ACT Math Parametric Equations: Eliminate Parameters to Find Cartesian Form

Parametric equations define x and y separately using a parameter (usually t). Example: x=2t, y=t+3. To find the Cartesian equation (y in terms of x), eliminate t. From x=2t, solve for t: t=x/2. Substitute into y: y=(x/2)+3. This is the Cartesian form. The process is mechanical: solve one parametric equation for the parameter, substitute into the other, simplify. Questions ask for the Cartesian form or ask you to identify which parametric equations produce a given curve. Graph-wise, parametric equations trace a path over time; Cartesian equations show the path's shape.

Another example: x=cos(t), y=sin(t). Square both: x²=cos²(t), y²=sin²(t). Add: x²+y²=cos²(t)+sin²(t)=1. This is the Cartesian equation of a circle (radius 1).

Mistake 1: Choosing the wrong equation to solve for the parameter. Pick the simpler one. If x=t and y=t², solve x equation for t (easier than y). Mistake 2: Forgetting to substitute completely. You solve for t but forget to replace it in the other equation. Mistake 3: Losing track of domain restrictions. If x=cos(t), then -1≤x≤1 always. This restriction carries to the Cartesian form. Parametric equations often have domain restrictions the Cartesian form might lose. Note these.

During practice, solve for the parameter step-by-step, then substitute carefully. Write the Cartesian equation clearly.

Problem 1: x=t, y=2t+1. Solve x for t: t=x. Substitute: y=2x+1 (linear equation). Problem 2: x=t+1, y=t². Solve x for t: t=x-1. Substitute: y=(x-1)² (parabola). Problem 3: x=2cos(t), y=3sin(t). Square and add: (x/2)²+(y/3)²=cos²(t)+sin²(t)=1. Cartesian: (x/2)²+(y/3)²=1 (ellipse). Problem 4: x=e^t, y=2e^t. From first: e^t=x. Substitute: y=2x. Note: x>0 (since e^t>0). Problem 5: x=3t-2, y=t/2. From y: t=2y. Substitute: x=3(2y)-2=6y-2. Rearranged: x=6y-2 or y=(x+2)/6. Convert each to Cartesian, noting domain restrictions.

Find five parametric-to-Cartesian conversion problems from a practice test. For each, eliminate the parameter and write the Cartesian form. By the fifth problem, elimination will feel mechanical.

Parametric equation questions appear on some ACT Math tests, usually in questions 50-60. They test algebraic manipulation. Students who systematically eliminate parameters pick up 1 point because the process is mechanical and straightforward once you follow the method.

Drill parameter elimination daily this week. Each day, convert five parametric equations to Cartesian form. By test day, you should eliminate any parameter and write the resulting equation within 90 seconds.