The figure below shows the graph of a rational function f. It has vertical asymp

Question

The figure below shows the graph of a rational function f. It has vertical asymptotes x = 1 and x = -3, and horizontal asymptote y = -3. The graph has x-intercept -1 and, it passes through the point (-2. 1). The equation for f(x) has one of the five forms shown below. Choose the appropriate form for f(x), and then write the equation. You can assume that f(x) is in simplest form. f(x) = a/x - b = f(x) = a(x - b)/x - e f(x) = a/(x - b)(x - e) f(x) = a(x - b)/(x - e)(x - d) f(x) = a(x - b)(x - c)/(x - d)(x - e)

Explanation / Answer

Since the vertical asymptotes are x = 1 and x = -3

for vertical asymptotes put the denominator of f(x) equals to zero and evalute the value for x

therefore the denominator of f(x) should be (x - 1) (x + 3)    //put it equals to zero then we can get vertical asymptote

or (x - 1) (x - (-3))

Since the horizontal asymptote is x = -3

Leading coefficient of 'x' divded by leading coefficient of 'y' should be '-3'

therefore the degree of numerator of f(x) should be x2

the graph of f(x) passes through (-2,1) and (-1,0)

f(x) = (-3x2 + bx + c) / (x-1)(x+3)

f(x) = (-3x2 + bx + c ) / (x2 + 2x - 3)

it passes through (-2,1)

1 = (-3*4 + 2b + c) / (4 -4 -3)

1 = (-12 + 2b + c) / (-3)

-3 = -12 + 2b +c

9 = 2b +c         (1)

it passes through (-1,0)

0 = (-3 -b + c) / (-2 * 2)

0 = - 3 -b +c

c = 3 + b       (2)

from (1) and (2)

9 = 2b + 3 + b

9 = 3b + 3

3 = b + 1

b = 2

from (2)

c = 3 + 2

c = 5

therefore f(x) = (-3x2 + 2x + 5) / ( (x-1) (x +3) )

f(x) = -(3x + 5) (x + 1)/ (x - 1) (x +3)

therefore option (5)

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