# How do you identify all asymptotes or holes for #f(x)=(x^3-7x^2+12x)/(-4x^2+8x)#?

First step is to factor the cubic and quadratic equation on the numerator and denominator, respectively.

To factor the cubic expression you can either use a graphing calculator, or find a factor and carry on with synthetic or long division to find the other two factors.

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = a/b. A and B being the leading coefficients of the numerator and denominator.

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Thus, in your equation, there will be no horizontal asymptote.

By signing up, you agree to our Terms of Service and Privacy Policy

To identify all asymptotes or holes for the function ( f(x) = \frac{x^3 - 7x^2 + 12x}{-4x^2 + 8x} ), we need to analyze its behavior near critical points and where the function is undefined.

First, we identify any vertical asymptotes by finding the values of ( x ) that make the denominator of the function zero, but not the numerator. These values represent the points where the function approaches infinity. So, for ( -4x^2 + 8x = 0 ), we find the critical points by setting the denominator equal to zero and solving for ( x ). This yields ( x = 0 ) and ( x = 2 ).

Next, we examine if any of these critical points are also zeros of the numerator. If they are, then they represent holes in the graph rather than vertical asymptotes. By substituting ( x = 0 ) and ( x = 2 ) into the numerator ( x^3 - 7x^2 + 12x ), we find that ( x = 0 ) is a zero of the numerator but ( x = 2 ) is not.

Therefore, ( x = 0 ) represents a hole in the graph.

In summary:

- Vertical asymptotes occur at ( x = 0 ) and ( x = 2 ).
- There is a hole in the graph at ( x = 0 ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- How do you determine if #f(x)= | x^3 |# is an even or odd function?
- How do you write an equation for a function, h(x), that has been translated 5 units down from #f(x)=2x^2-8x+1#?
- Let #f(x) = 1 + 2x# and #g(x) = x/(x-1)#, how do you find each of the compositions and domain and range?
- How do you find the vertical, horizontal or slant asymptotes for #f(x) = (-x^2 + 4x)/(x+2)#?
- How do you find the vertical, horizontal or slant asymptotes for #f(x) = (x^2 + x + 3) /( x-1)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7