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

Answer 1

No holes, one VA at #x=4#, #x#- and #y#-intercept at the origin, and EBA is #y=x+4#.

A vertical asymptote occurs in a rational function when there is #0# value in the denominator. Let's take a look our function:
#f(x)=x^2/(x-4)#
If we set the denominator equal to #0#, we can find out when our asymptote will occur:
#x-4=0=>#
#x=4#
Now we know that the function has a vertical asymptote at #x=4#.

We also know that there are no holes in the function because holes occur when there are factors in common in the numerator and the denominator; this function doesn't have any common factors.

An #x#-intercept occurs when the numerator of the rational function is equal to #0#. We can solve for the x-intercept(s):
#x^2=0=>#
#x=0#
There's an #x#-intercept at #(0,0)#. Since the #y#-value is #0#, this point is also the #y#-intercept. Additionally, we know that this #x#-intercept only "bounces" off the #x#-axis (and doesn't cross it) because the multiplicity of #x^2# is #2#, which is even.

Lastly, find the EBA (end behavior asymptote). Since the power in the numerator is greater than the power in the denominator, we have to divide the two using synthetic division:

#4# | #1" "0" "0# #color(white)(w)#| #" ---------------"# #" "|#
#4# | #1" "0" "0# #color(white)(w)#| #" ---------------"# #" "1" "|#
#4# | #1" "0" "0# #color(white)(w)#|#" "4# #" ---------------"# #" "1" "4" "|#
#4# | #1" "0" "0# #color(white)(w)#|#" "4" "16# #" ---------------"# #" "1" "4" | "16#
The quotient is #x+4#, so the EBA is the line #y=x+4#.

The final graph looks like this:

graph{x^2/(x-4) [-47.03, 56.97, -18.04, 34]}

As you can see, there are no holes, 1 #x#- and #y#-intercept, a vertical asymptote at #x=4# and a diagonal asymptote (EBA) of #y=x+4#.
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Answer 2

To graph the function f(x) = x^2/(x-4), we can start by identifying the vertical and horizontal asymptotes, holes, x-intercepts, and y-intercept.

Vertical asymptote: The vertical asymptote occurs when the denominator of the function becomes zero. In this case, the denominator (x-4) becomes zero when x = 4. Therefore, the vertical asymptote is x = 4.

Horizontal asymptote: To determine the horizontal asymptote, we need to compare the degrees of the numerator and denominator. The degree of the numerator is 2 (x^2) and the degree of the denominator is 1 (x-4). Since the degree of the numerator is greater, there is no horizontal asymptote.

Hole: To find the hole, we need to factor the numerator and denominator. The numerator x^2 cannot be factored further, but the denominator (x-4) can be factored as (x-4). Therefore, there is a hole at x = 4.

X-intercept: To find the x-intercept, we set y = 0 and solve for x. So, 0 = x^2/(x-4). This equation is satisfied when x = 0. Therefore, the x-intercept is (0, 0).

Y-intercept: To find the y-intercept, we set x = 0 and evaluate the function. So, f(0) = 0^2/(0-4) = 0. Therefore, the y-intercept is (0, 0).

To summarize:

  • Vertical asymptote: x = 4
  • Horizontal asymptote: None
  • Hole: (4, f(4))
  • X-intercept: (0, 0)
  • Y-intercept: (0, 0)

Using this information, you can plot these points on a graph and draw the curve accordingly.

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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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