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

No holes, one VA at

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.

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:

The final graph looks like this:

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

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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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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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