How do I graph the rational function: #y=6/x+4#?
I like to identify the following things first, when asked to graph a rational function:
 yintercept(s)
 xintercept(s)
 vertical asymptote(s)
 horizontal asymptote(s)

To identify the yintercept(s), ask yourself "what is the value of y when x=0"?
#y = 6/0+4#
Since#6/0# is undefined, there is no yint
yintercept: none 
To identify the xintercept(s), ask yourself "what is the value of x when y=0"?
#0 = 6/x+4#
#4 = 6/x#
#4x = 6#
#x = 6/4 = 3/2#
xintercept:#(3/2,0)# 
To identify the vertical asymptotes, we first try and simplify the function as much as possible and then look at where it is undefined
#y = 6/x+4# is already simplified
Undefined when#x=0#
Vertical asymptotes:#x=0# 
To identify the horizontal asymptotes, we think of the limiting behavior (ie: what happens as x gets HUGE)
#y = 6/"HUGE" +4 > 0 + 4 > 4#
Horizontal asymptote:#y=4# Now you might pick a couple additional points to the left/right of your horizontal asymptote to get a sense of the graph shape.
 Pick a point to the left of the
#x=0# asymptote, ie:#x=6#
#y = 6/6 + 4 = 1 + 4 = 3#
Point 1:#(−6,3)#  Pick a point to the right of the
#x=0# asymptote, ie:#x=6#
#y = 6/6 + 4 = 1 + 4 = 5#
Point 2:#(6,5)#
Domain:
#(oo,0),(0,oo)#
Range:#(oo,4),(4,oo)#  Pick a point to the left of the
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To graph the rational function y = 6/(x + 4), follow these steps:

Determine any vertical asymptotes by setting the denominator (x + 4) equal to zero and solving for x. In this case, x + 4 = 0 gives x = 4. So, there is a vertical asymptote at x = 4.

Determine any horizontal asymptotes by comparing the degrees of the numerator and denominator. Since the degree of the numerator (6) is less than the degree of the denominator (1), there is a horizontal asymptote at y = 0.

Find the xintercept by setting y = 0 and solving for x. In this case, 6/(x + 4) = 0 gives x = 4. So, there is an xintercept at (4, 0).

Plot the vertical asymptote, horizontal asymptote, and xintercept on the coordinate plane.

Choose additional xvalues to evaluate the function and find corresponding yvalues. For example, you can choose x = 5, 3, 2, 0, 1, 2, etc.

Plot the points obtained from step 5 on the graph.

Connect the points smoothly to form the graph of the rational function.
Remember to label the axes and provide a title for the graph.
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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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 How do you solve #5/(x^24)+2/x=2/(x2)#?
 How do you solve #8x+ 9x = 34#?
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