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# Sketching Curves with Asymptotes – Example 3

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 x-intercept 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 x-axis.

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:

x-intercept : 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 = x – y = x + 1 – y = x – 1 –
y = = = = =  –

In region I, the curve is below the x-axis

Region II:

 y = x – y = x + 1 + y = x – 1 –
y = = =  +

In region II, the curve is above the x-axis

Region III:

 y = x + y = x + 1 + y = x – 1 –
y = = =  –

In region III, the curve is below the x-axis

Region IV:

 y = x + y = x + 1 + y = x – 1 +
y = = =  +

In region IV, the curve is above the x-axis

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, -x2 – 1 = 0

Or x2 = -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 x2 + 6 = 0

x2 + 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 x-intercept

x = 1 is vertical asymptotes

y = 0 is a horizontal asymptote 