANS, ON THE OPEN 
INTERVAL FROM 2 TO INFINITY
THE LINEAR FUNCTION THAT GIVES 
THE SLOPE OF THE TANGENT LINES
OR THE DERIVATIVE FUNCTION MUST 
BE NEGATIVE ON THIS INTERVAL.
SO LOOKING AT LINES C AND D 
ON THE SAME INTERVAL,
NOTICE HOW LINE C DOES HAVE
NEGATIVE FUNCTION VALUES ON THIS 
INTERVAL,
BUT LINE D HAS BOTH POSITIVE 
AND NEGATIVE FUNCTION VALUES,
WHICH IS ANOTHER REASON 
WHY LINE D IS NOT CORRECT.
LINE C IS A LINEAR FUNCTION
THAT GIVES THE SLOPE 
OF THE TANGENT LINES,
WHICH WOULD ALSO BE THE 
DERIVATIVE FUNCTION OF F OF X.
I HOPE YOU FOUND THIS HELPFUL.
