## SAT SUBJECT TEST MATH LEVEL 2

## PART 2

## REVIEW OF MAJOR TOPICS

## CHAPTER 1

Functions

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1.6 Miscellaneous Functions

### PARAMETRIC EQUATIONS

At times, it is convenient to express a relationship between *x *and *y *in terms of a third variable, usually denoted by a **parameter** *t*. For example, **parametric equations** *x *= *x(t*), *y *= *y(t*) can be used to locate a particle on the plane at various times *t*.

**EXAMPLES**

**1. Graph the parametric equations**

Select MODE on your graphing calculator, and select PAR. Enter 3*t *+ 4 into *X*1_{T} and *t *– 5 into *Y*1_{T}. The standard window uses 0 for Tmin and 6.28… (2π) for Tmax along with the usual ranges for *x *and *y*. The choice of 0 for Tmin reflects the interpretation of *t *as “time.” With the standard window, the graph looks like the figure below.

If you use TRACE, the cursor will begin at *t *= 0, where (*x*,*y*) = (4,–5). As *t *increases from 0, the graph traces out a line that ascends as it moves right.

It may be possible to eliminate the parameter and to rewrite the equation in familiar *xy*-form. Just remember that the resulting equation may consist of points not on the graph of the original set of equations.

**2.** **Eliminate the parameter and sketch the graph**

Substituting *x *for *t*^{2} in the second equation results in *y *= 3*x *+ 1, which is the equation of a line with a slope of 3 and a *y*-intercept of 1. However, the original parametric equations indicate that *x * 0 and *y * 1 since *t*^{2} cannot be negative. Thus, the proper way to indicate this set of points without the parameter is as follows: *y *= 3*x *+ 1 and *x * 0. The graph is the ray indicated in the figure.

**3. Sketch the graph of the parametric equations**

Replace the parameter with *t*, and enter the pair of equations. The graph has the shape of an ellipse, elongated horizontally, as shown in this diagram.

It is possible to eliminate the parameter, , by dividing the first equation by 4 and the second equation by 3, squaring each, and then adding the equations together.

Here, =1 is the equation of an ellipse with its center at the origin, *a *= 4, and *b *= 3 (see Coordinate Geometry). Since –1 cos 1 and –1 sin 1, –4 *x * 4 and –3 *y * 3 from the two parametric equations. In this case the parametric equations do not limit the graph obtained by removing the parameter.

**EXERCISES**

1. In the graph of the parametric equations

(A) *x * 0

(B)

(C) *x *is any real number

(D) *x * –1

(E) *x * 1

2. The graph of is a

(A) straight line

(B) line segment

(C) parabola

(D) portion of a parabola

(E) semicircle

3. Which of the following is (are) a pair of parametric equations that represent a circle?

I.

II.

III.

(A) only I

(B) only II

(C) only III

(D) only II and III

(E) I, II, and III