H2 Maths 1.2 - Sketch a Rational Function with an Oblique Asymptote

Step 1 - The question

Sketch y equals x squared minus four over x, labelling all asymptotes and axial intercepts.

Step 2 - Rewrite and find the oblique asymptote

Rewrite the fraction by splitting the numerator term by term.

Step 2 - Rewrite and find the oblique asymptote

x squared over x is x, and negative four over x stays as negative four over x.

Step 2 - Rewrite and find the oblique asymptote

So y equals x minus four over x.

Step 2 - Rewrite and find the oblique asymptote

As x tends to plus or minus infinity, the term negative four over x vanishes, and y approaches x.

Step 2 - Rewrite and find the oblique asymptote

The oblique asymptote is y equals x.

Step 3 - Vertical asymptote and y-intercept

The denominator is zero at x equals zero, so x equals zero is the vertical asymptote.

Step 3 - Vertical asymptote and y-intercept

There is no y-intercept because x equals zero is not in the domain.

Step 4 - x-intercepts

For the x-intercepts, set y to zero: x squared minus four equals zero.

Step 4 - x-intercepts

This factors as x minus two times x plus two, giving x equals two and x equals negative two.

Step 5 - Odd function and symmetry

Check whether the function is odd or even.

Step 5 - Odd function and symmetry

Replace x with negative x: y becomes negative x minus four over negative x, which equals negative x plus four over x.

Step 5 - Odd function and symmetry

That is the negative of the original function, so y equals x minus four over x is an odd function.

Step 5 - Odd function and symmetry

The graph has rotational symmetry of order two about the origin.

Step 6 - Branch directions and sketch

Check the branch directions near the vertical asymptote using the rewritten form.

Step 6 - Branch directions and sketch

As x approaches zero from the right, negative four over x tends to negative infinity, so y tends to negative infinity.

Step 6 - Branch directions and sketch

By odd symmetry, as x approaches zero from the left, y tends to positive infinity.

Step 6 - Branch directions and sketch

The right branch starts below the x-axis near the origin and rises to the upper right, crossing at x equals two.

Step 6 - Branch directions and sketch

The left branch is the rotation of the right branch - it falls to the lower left, crossing at x equals negative two.

Step 6 - Branch directions and sketch

Always state the oblique asymptote equation and mark the x-intercepts with exact coordinates.