How do you graph #f(x)=(x^32x^23x)/(4x^2+8x)# using holes, vertical and horizontal asymptotes, x and y intercepts?
Factor then analyze!
We can factor both the top and the bottom to get asymptotes, some intercepts, and holes.
From all of that analysis, you should be able to sketch a plot similar to this: graph{x(x3)(x+1)/(4x(x+2)) [13.86, 13.86, 6.93, 6.93]}
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To graph the function f(x) = (x^3  2x^2  3x) / (4x^2 + 8x), we can analyze its properties:

Holes: To find any potential holes, we need to factor the numerator and denominator. Factoring the numerator, we get x(x + 1)(x  3). Factoring the denominator, we get 4x(x + 2). We notice that x = 0 and x = 2 are common factors in both the numerator and denominator. Therefore, there are holes at x = 0 and x = 2.

Vertical Asymptotes: Vertical asymptotes occur when the denominator equals zero, but the numerator does not. Setting the denominator equal to zero, we find x = 0 as a vertical asymptote.

Horizontal Asymptotes: To determine the horizontal asymptote, we compare the degrees of the numerator and denominator. Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote.

xintercepts: To find the xintercepts, we set the numerator equal to zero and solve for x. Setting x(x + 1)(x  3) = 0, we find x = 0, x = 1, and x = 3 as xintercepts.

yintercept: To find the yintercept, we substitute x = 0 into the function. Evaluating f(0), we get y = 0.
By considering these properties, we can plot the graph of f(x) accordingly.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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