Ex - Find the average rate of change given a function rule

- WE WANT TO FIND THE AVERAGE RATE OF CHANGE OF THE FUNCTION F OF X = -X SQUARED + 9 DIVIDED BY X ON THE CLOSED INTERVAL FROM 1 TO 6. TO FIND THE AVERAGE RATE OF CHANGE OF A FUNCTION, WE WANT TO FIND THE CHANGE IN THE OUTPUT OR CHANGE IN THE FUNCTION VALUES AND THEN DIVIDE BY THE CHANGE IN INPUT WHICH IS THE CHANGE IN THE X VALUES. NOTICE HOW DETERMINING THE AVERAGE RATE OF CHANGE IS THE SAME AS DETERMINING THE SLOPE BETWEEN TWO POINTS. SO FOR THE FIRST STEP, WE WANT TO DETERMINE THE FUNCTION VALUES WHEN X = 1 AND X = 6. SO F OF 1 IS GOING TO BE EQUAL TO -1 SQUARED + 9 DIVIDED BY 1. THIS WILL BE -1 + 9 = 8. SO F OF 1 = 8 WHICH MEANS THE INPUT IS 1 AND THE OUTPUT IS 8. THIS ALSO REPRESENTS A POINT ON THE GRAPH WHERE THE X COORDINATE WOULD BE 1 AND THE Y COORDINATE WOULD BE 8. NOW, WE NEED TO FIND F OF 6, SO WE WOULD HAVE NEGATIVE OR THE OPPOSITE OF 6 SQUARED + 9 DIVIDED BY 6. WELL, THIS WOULD BE -36 WHICH I'LL WRITE AS -36/1 + [9,6] SIMPLIFIES TO 3 HALVES. SO TO ADD THESE, OUR COMMON DENOMINATOR IS GOING TO BE 2. SO WE'D MULTIPLY THIS FRACTION BY 2/2, SO THE DENOMINATOR'S GOING TO BE 2. THE NUMERATOR IS GOING TO BE-- THIS WILL BE -72 + 3 WHICH IS -69. HERE OUR INPUT IS 6 AND THE OUTPUT IS -69 HALVES WHICH AGAIN, IF WE WANT TO, WE COULD WRITE AS AN ORDERED PAIR WHERE THE X COORDINATE WOULD BE 6 AND THE Y COORDINATE WOULD BE -69 DIVIDED BY 2. NOW, WE HAVE ALL THE INFORMATION WE NEED TO DETERMINE THE AVERAGE RATE OF CHANGE ON THIS INTERVAL. WE'LL FIRST DETERMINE THE CHANGE IN THE OUTPUTS OR THE CHANGE IN THE FUNCTION VALUES. SO WE'D HAVE -69 HALVES - 8 WHICH I'M GOING TO WRITE AS 8/1. AGAIN, NOTICE HOW THIS IS A DIFFERENCE IN THE FUNCTION VALUES IF WE WANT THE DIFFERENCE IN THE Y COORDINATES AND WE'LL DIVIDE THIS BY THE CHANGE IN THE INPUTS OR THE CHANGE IN THE X VALUES WHICH WOULD BE 6 - 1. OR AGAIN, IF WE WANTED TO, WE COULD USE THE X COORDINATES FROM THE POINTS. NOW, WE NEED TO SIMPLIFY THIS. LOOKING AT THE NUMERATOR, WE HAVE TO HAVE A COMMON DENOMINATOR WHICH WOULD BE 2, SO MULTIPLY THIS FRACTION BY 2/2. SO WE'RE GOING TO HAVE A DENOMINATOR OF 2. WE'LL HAVE -69 MINUS THIS WILL BE 16, THAT'S GOING TO BE -85, AND THEN 6 - 1 = 5. LET'S GO AHEAD AND REWRITE THIS AS -85 HALVES. THEN INSTEAD OF DIVIDING BY 5, LET'S MULTIPLY BY THE RECIPROCAL, SO WE'RE GOING TO MULTIPLY BY 1/5. IN THIS FORM, WE CAN SEE 5 AND - 85 SIMPLIFIES. THERE'S [1,5] AND 5 AND -17 FIVES AND 85, SO WE CAN SEE OUR RATE OF CHANGE IS GOING TO BE -17 HALVES. SO THE AVERAGE RATE OF CHANGE OF F OF X ON THE INTERVAL FROM 1 TO 6 IS -17 HALVES. TO INTERPRET THIS, REMEMBER THE -17 REPRESENTS THE CHANGE IN THE OUTPUT OR CHANGE IN FUNCTION VALUES AND THE 2 REPRESENTS THE CHANGE IN THE INPUT OR CHANGE IN X VALUE. SO ON AVERAGE ON THIS INTERVAL, THE FUNCTION VALUE WILL DECREASE 17 EVERY TIME THE INPUT INCREASES 2. AND AS I MENTIONED EARLIER, THIS AVERAGE RATE OF CHANGE WOULD BE THE SAME AS A SLOPE BETWEEN THESE 2 POINTS. SO LET'S FINISH BY TAKING A LOOK AT THIS GRAPHICALLY. HERE'S A GRAPH OF OUR FUNCTION. HERE'S THE POINT WHEN X = 1 AND HERE'S THE POINT WHEN X = 6. THE SLOPE OF THIS RED LINE IS EQUAL TO THE AVERAGE RATE OF CHANGE WHICH WE JUST FOUND AS -17 HALVES. I HOPE YOU FOUND THIS EXAMPLE HELPFUL.