Parametric equations

Points and sketches

x = f(t), y = g(t). Each value of t gives one point (x, y). To find where the curve meets the y-axis, solve x = 0 for t, then find y.

Converting to Cartesian

Eliminate t:

• Rearrange one equation for t and substitute: x = 2t + 1, y = t² gives y = (x - 1)²/4.
• With trig, use an identity: x = 3cos t, y = 3sin t gives x² + y² = 9; x = sin t, y = cos 2t gives y = 1 - 2x².

Domain and range

The range of t limits x and y. For x = 2 + cos t, x can only take values from 1 to 3, so the Cartesian equation has domain 1 ≤ x ≤ 3. State it when asked.

Worked example

Find a Cartesian equation for x = 2t + 1, y = t².

1. t = (x - 1)/2
2. y = ((x - 1)/2)²

Answer: y = (x - 1)²/4

Check yourself

1. x = t² - 1, y = 3t with t > 0. Find y when x = 8.
   Show answer

   9. t² - 1 = 8 gives t = 3 (t > 0), so y = 9.

2. Find a Cartesian equation for x = 3cos t, y = 3sin t.
   • x² + y² = 3
   • x² + y² = 9
   • x + y = 3
   • y = tan x
   Show answer

   x² + y² = 9. cos t = x/3 and sin t = y/3; cos² t + sin² t = 1 gives x²/9 + y²/9 = 1.

3. Eliminate the parameter from x = 2cos t, y = 5sin t.
   • x² + y² = 29
   • x²/2 + y²/5 = 1
   • 4x² + 25y² = 1
   • x²/4 + y²/25 = 1
   Show answer

   x²/4 + y²/25 = 1. cos t = x/2 and sin t = y/5. Then (x/2)² + (y/5)² = 1.

4. A curve has x = t + 1 and y = 1/t. Which equation links x and y?
   • y = 1/(x - 1)
   • y = 1/(x + 1)
   • y = x - 1
   • y = 1/x - 1
   Show answer

   y = 1/(x - 1). t = x - 1, so y = 1/(x - 1).

5. Show that the curve x = sin t, y = cos 2t lies on a parabola. Which is its equation?
   • y = 2x² - 1
   • y = 1 - x²
   • y = 1 - 2x²
   • y = cos 2x
   Show answer

   y = 1 - 2x². cos 2t = 1 - 2sin² t = 1 - 2x².

6. x = t² - 4, y = 2t + 1. Find the y-coordinates where the curve crosses the y-axis.
   Show answer

   y = 5 or y = -3. x = 0 gives t² = 4, so t = ±2. Then y = 5 or -3.

7. x = 2 + cos t, y = sin t, for 0 ≤ t ≤ 2π. What values can x take?
   • 0 ≤ x ≤ 2π
   • -1 ≤ x ≤ 1
   • 1 ≤ x ≤ 3
   • x ≥ 2
   Show answer

   1 ≤ x ≤ 3. cos t lies between -1 and 1, so x lies between 1 and 3.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 4 scheme of work

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