Sketching Graphs of Parametric Equations

Parametric Sketching is the process of drawing a curve defined by equations $x = f(t)$ and $y = g(t)$, where $t$ is the parameter. Instead of a direct $y = f(x)$ relationship, the coordinates are linked indirectly through $t$.

The Direction of Motion is a unique feature of parametric graphs; as the parameter $t$ increases, the curve is traced in a specific direction, often indicated by arrows on the sketch.

The Domain and Range of the resulting graph are determined by the outputs of the individual functions $f(t)$ and $g(t)$ over the allowed interval of the parameter.

y-axis Intercepts: These occur when the horizontal component is zero ($x = 0$). To find them, solve $f(t) = 0$ for $t$, then substitute these $t$-values into $g(t)$ to find the corresponding $y$-coordinates.

x-axis Intercepts: These occur when the vertical component is zero ($y = 0$). Solve $g(t) = 0$ for $t$, then substitute these values into $f(t)$ to find the $x$-coordinates.

Asymptotes: Vertical asymptotes occur if $x$ approaches a constant while $y o \pm\infty$ as $t$ approaches a certain value. Horizontal asymptotes occur if $y$ approaches a constant while $x o \pm\infty$.

Check the Domain: Always verify if the parameter $t$ is restricted (e.g., $0 \le t \le 2\pi$). This determines if the graph is a closed loop, a segment, or an infinite curve.

Verify Intercepts: A common mistake is solving for $t$ and forgetting to find the actual $x$ or $y$ coordinate. Always perform the second substitution step.

Use Cartesian as a Check: If you have time, convert to Cartesian form to confirm the general shape (e.g., if you get $y = x^2$, your sketch should look like a parabola).

Ignoring Function Limits: If $x = \sin(t)$, the graph cannot exist outside $-1 \le x \le 1$. Students often draw curves extending infinitely when the parametric components are bounded.

Incorrect Intercept Logic: Students sometimes set $t=0$ to find intercepts. While $t=0$ is a point on the curve, it is only an intercept if $f(0)=0$ or $g(0)=0$.

Missing Asymptotes: When a denominator in a parametric expression approaches zero, an asymptote is likely present; failing to check for these leads to incomplete sketches.