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Understanding parametric equations

Understand parametric equations as two coordinate rules for one parameter. Build a table, trace a curve manually and distinguish a point, its parameter and the direction of travel.

Before you startCoordinates, functions and basic graph shapes

01 / Use one parameter in both coordinate rules

A single input produces an ordered pair.

A usual graph rule gives y from x. Parametric equations instead give both x and y from another variable, often called t. Substitute the same value into both rules to locate one point.

x = p(t), y = q(t)
Point on the curve: (p(t), q(t))

The parameter is a label for a position. It may represent time, an angle or another quantity, but no particular meaning is automatic.

One input, two coordinatesExplore

Coordinate rulesParameter t

x = t + 1, y = t² − 1; −2 ≤ t ≤ 2.

t = −2.00 gives (x, y) = (−1.00, 3.00).

The next larger parameter traces the curve down and right.

Grey shows the whole permitted curve. Blue shows the part traced up to your chosen parameter. Retraced portions overlap. Move t yourself; nothing advances automatically.

02 / Keep the parameter separate from the coordinates

The point is (x, y), not (t, y).

Find the point with parameter t = −2 on x = 3t + 2, y = t² + t.Worked example

x = 3(−2) + 2 = −4

Use the x-rule for the first coordinate.

y = (−2)² + (−2) = 2

Use the same parameter in the y-rule.

The point is (−4, 2)

The parameter −2 is not either coordinate in this example.

Parentheses matter when substituting a negative value into a square. (−2)² is 4; −2² is −4.

03 / Make a table of paired coordinates

Read down a column to obtain one point.

For x = t + 1 and y = t² − 1, with −2 ≤ t ≤ 2, a short table locates useful points.

t: −2, −1, 0, 1, 2
x: −1, 0, 1, 2, 3
y: 3, 0, −1, 0, 3

The five points are (−1, 3), (0, 0), (1, −1), (2, 0) and (3, 3). Join them in increasing parameter order with the smooth curve implied by the rules. A small table helps sketch the shape; it does not prove that no features occur between sampled points.

Watch one parameter generate a coordinate pair

Pause, replay or seek freely. The notes explain the same idea and stay in view.

04 / Trace in increasing parameter order

A locus does not record the direction by itself.

In the first model, x increases as t increases. Starting at t = −2, the point travels down and right to (1, −1), then up and right to (3, 3). The turning point comes from y reaching its minimum at t = 0.

Increasing t describes an order along the curve.
It describes physical motion only if t has a time meaning.

In a sketch, a small arrow can show this order. Do not assume every curve is traced from left to right: the x-rule may decrease or change direction.

05 / The permitted parameter values determine the actual curve

A formula alone does not specify the whole question.

Use x = t + 1, y = t² − 1 with different parameter domains.Worked example

−2 ≤ t ≤ 2

Both endpoints (−1, 3) and (3, 3) are included.

0 ≤ t ≤ 2

Only the branch from (1, −1) to (3, 3) is traced.

0 < t ≤ 2

The point (1, −1) is approached but excluded; (3, 3) remains included.

Open or closed endpoint symbols describe the actual point set. Sometimes another permitted parameter produces an endpoint too, so check all possible parameter values before declaring a point excluded.

06 / A parametric curve need not be a function of x

One x-coordinate can occur with different y-coordinates.

Consider x = t², y = t, with −2 ≤ t ≤ 2.Worked example

At t = −1, the point is (1, −1)

The square makes x positive.

At t = 1, the point is (1, 1)

The same x now has a different y.

The whole curve is not a single-valued y = f(x)

It fails the vertical line test.

It can still be described by x = y², −2 ≤ y ≤ 2

A Cartesian relation need not have y alone on one side.

Restricting to 0 ≤ t ≤ 2 gives y = √x on 0 ≤ x ≤ 4. Restricting to −2 ≤ t ≤ 0 gives y = −√x on the same x-range. The parameter domain selects the branch.

07 / Different parameter values may produce the same point

Count positions separately from parameter values.

For x = t² and y = t² with −2 ≤ t ≤ 2, the points at t = −1 and t = 1 are both (1, 1). As t increases from −2 to 0, the point moves from (4, 4) to (0, 0). It then travels back along the same segment.

The locus is y = x, 0 ≤ x ≤ 4.
The segment is retraced; there are not two different copies of it.

A curve can cross itself at a point, retrace an entire portion, or return to its starting point. These are different behaviours; the coordinate rules and parameter values tell you which is happening.

08 / The same curve can have different parameterisations

Changing the clock does not necessarily change the path.

Compare A: x = t, y = t² for 0 ≤ t ≤ 2 with B: x = 2u, y = 4u² for 0 ≤ u ≤ 1.Worked example

In B, set t = 2u

Both rules become the coordinates in A.

Both trace y = x² for 0 ≤ x ≤ 2

The point sets are identical.

Why is plotting only a few parameter values not enough to prove the complete shape of an unfamiliar curve?

Hint

Think about what can happen between samples.

Worked solution

Extra turns, crossings or excluded inputs may lie between samples. Use the rules, domains, identities and further analysis to justify the sketch, rather than treating a sparse table as proof.

10 / Recap

A parameter controls a pair, a domain and an order.

Use the same parameter value in both coordinate rules.
Report the point as (x, y), keeping t separate.
Trace only permitted parameter values, in the requested order.
Distinguish a curve from a function of x and points from parameter values.
Read time, units and physical meaning from the context.

Section 1 of 10 · Use one parameter in both coordinate rules