H2 Maths 1.2 - Sketch a Linear Rational Function

Step 1 - The question

Sketch y equals two x minus three over x plus one, labelling all asymptotes and axial intercepts.

Step 2 - Vertical asymptote

Start with the vertical asymptote.

Step 2 - Vertical asymptote

The denominator is zero when x plus one equals zero, so x equals negative one.

Step 3 - Horizontal asymptote

For the horizontal asymptote, rewrite the fraction using polynomial division.

Step 3 - Horizontal asymptote

Two x minus three equals two times x plus one, minus five.

Step 3 - Horizontal asymptote

So y equals two minus five over x plus one.

Step 3 - Horizontal asymptote

As x tends to plus or minus infinity, the fraction vanishes and y tends to two.

Step 4 - Axial intercepts

For the x-intercept, set y to zero: two x minus three equals zero gives x equals three halves.

Step 4 - Axial intercepts

For the y-intercept, set x to zero: y equals negative three.

Step 5 - Sketch and annotate

Draw the asymptote lines x equals negative one and y equals two as dashed lines.

Step 5 - Sketch and annotate

Mark the intercepts: x equals three halves on the x-axis and negative three on the y-axis.

Step 5 - Sketch and annotate

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

Step 5 - Sketch and annotate

As x approaches negative one from the right, y tends to negative infinity - the right branch dips down.

Step 5 - Sketch and annotate

As x approaches negative one from the left, y tends to positive infinity - the left branch rises up.

Step 5 - Sketch and annotate

Always label the asymptote equations and exact intercept coordinates for full marks.