 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

 The set of all ordered pairs (x, y) is defined by the equations x = t + 1 and y = t 2 + 2t a. Find the points determined by t =-2, -1, 0, 1, and 2 tx = t + 1y = t 2 + 2t(x, y) -20(-1, 0) 0 (0, -1) 010(1, 0) 123(2, 3) 238(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 t x = t + 1, so t = x Substitute your new equation into the y equation: y = t 2 + 2t y = (x – 1) 2 + 2(x – 1)now simplify y = x 2 – 2x x – 2 y = x 2 - 1

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

 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 2 nd window: have TblStart = 0, indpnt: auto, depend: auto  Hit graph  Hit 2 nd 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 = t x = 5t – 7 and y = 17 – 3t 3. x = |t + 3| and y = 1/t

 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

 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

 Find the inverse of each function 1. f(x) = x/(x + 1) 2. f(x) = 3x – 6 3. f(x) = x - 3

 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, and g(f(x)) = x for every x in the domain of f

 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.

 p. 126 #5-7 odd, odd, odd, all