Sketch the graph of the reciprocal function

All reciprocal functions follow this pattern:

\(y=a\ \frac{1}{k\left(x-d\right)}+c\)

As does your equation: \(y=\frac{4}{x+1}+5\)

Notice that:

a=4 > vertical compression by a factor of 4
d=+1 > horizontal shift to the left by 1
c=5 > vertical shift up 2 units

When it comes to graphing transformations, start by graphing y=1/x. You should know how to graph it using a table of values:

x	y
-2	-0.5
-1	-1
0	DNE
1	1
2	0.5

Some key anchor points are (-1,-1) and (1,1).

Next apply the 'a' value of 4 by multiplying only the y-outputs by 4:

x	4[y]
-2	-2
-1	-4
0	DNE
1	4
2	2

Next, we apply the 'd' value of +1. This means we'll subtract 1 from every x-value, while leaving the y-outputs the same:

x-1	4[y]
-3	-2
-2	-4
-1	DNE
0	4
1	2

Finally, we apply the 'c' value of +5 to the y's only:

x	4[y]+5
-3	3
-2	1
-1	DNE
0	9
1	7

By plotting these key points successively, you get your transformation.

Please review the following video: