- IN THIS PROBLEM, WE WANT TO FIND AN EQUATION
FOR THE RATIONAL FUNCTION BASED UPON THE GRAPH.
TO DO THIS, WE'LL FIND THE FUNCTION IN FACTORED FORM
OR THIS FORM HERE, WHERE "A" IS A CONSTANT
AND WE'LL HAVE FACTORS IN THE NUMERATOR
AND FACTORS IN THE DENOMINATOR.
BEFORE WE DO THIS
LET'S IDENTIFY SOME OF THE KEY COMPONENTS FROM OUR GRAPH.
LET'S START WITH THE INTERCEPTS.
NOTICE HOW WE HAVE AN X INTERCEPT OF 3,
THIS WOULD BE THE POINT (3, 0)
AND WE HAVE A Y INTERCEPT OF -2.
THIS WOULD BE THE POINT (0, -2).
NOTICE HOW WE ALSO HAVE A VERTICAL ASYMPTOTE HERE
WHICH IS X = -1.
THIS IS THE INFORMATION WE NEED
TO DETERMINE OUR RATIONAL FUNCTION
BUT WE DO HAVE TO REMEMBER,
KNOWING THE VERTICAL ASYMPTOTE IS X = -1
GIVES US INFORMATION ABOUT A FACTOR IN THE DENOMINATOR
AND KNOWING THAT WE HAVE AN X INTERCEPT OF 3
GIVES US INFORMATION ABOUT A FACTOR IN THE NUMERATOR.
AND THEN WE CAN USE THE Y INTERCEPT OR THIS POINT HERE
TO DETERMINE THE VALUE OF OUR CONSTANT "A".
SO LET'S BEGIN TO SET THIS UP.
WE'RE GOING TO HAVE F OF X =
WE'LL FIND "A" IN THE SECOND STEP.
SO WE'LL HAVE OUR CONSTANT "A" HERE
AND NOW LET'S CONSIDER OUR VERTICAL ASYMPTOTE.
REMEMBER VERTICAL ASYMPTOTES ARE FOUND
BY DETERMINING WHICH X VALUES MAKE THE DENOMINATOR
EQUAL TO 0.
SO IF X = -1 MAKES OUR DENOMINATOR EQUAL TO 0
THAT MEANS WE MUST HAVE A FACTOR OF X + 1
IN THE DENOMINATOR.
NOTICE IF X IS -1 WE'D HAVE A ZERO IN THE DENOMINATOR
WHICH WOULD GIVE US THIS VERTICAL ASYMPTOTE.
NEXT, IF WE KNOW THAT THE X INTERCEPT IS (3, 0)
THAT MEANS F OF 3 MUST EQUAL 0,
MEANING WHEN X IS 3 THE FUNCTION VALUE IS 0.
WELL A RATIONAL FUNCTION IS EQUAL TO 0
WHEN THE NUMERATOR IS EQUAL TO 0.
THEREFORE, IF WE KNOW THE X INTERCEPT IS +3
OUR NUMERATOR MUST HAVE A FACTOR OF X - 3.
AND NOW WE ALMOST HAVE OUR FUNCTION
BUT WE STILL HAVE TO FIND THE VALUE OF "A".
AND WE CAN FIND "A" BY USING THE Y INTERCEPT
OR THIS POINT HERE.
IF THIS POINT ON THE FUNCTION HAS COORDINATES (0, -2)
THEN F OF 0 = -2.
SO IF WE SUBSTITUTE 0 FOR X
THIS FUNCTION VALUE MUST EQUAL -2.
SO LET'S GO AHEAD AND DO THAT.
F OF 0 WOULD BE EQUAL TO "A" x 0 - 3 IS -3, 0 + 1 = 1.
THIS MUST EQUAL -2.
SO WE HAVE WHAT, -3A = -2 DIVIDE BOTH SIDES BY -3
AND WE HAVE THE VALUE OF "A."
"A" IS EQUAL TO +2/3.
NOW LOOKING BACK AT THIS FUNCTION HERE,
THIS IS THE SAME AS "A" TIMES THIS FRACTION
AND SINCE "A" IS 2/3 WE CAN WRITE THE FUNCTION
AS F OF X = 2 x THE QUANTITY X - 3 DIVIDED BY 3
x THE QUANTITY X + 1.
NOTICE HOW WE HAVE 2/3 x THE FRACTION,
WHICH FITS THE FORM THAT WE HAVE HERE.
WE REALLY SHOULDN'T LEAVE THE FRACTION OF 2/3
IN THE NUMERATOR.
WE SHOULD WRITE IT THIS WAY.
THIS FUNCTION HERE HAS ALL THE CHARACTERISTICS
GIVEN BY THE GRAPH.
ALSO NOTICE THAT TO DOUBLE CHECK THIS
NOTICE HOW THE DEGREE OF THE NUMERATOR AND DENOMINATOR
ARE BOTH 1,
WHICH MEANS A HORIZONTAL ASYMPTOTE WOULD BE THE RATIO
OF THE LEADING COEFFICIENTS
WHICH IN THIS CASE WOULD BE Y = 2/3.
LOOKING AT OUR GRAPH, IT DOES APPEAR
THAT OUR HORIZONTAL ASYMPTOTE IS Y = 2/3.
OKAY. I HOPE THIS EXPLANATION WAS HELPFUL.