How do you graph #f(x)=(-3x^2-12x-9)/(x^2+5x+4)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

see explanation.

First, factorise and simplify f(x)

#f(x)=(-3cancel((x+1))(x+3))/(cancel((x+1))(x+4))=(-3(x+3))/(x+4)#

with exclusion x ≠ - 1 which indicates a hole at x = - 1

The graph of #f(x)=(-3(x+3))/(x+4)# is the same as
#(-3x^2-12x-9)/(x^2+5x+4)# but without the hole.
#color(blue)"Asymptotes"#

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve : #x+4=0rArrx=-4" is the asymptote"#

Horizontal asymptotes occur as

#lim_(xto+-oo),f(x)toc" ( a constant)"#

divide terms on numerator/denominator by x

#f(x)=((-3x)/x-9/x)/(x/x+4/x)=(-3-9/x)/(1+4/x)#
as #xto+-oo,f(x)to(-3-0)/(1+0)#
#rArry=-3" is the asymptote"#
#color(blue)"Approaches to asymptotes"#
#"horizontal asymptote " y=-3#
as #xto+oo,f(x)toy=-3" from above"#
as #xto-oo,f(x)toy=-3" from below"#
#"vertical asymptote "x=-4#
#lim_(xto-4^-)f(x)to-oo#
#lim_(xto-4^+)f(x)to+oo#
#color(blue)"Intercepts"#
#x=0toy=-9/4larr" y-intercept"#
#y=0tox+3=0tox=-3larr" x-intercept"# graph{(-3(x+3))/(x+4) [-10, 10, -5, 5]}
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Answer 2

To graph the function f(x)=(-3x^2-12x-9)/(x^2+5x+4), we can analyze its holes, vertical and horizontal asymptotes, as well as the x and y intercepts.

  1. Holes: To find the holes, we need to factor the numerator and denominator. The numerator can be factored as -3(x+1)(x+3), and the denominator can be factored as (x+1)(x+4). We can cancel out the common factor (x+1), which results in f(x)=-3(x+3)/(x+4). The hole occurs at x=-1, as it was canceled out from the numerator and denominator.

  2. Vertical Asymptotes: Vertical asymptotes occur where the denominator equals zero. By factoring the denominator, we find that it equals zero at x=-1 and x=-4. Therefore, the vertical asymptotes are x=-1 and x=-4.

  3. Horizontal Asymptotes: To determine the horizontal asymptote, we compare the degrees of the numerator and denominator. Since the degree of the numerator (-3x^2-12x-9) is less than the degree of the denominator (x^2+5x+4), the horizontal asymptote is y=0.

  4. X-intercepts: X-intercepts occur where the function crosses the x-axis. To find them, we set f(x) equal to zero and solve for x. In this case, we set -3(x+3)/(x+4) equal to zero, which gives us x=-3. Therefore, the x-intercept is (-3, 0).

  5. Y-intercept: The y-intercept occurs where the function crosses the y-axis. To find it, we set x equal to zero in the function f(x). Plugging in x=0, we get f(0)=-3(0+3)/(0+4), which simplifies to f(0)=-9/4. Therefore, the y-intercept is (0, -9/4).

By considering these aspects, we can graph the function f(x)=(-3x^2-12x-9)/(x^2+5x+4) accurately.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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