- DETERMINE THE POINTS WHERE THE TANGENT LINES
OF THE GRAPH OF F OF X = -1 DIVIDED BY X ARE PARALLEL
TO THE LINE Y = 2X + 4.
SO THE FIRST THING WE SHOULD WE RECOGNIZE
IS THAT THE SLOPE OF THE GIVEN LINE IS 2.
AND IF LINES ARE PARALLEL, THEY HAVE THE SAME SLOPE.
SO NOW LOOKING AT THE GRAPH,
WE HAVE THE FUNCTION GRAPHED HERE IN BLUE
AND THE GIVEN LINE GRAPHED HERE IN RED.
WE'RE LOOKING FOR THE POINTS ON THE BLUE FUNCTION
WHERE IF WE SKETCH A TANGENT LINE,
THE TANGENT LINE WILL BE PARALLEL TO THE GIVEN LINE,
MEANING THE TANGENT LINE WOULD HAVE A SLOPE OF 2.
SO TO BEGIN, WE'LL FIND THE DERIVATIVE OF OUR FUNCTION,
WHICH WILL GIVE US THE SLOPES OF THE TANGENT LINES
AT A GIVEN VALUE OF X.
SO OUR FUNCTION IS F OF X = -1 DIVIDED BY X,
WHICH I'M GOING TO WRITE AS -1 DIVIDED BY X TO THE FIRST
BECAUSE WE WANT TO CHANGE THE FORM,
SO WE CAN APPLY THE POWER RULE OF DIFFERENTIATION.
IF WE MOVE THIS UP TO THE NUMERATOR
IT'LL CHANGE THE SIGN OF THE EXPONENT.
THIS IS EQUIVALENT TO -1 x X TO THE POWER OF -1
OR JUST -X TO THE POWER OF -1.
IN THIS FORM WE CAN NOW FIND THE DERIVATIVE
USING THE POWER RULE OF DIFFERENTIATION.
SO F PRIME OF X WOULD BE EQUAL TO--
WE'RE GOING TO MULTIPLY BY THE EXPONENT
THAT'S GOING TO MAKE IT POSITIVE.
X TO THE POWER OF -1 - 1 OR X TO THE POWER OF -2,
WHICH WE COULD ALSO WRITE AS 1/X SQUARED.
SO THIS FUNCTION WILL GIVE US THE SLOPE OF THE TANGENT LINE
AT A GIVEN VALUE OF X.
SO NOW WE'RE GOING TO SET 1 DIVIDED BY X SQUARED EQUAL TO 2
TO DETERMINE THE X VALUES
WHERE THE SLOPE OF THE TANGENT LINES WOULD BE 2.
SO NOW WE'LL HAVE 1 DIVIDED BY X SQUARED = 2,
WHICH I'LL WRITE AS 2/1.
AND NOW TO SOLVE FOR X, WE'LL CROSS MULTIPLY.
X SQUARED x 2 MUST EQUAL 1 x 1.
SO 2X SQUARED = 1, DIVIDING BOTH SIDES BY 2,
WE HAVE X SQUARED = 1/2.
NOTICE HERE WE'LL HAVE A POSITIVE AND NEGATIVE SOLUTION.
SO TO SOLVE FOR X WE'LL TAKE THE SQUARE ROOT OF BOTH SIDES,
BUT WE'LL INCLUDE A PLUS OR MINUS SIGN HERE
TO MAKE SURE WE OBTAIN BOTH SOLUTIONS.
SO THE SQUARE ROOT OF X SQUARED IS JUST X,
THEN WE HAVE PLUS OR MINUS THE SQUARE ROOT OF 1
DIVIDED BY THE SQUARE ROOT OF 2 OR JUST PLUS OR MINUS 1
DIVIDED BY SQUARE ROOT 2.
SO NOTICE HOW THERE ARE TWO X VALUES
WHERE THE SLOPE OF THE TANGENT LINE WOULD BE 2.
AGAIN, WE'RE LOOKING FOR THE POINTS ON THE FUNCTION.
SO WE NEED TO FIND BOTH THE X COORDINATES
AND THE Y COORDINATES.
SO THERE'LL BE TWO POINTS
WHERE ONE POINT HAS AN X COORDINATE OF -1
DIVIDED BY SQUARE ROOT 2.
THE OTHER POINT WOULD HAVE X COORDINATE OF +1
DIVIDED BY SQUARE ROOT 2.
TO FIND THE CORRESPONDING Y COORDINATES
WE'LL HAVE TO SUBSTITUTE THESE X VALUES
BACK INTO THE ORIGINAL FUNCTION F OF X = -1 DIVIDED BY X.
SO FOR F OF -1 DIVIDED BY SQUARE ROOT 2
WE'D HAVE -1 DIVIDED BY X, WHICH IS -1/SQUARE ROOT 2.
THIS IS EQUIVALENT TO -1
DIVIDED BY THIS FRACTION OR -1 x THE RECIPROCAL,
WHICH WOULD JUST BE -SQUARE ROOT 2/1 OR -SQUARE ROOT 2
GIVING US +SQUARE ROOT 2.
SO THIS Y COORDINATE IS +SQUARE ROOT 2,
AND WHEN X IS +1 DIVIDED BY SQUARE ROOT 2,
IT'S JUST GOING TO CHANGE THE SIGN OF THE OUTPUT.
IT WOULD BE -SQUARE ROOT 2.
NOW, TO CHECK THESE POINTS ON THE COORDINATE PLANE,
LET'S GO AND CONVERT THESE TO DECIMALS
SO WE CAN LOCATE THEM ON THE COORDINATE PLANE.
BUT WHEN GIVING OUR FINAL ANSWER
WE DO WANT TO GIVE THE EXACT COORDINATES IN THIS FORM HERE.
TO SAVE SOME TIME, I'VE ALREADY CONVERTED THESE TO DECIMALS.
NOTICE -1 DIVIDED BY SQUARE ROOT 2 IS APPROXIMATELY -0.707,
WHICH MEANS THE SECOND X COORDINATE,
WHICH WOULD BE +0.707 OR APPROXIMATELY.
AND THEN SQUARE ROOT 2 IS APPROXIMATELY 1.414.
SO THIS WOULD BE APPROXIMATELY 1.414
AND THIS WOULD BE APPROXIMATELY -1.414.
SO NOW GOING OVER TO THE COORDINATE PLANE,
NOTICE HOW THIS FIRST PLANE WOULD BE APPROXIMATELY HERE,
AND THE SECOND POINT WOULD BE APPROXIMATELY HERE.
SO IF WE SKETCH THE TANGENT LINES AT THESE POINTS
THEY SHOULD BE PARALLEL TO THIS RED LINE.
AND OF COURSE THESE ARE HAND DRAWN,
BUT IT DOES LOOK LIKE THESE TANGENT LINES ARE PARALLEL
TO THE GIVEN LINE Y = 2X + 4.
SO THESE WOULD BE THE COORDINATES OF THE TWO POINTS
WHERE THE TANGENT LINES WILL BE PARALLEL TO THE GIVEN LINE.
I HOPE YOU FOUND THIS HELPFUL.