another natural way to define relations is to define both elements of the ordered pair (x, y), in...

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