Parametric Equations Grapher (AP Precalculus)

How to use the simulator

The grid runs from −5 to 5 on both axes. The whole path is drawn faintly in the background, the part traced so far is coral, and a white dot marks the current point. Dashed teal lines run from the dot to both axes.

Preset menu:
Circle: $x = 3\cos t,\ y = 3\sin t$ , $0 \le t \le 2\pi$ .
Ellipse (default): $x = 3.5\cos t,\ y = b\sin t$ , $0 \le t \le 2\pi$ .
Lissajous: $x = 3.4\sin(3t + \frac{\pi}{2}),\ y = 3.4\sin(qt)$ , $0 \le t \le 2\pi$ .
Line segment: $x = -4 + 8t,\ y = -3 + 6t$ , $0 \le t \le 1$ .
height b (ellipse only): from 1 to 4 in steps of 0.5 (default 2).
y-freq q (Lissajous only): a whole number from 1 to 5.
trace t: moves the dot from the start to the end of the t-interval. The number beside it is the current t, to two decimals.
▶ Animate: restarts the trace at the beginning and runs it to the end; press ■ Stop to pause.

The readout below the equations shows, for example, t = 1.57 → (0, 2). Animate each preset once and note the direction of travel: the circle and ellipse both go counterclockwise from

(3, 0)

or

(3.5, 0)

, and the segment runs from

(-4, -3)

to

(4, 3)

.

The equations

A parametric function assigns to each value of the parameter t a point

\big(x(t),\ y(t)\big)

. The graph is the set of those points, but the parametrization carries more: where the point starts, which way it moves, and how fast.

Circle of radius r centered at $(h, k)$ : $x = h + r\cos t,\ y = k + r\sin t$ , counterclockwise for $0 \le t \le 2\pi$ .
Ellipse: $x = A\cos t,\ y = B\sin t$ satisfies $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ .
Line segment from $(x_1, y_1)$ to $(x_2, y_2)$ : $x = x_1 + (x_2 - x_1)t,\ y = y_1 + (y_2 - y_1)t$ , $0 \le t \le 1$ .
Eliminating the parameter: solve one equation for t (or use $\cos^2 t + \sin^2 t = 1$ ) and substitute to get an equation in x and y.
Average rate of change: over $[t_1, t_2]$ , $\frac{x(t_2) - x(t_1)}{t_2 - t_1}$ and $\frac{y(t_2) - y(t_1)}{t_2 - t_1}$ give the horizontal and vertical rates separately.

Worked example

For the line-segment preset, $x(t) = -4 + 8t$ , $y(t) = -3 + 6t$ , $0 \le t \le 1$ : find the point at $t = 0.25$ , eliminate the parameter, and describe the motion.

Step 1: evaluate.

x(0.25) = -4 + 2 = -2

and

y(0.25) = -3 + 1.5 = -1.5

, so the point is

(-2, -1.5)

.

Step 2: eliminate t. From

x = -4 + 8t

,

t = \frac{x + 4}{8}

. Then

y = -3 + 6\cdot\frac{x + 4}{8} = -3 + 0.75x + 3 = 0.75x

. The path is part of the line

y = 0.75x

, for

-4 \le x \le 4

.

Step 3: rates. x increases by 8 per unit of t and y by 6 per unit of t, so the slope of the path is

\frac{6}{8} = 0.75

, and the point travels

\sqrt{8^2 + 6^2} = 10

units from

t = 0

to

t = 1

.

Check it in the simulator: choose Line segment and drag trace t until the value reads 0.25. The readout shows t = 0.25 → (-2, -1.50).

Follow-up (ellipse): choose Ellipse with b = 2. Eliminating t with

\cos^2 t + \sin^2 t = 1

gives

\frac{x^2}{3.5^2} + \frac{y^2}{2^2} = 1

. At

t = \frac{\pi}{2} \approx 1.57

, the point is

(3.5\cos\frac{\pi}{2},\ 2\sin\frac{\pi}{2}) = (0, 2)

, the top of the ellipse; set the trace to a quarter of its length and the readout shows t = 1.57 → (0, 2).

Common mistakes on the AP exam

Unit 4 of AP Precalculus, which contains parametric functions, is not assessed on the AP Precalculus exam; it is taught for class assessments and as preparation for AP Calculus BC. These are the errors that show up on unit tests and later in calculus:

Treating t as x. t is the input; x and y are both outputs. A point on the curve is $(x(t), y(t))$ , never $(t, y)$ .
Losing the direction. Eliminating the parameter gives the shape but erases orientation and the start and end points. State them separately.
Forgetting the restricted domain. The segment above is only the part of $y = 0.75x$ with $-4 \le x \le 4$ , not the whole line.
Mixing up the coefficients. In $x = 3.5\cos t,\ y = 2\sin t$ , the ellipse is 7 wide and 4 tall: the coefficients are semi-axes.
Assuming a curve is traced once. Animate the Lissajous preset with q = 3: then $x = 3.4\cos 3t$ and $y = 3.4\sin 3t$ , so the point goes around one circle three times.

When the AP exam uses this

Because Unit 4 is outside the exam's scope, parametric questions will not appear on the AP Precalculus exam itself. The ideas return directly in AP Calculus BC, where you find

\frac{dy}{dx}

for parametric curves, speed and distance traveled, and in AP Physics, where projectile motion is a pair of parametric equations in time.

Parametric Equations Grapher

How to use the simulator

The equations

Worked example

Common mistakes on the AP exam

When the AP exam uses this

More AP Precalculus simulators