Reciprocal graphs and intersections

y = k/x

Two separate branches. Asymptotes are x = 0 and y = 0.

• k > 0: branches in the first and third quadrants
• k < 0: branches in the second and fourth quadrants

y = k/x²

x² is always positive, so for k > 0 both branches are in the first and second quadrants (y > 0). Asymptotes are again x = 0 and y = 0.

Using intersections

The solutions of f(x) = g(x) are the x-coordinates where y = f(x) and y = g(x) meet. A sketch tells you how many solutions there are; algebra finds them.

Worked example

Find where y = 4/x meets y = x.

1. 4/x = x, so x² = 4
2. x = 2 or x = -2
3. y = x gives the y-values

Answer: (2, 2) and (-2, -2)

Check yourself

1. What are the asymptotes of y = 3/x?
   • x = 3 and y = 0
   • x = 0 and y = 0
   • x = 0 and y = 3
   • x = 0 only
   Show answer

   x = 0 and y = 0. y = 3/x is undefined at x = 0 and never equals 0.

2. In which quadrants does the graph of y = -2/x lie?
   • First and third
   • First and second
   • Third and fourth
   • Second and fourth
   Show answer

   Second and fourth. k = -2 < 0. Positive x gives negative y (fourth); negative x gives positive y (second).

3. Where does the graph of y = 5/x² lie?
   • In the first and third quadrants
   • In the first and second quadrants
   • In the third and fourth quadrants
   • In the first quadrant only
   Show answer

   In the first and second quadrants. x² > 0, so y > 0 for every x not 0. Both branches are over the x-axis.

4. Find the x-coordinates where y = 8/x meets y = 2x.
   Show answer

   x = 2 or x = -2. 8/x = 2x gives x² = 4.

5. How many times does the graph of y = x³ meet the graph of y = 1/x?
   Show answer

   2. x³ = 1/x gives x⁴ = 1, so x = 1 or x = -1: two points.

6. y = 6/x meets y = x + 1 in the first quadrant. Find the y-coordinate of this point.
   Show answer

   3. 6/x = x + 1 gives x² + x - 6 = 0, so (x + 3)(x - 2) = 0. In the first quadrant x = 2, so y = 3.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.