# How do you graph #f(x)=(x^2-16)/(x^2-6x+8)# using holes, vertical and horizontal asymptotes, x and y intercepts?

First, factor:

To find the vertical asymptote, set the simplest form of the denominator equal to zero:

**Here, the degrees are the same so #1/1# is our horizontal asymptote
(#y=1#).To find the x-intercept, plug in #y=0#:#(x+4)/(x-2)=0##x=-4#The x-intercept is at #(-4,0)#To find the y-intercept, plug in #x=0##(0+4)/(0-2)=y##y=-2#The y-intercept is at #(0,-2)#**

Graph it:

graph{(x^2-16)/(x^2-6x+8) [-15.26, 17.28, -6.31, 9.96]}

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To graph the function f(x) = (x^2-16)/(x^2-6x+8), we can analyze its holes, vertical and horizontal asymptotes, as well as the x and y intercepts.

First, let's find the x-intercepts by setting f(x) equal to zero and solving for x: (x^2-16)/(x^2-6x+8) = 0 x^2-16 = 0 (x-4)(x+4) = 0 x = 4 or x = -4

Therefore, the x-intercepts are x = 4 and x = -4.

Next, let's find the y-intercept by substituting x = 0 into the function: f(0) = (0^2-16)/(0^2-6(0)+8) f(0) = -16/8 f(0) = -2

Therefore, the y-intercept is y = -2.

To determine the holes in the graph, we need to factor the numerator and denominator and cancel out any common factors: f(x) = (x^2-16)/(x^2-6x+8) f(x) = [(x-4)(x+4)]/[(x-4)(x-2)]

We can see that (x-4) cancels out, leaving us with: f(x) = (x+4)/(x-2)

Therefore, there is a hole at x = 4.

To find the vertical asymptotes, we need to determine the values of x that make the denominator equal to zero: x^2-6x+8 = 0 (x-2)(x-4) = 0 x = 2 or x = 4

Therefore, there are vertical asymptotes at x = 2 and x = 4.

To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator: The degree of the numerator is 1, and the degree of the denominator is also 1.

Therefore, there is a horizontal asymptote at y = 1.

To summarize:

- x-intercepts: x = 4 and x = -4
- y-intercept: y = -2
- Hole: x = 4
- Vertical asymptotes: x = 2 and x = 4
- Horizontal asymptote: y = 1

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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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