Now let us look at another method of curve sketching with asymptotes. This method involves dividing the number plane into regions by drawing dotted lines through the critical points. A critical point is a xintercept point or a point where there is a vertical asymptote.
The following steps are involved in this method of sketching curves with asymptotes:
Step 1: Draw dotted lines through critical points. The critical point/s will divide the number planes to regions. Name each region as region1, region2, and so on.
Step 2: Draw lines through the critical points as y = x + critical points
Step 3: Now look at the sign of the curve in each region. A graph is positive if it is above the xaxis.
Step 4: Find any horizontal asymptotes i.e. when x , find the limiting value of y.
Use the above information to sketch the graph. If you need more accurate sketching, you can also follow these additional steps:
Step 5: Find the stationary points and its nature
Step 6: Find the point of inflection – if it exists – and check for its concavity
Step 7: Sketch the curve
Example of sketching curves with asymptotes:
Sketch the curve for y =
Step 1: The critical points are:
xintercept : x = 0, y = 0 (0, 0)
Vertical asymptotes at x = because x² – 1 ≠ 0
The number plane is divided into 4 regions as shown in the diagram below:
Step 2: Now draw lines through the critical points as shown below.
The lines are y = x, y = x + 1 and y = x – 1
Step 3: Find the sign of y.
Region I:

y = = = = = – 
In region I, the curve is below the xaxis
Region II:

y = = = + 
In region II, the curve is above the xaxis
Region III:

y = = = – 
In region III, the curve is below the xaxis
Region IV:

y = = = + 
In region IV, the curve is above the xaxis
Step 4: Let’s find the horizontal asymptotes
We can find them by finding y when x
=
When x , 0. So
=
= 0+
When x , 0+ from above.
When x , 0. So
=
= 0 from below
Step 5: We next need to find the stationary points
=
=
=
y’ = 0 for stationary point.
When y’ = 0, x^{2} – 1 = 0
Or x^{2} = 1
So x has no real value. Hence there is no stationary points.
Step 6: We now check for any points of inflection.
y” = =
=
=
=
=
=
To obtain the point of inflection, y” = 0, so
= 0
x = 0 or x^{2} + 6 = 0
x^{2} + 6 = 0 does not give a real solution, hence it cannot be a point of inflection.
Let us check if x = 0 is a point of inflection by checking for y” at points adjacent to 0, viz and .
f”() = =
= = a positive number
f”() = =
= = a negative number
This can be shown in the table below:
x  0  
f”(x)  +  0  – 
The concavity of the curve changes, hence x = 0 is a point of inflection
Step 7: Using the points and region, we sketch the graph
(0, 0) is a point of inflection
(0, 0) is also the xintercept
x = 1 is vertical asymptotes
y = 0 is a horizontal asymptote
See also other examples of sketching curves with asymptotes –
Sketching curves with asymptotes – Example 1