Eliminating a parameter

Textbook reading

The problem that opens the lesson

Eliminate $t$ from $x=t+1, y=t^2-4$ for $-2\le t\le 3$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. Eliminate t, translate the parameter interval into restrictions on $x$ or y, and record direction separately. The relevant conditions are not optional bookkeeping: A parameter may not be globally solvable as one expression, so elimination can require cases or may not produce a simple Cartesian form. Following that structure gives $y=(x-1)^2-4$ with $-1\le x\le 4$ ; orientation follows increasing $x$ .

Why this works

The Cartesian equation may describe more points than the chosen parameter interval traces, and it usually loses orientation and timing. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

Eliminating a parameter produces a Cartesian relation containing the parametric path.

Solving one equation for $t$ and substituting into the other is the usual method. Trigonometric parametrizations often eliminate through identities such as $sin^2 t+cos^2 t=1$ .

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

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Why the relationship works

The Cartesian equation may describe more points than the chosen parameter interval traces, and it usually loses orientation and timing.

Textbook reading

A reliable way to work

Eliminate t, translate the parameter interval into restrictions on $x$ or y, and record direction separately.

A parameter may not be globally solvable as one expression, so elimination can require cases or may not produce a simple Cartesian form.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is reporting the full Cartesian curve while ignoring the restricted segment or repeated traversal.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

Eliminate $t$ from $x=t+1, y=t^2-4$ for $-2\le t\le 3$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. Eliminate t, translate the parameter interval into restrictions on $x$ or y, and record direction separately. The relevant conditions are not optional bookkeeping: A parameter may not be globally solvable as one expression, so elimination can require cases or may not produce a simple Cartesian form. Following that structure gives $y=(x-1)^2-4$ with $-1\le x\le 4$ ; orientation follows increasing $x$ .

Why this works

The Cartesian equation may describe more points than the chosen parameter interval traces, and it usually loses orientation and timing. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Eliminate trig parameters from $x=3cos t,y=3sin t$ .

Worked development

Eliminate t, translate the parameter interval into restrictions on $x$ or y, and record direction separately. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Solving one equation for $t$ and substituting into the other is the usual method. Trigonometric parametrizations often eliminate through identities such as $sin^2 t+cos^2 t=1$ . Then apply the conditions explicitly: A parameter may not be globally solvable as one expression, so elimination can require cases or may not produce a simple Cartesian form. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Elimination helps compare parametric and implicit representations and solve intersections.

Reasoning example

Problem

Identify a Cartesian equation that includes more than the parametric path.

Worked development

Eliminate t, translate the parameter interval into restrictions on $x$ or y, and record direction separately. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Solving one equation for $t$ and substituting into the other is the usual method. Trigonometric parametrizations often eliminate through identities such as $sin^2 t+cos^2 t=1$ . Then apply the conditions explicitly: A parameter may not be globally solvable as one expression, so elimination can require cases or may not produce a simple Cartesian form. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Elimination helps compare parametric and implicit representations and solve intersections.

Worked example 4: quick check

Eliminate $t$ from $x=t^2,y=t$ for $-1\le t\le 2$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. Eliminate t, translate the parameter interval into restrictions on $x$ or y, and record direction separately. The relevant conditions are not optional bookkeeping: A parameter may not be globally solvable as one expression, so elimination can require cases or may not produce a simple Cartesian form. Following that structure gives $x=y^2$ with $-1\le y\le 2$ .

Why this works

The Cartesian equation may describe more points than the chosen parameter interval traces, and it usually loses orientation and timing. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

Elimination helps compare parametric and implicit representations and solve intersections.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice