1.5 Parametric Relations and

Inverses

Another natural way to define relations is to define both elements of the ordered pair (x, y), in terms of another variable t, called a parameter

Parametric equations: equations in the form x = f(t) and y = g(t) for all t in the interval

I. The variable t is the parameter and I .is the parameter interval

What is a parameter

The set of all ordered pairs (x, y) is defined by the equationsx = t + 1 and y = t2 + 2t

a. Find the points determined by t =-2, -1, 0, 1, and 2

Define a functions parametrically

t x = t + 1 y = t2 + 2t (x, y)

-2 -1 0 (-1, 0)

-1 0 -1 (0, -1)

0 1 0 (1, 0)

1 2 3 (2, 3)

2 3 8 (3, 8)

b. find an algebraic relationship between x and y. (can be called “eliminating the parameter”). Is y a function of x?1. Solve the x equation for tx = t + 1, so t = x -12. Substitute your new equation into the y

equation:y = t2 + 2ty = (x – 1)2 + 2(x – 1) now simplifyy = x2 – 2x + 1 + 2x – 2y = x2 - 1

Cont’d

C. graph the relation in the (x, y) plane

Cont’d

Using a calculator in parametric mode

Mode: arrow down 3, change from FUNC to PAR

Hit y = Enter your x and y equations Go to window: tmin: -4, tmax: 2, tstep:.1,

xmin: -5, xmax: 5, ymin:-5, ymax:5 Go to 2nd window: have TblStart = 0, indpnt:

auto, depend: auto Hit graph Hit 2nd graph to get your table of values

find a) find the points determined by t = -3, -2, -1, 0, 1, 2, 3

b) Find the direct algebraic relationship (rewrite the equation in terms of t)

c) Graph the relationship (this can be done either by hand or on the calculator)

1. x = 3t and y = t2 + 52. x = 5t – 7 and y = 17 – 3t3. x = |t + 3| and y = 1/t

Examples

The ordered pair (a, b) is in a relation if and only if the ordered pair (b, a) is in the inverse relation

Inverse functions: if f is a one-to-one function with domain D and range R, then the inverse function of f, denoted f-1, is the function with domain R and range D defined by : f-1 (b) = a if and only if f (a) = b

Inverse functions

Change f(x) to y Switch your x and y Solve for y Rewrite as f-1(x) Determine if f-1(x) is a function

Finding an inverse

Find the inverse of each function1. f(x) = x/(x + 1)

2. f(x) = 3x – 6

3. f(x) = x - 3

Examples

The points (a, b) and (b, a) in the coordinate plane are symmetric with respect to the line y = x. The points (a, b) and (b, a) are reflections of each other across the line y = x.

Inverse Composition Rule: a function is one-to-one with inverse function g if and only if:f(g(x)) = x for every x in the domain of g, andg(f(x)) = x for every x in the domain of f

Inverse Reflection Principle

Algebraically: use the Inverse Composition Rule, find both f(g(x)) and g(f(x)) and if the answers are the same, the functions are inverses

Graphically: in parametric mode, graph and compare the graphs of the 2 sets of parametric equations.

Verifying inverse functions

p. 126 #5-7 odd, 13-21 odd, 27-31 odd, 34- 36 all

Homework