## Calculus AB and Calculus BC

## CHAPTER 1 Functions

### F. PARAMETRICALLY DEFINED FUNCTIONS

**BC ONLY**

If the *x*- and *y*-coordinates of a point on a graph are given as functions *f* and *g* of a third variable, say *t*, then

*x* = *f* (*t*), *y* = *g*(*t*)

are called *parametric equations* and *t* is called the *parameter.* When *t* represents time, as it often does, then we can view the curve as that followed by a moving particle as the time varies.

**Examples 12–18 are** **BC ONLY****.**

**EXAMPLE 12**

Find the Cartesian equation of, and sketch, the curve defined by the parametric equations

*x* = 4 sin *t*, *y* = 5 cos *t* (0 *t* 2π).

**SOLUTION:** We can eliminate the parameter *t* as follows:

Since sin^{2} *t* + cos^{2} *t* = 1, we have

The curve is the ellipse shown in Figure N1–9.

**FIGURE N1–9**

Note that, as *t* increases from 0 to 2π, a particle moving in accordance with the given parametric equations starts at point (0, 5) (when *t* = 0) and travels in a clockwise direction along the ellipse, returning to (0, 5) when *t* = 2π.

**EXAMPLE 13**

Given the pair of parametric equations,

*x* = 1 − *t*, *y* = (*t* 0),

write an equation of the curve in terms of *x* and *y*, and sketch the graph.

**SOLUTION:** We can eliminate *t* by squaring the second equation and substituting for *t* in the first; then we have

*y*^{2} = *t* and *x* = 1 − *y*^{2}.

We see the graph of the equation *x* = 1 − *y*^{2} on the left in Figure N1–10. At the right we see only the upper part of this graph, the part defined by the parametric equations for which *t* and *y* are both restricted to nonnegative numbers.

**FIGURE N1–10**

The function defined by the parametric equations here is *y* = *F*(*x*) = whose graph is at the right above; its domain is *x* 1 and its range is the set of nonnegative reals.

**EXAMPLE 14**

A satellite is in orbit around a planet that is orbiting around a star. The satellite makes 12 orbits each year. Graph its path given by the parametric equations

*x* = 4 cos *t* + cos 12*t*,

*y* = 4 sin *t* + sin 12*t*.

**SOLUTION:** Shown below is the graph of the satellite’s path using the calculator’s parametric mode for 0 ≤ *t* ≤ 2π.

**FIGURE N1–11**

**EXAMPLE 15**

Graph *x* = *y*^{2} − 6*y* + 8.

**SOLUTION:** We encounter a difficulty here. The calculator is constructed to graph *y* as a function of *x*: it accomplishes this by scanning horizontally across the window and plotting points in varying vertical positions. Ideally, we want the calculator to scan *down* the window and plot points at appropriate horizontal positions. But it won’t do that.

One alternative is to interchange variables, entering *x* as Y_{1} and *y* as X, thus entering Y_{1}, = X^{2} − 6X + 8. But then, during all subsequent processing we must remember that we have made this interchange.

Less risky and more satisfying is to switch to parametric mode: Enter *x* = *t*^{2} − 6*t* + 8 and *y* = *t*. Then graph these equations in [−10,10] × [−10,10], for *t* in [−10,10], See Figure N1–12.

**FIGURE N1–12**

**EXAMPLE 16**

Let *f* (*x*) = *x*^{3} + *x*; graph *f* ^{−1}(*x*).

**SOLUTION:** Recalling that *f* ^{−1} interchanges *x* and *y*, we use parametric mode to graph

*f*: *x* = *t*, *y* = *t*^{3} + *t*

and *f* ^{−1}: *x* = *t*^{3} + *t*, *y* = *t*.

Figure N1–13 shows both *f* (*x*) and *f* ^{−1}(*x*).

**FIGURE N1–13**

Parametric equations give rise to vector functions, which will be discussed in connection with motion along a curve in Chapter 4.