How do you graph #f(x)=(x^2+3x)/(4x+4)# using holes, vertical and horizontal asymptotes, x and y intercepts?
First, I decided to determine the vertical asymptote! You can determine the vertical asymptote by taking the denominator and setting it equal to zero.
There's a vertical asymptote at
Now for the horizontal asymptote! The horizontal asymptote is determined by looking at the leading coefficients in the numerator and denominator. Here's a helpful visual.
In our case, the leading coefficient in the numerator (
In some cases, a slant asymptote may be present in a graph. These are sometimes present in graphs that have a stronger leading coefficient in the numerator than the denominator. To determine the slant asymptote, we need to divide the numerator by the denominator.
If you don't know how to do polynomial long division, I'd recommend watching this helpful YouTube video:
Here's how I worked it out:
The slant asymptote is
Holes are determined by finding numbers that when plugged into x, return undefined. On a graphing calculator these values will give us an error. Luckily for us, there are no holes, so we can move on.
Xintercepts are values that make the numerator equal zero. We can find these by setting the numerator equal to zero and solving.
Another number that makes the numerator equal zero is, well, zero!
So we know our xintercepts are 0 and 3.
To find the yintercepts, set x equal to zero and solve the equation!
So we have a yintercept at
Now that we have everything we can graph the equation!
I'm really bad at drawing lines, but you get the idea! Be sure to have your line go through the vertical and horizontal intercepts!
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To graph the function f(x) = (x^2 + 3x)/(4x + 4), we can analyze its properties:

Holes: To find any holes in the graph, we need to identify values of x that make the denominator zero. In this case, the denominator 4x + 4 equals zero when x = 1. Therefore, there is a hole at x = 1.

Vertical Asymptotes: Vertical asymptotes occur when the denominator of a rational function equals zero, but the numerator does not. In this case, the denominator 4x + 4 equals zero when x = 1. Thus, there is a vertical asymptote at x = 1.

Horizontal Asymptotes: To determine the horizontal asymptote(s), we compare the degrees of the numerator and denominator. The degree of the numerator is 2 (highest power of x), and the degree of the denominator is also 1. Since the degree of the numerator is greater, there is no horizontal asymptote.

xintercepts: To find the xintercepts, we set the numerator equal to zero and solve for x. In this case, x^2 + 3x = 0. Factoring out an x, we get x(x + 3) = 0. Therefore, the xintercepts are x = 0 and x = 3.

yintercept: To find the yintercept, we substitute x = 0 into the function. f(0) = (0^2 + 3(0))/(4(0) + 4) = 0/4 = 0. Hence, the yintercept is at y = 0.
By considering these properties, we can plot the graph of f(x) = (x^2 + 3x)/(4x + 4) 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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