# How do you graph #f(x)=(x^2-5x+6)/(x^2-4x+3)# using holes, vertical and horizontal asymptotes, x and y intercepts?

See below.

It would be very hard to graph this equation correctly as without factoring both the numerator and denominator it would be hard to account for all of the information.

Then, following the same process for the bottom, you get

Based on this, you can rewrite f(x) as:

That's all. Hope that helps!

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See graph

graph{(x^2-5x+6)/(x^2-4x+3) [-10, 10, -5, 5]}

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To graph the function f(x) = (x^2-5x+6)/(x^2-4x+3), 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^2-5x+6)/(x^2-4x+3) = 0 x^2-5x+6 = 0 (x-2)(x-3) = 0 x = 2 or x = 3

Therefore, the x-intercepts are (2, 0) and (3, 0).

Next, let's find the y-intercept by evaluating f(x) when x = 0: f(0) = (0^2-5(0)+6)/(0^2-4(0)+3) f(0) = 6/3 f(0) = 2

Therefore, the y-intercept is (0, 2).

To determine the holes, we need to factor the numerator and denominator: x^2-5x+6 = (x-2)(x-3) x^2-4x+3 = (x-1)(x-3)

We can see that (x-3) is a common factor in both the numerator and denominator. Therefore, there is a hole at x = 3.

To find the vertical asymptotes, we need to determine the values of x that make the denominator equal to zero: x^2-4x+3 = 0 (x-1)(x-3) = 0 x = 1 or x = 3

Therefore, there are vertical asymptotes at x = 1 and x = 3.

To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator. Since both have a degree of 2, we divide the leading coefficients: Leading coefficient of numerator: 1 Leading coefficient of denominator: 1

Therefore, the horizontal asymptote is y = 1.

To summarize:

- x-intercepts: (2, 0) and (3, 0)
- y-intercept: (0, 2)
- Hole: x = 3
- Vertical asymptotes: x = 1 and x = 3
- 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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