Rational Functions

Rational Functions

Example: Sketch the graph of

Solution:

Split into partial fractions.

From here, we can tell that

Vertical asymptote: x = -1

Horizontal asymptote: y = 1

Intercepts:

when y = 0, x = 1

when x = 0, y = -1

Turning point:

dy/dx = 0

---> [x + 1 - (x - 1)] / (x + 1)² = 0

---> 2 / (x + 1)²= 0 --> no solution

There are no turning points.

Gradient:

dy/dx is always positive.

The gradient is always increasing.

Actually, this example is exactly the same as the one we used in our section Rectangular Hyperbolas. Hence, you can see the two methods that can be used for deriving the shape of this graph. Obviously, the method described in this page is more general as it can apply to graphs of any rational function, unlike the other method described earlier.

Below is another example to illustrate what we mean.

Example: Sketch the graph of

Solution:

Express as partial fractions.

From here, we can derive the asymptotes--

Vertical: x = 2

Oblique : y = 2x

Intercepts:

when x = 0, y = -3/2

when y = 0, x = 2x² - 2x + 3

Discriminant = -20 < 0 --> no real roots.

There are no x-intercepts.

Turning points:

Find the corresponding y-values.