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1.5 Parametric Relations and Inverses

1.5 Parametric Relations and Inverses . What is a parameter. 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

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1.5 Parametric Relations and Inverses

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  1. 1.5 Parametric Relations and Inverses

  2. What is a parameter • 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

  3. Define a functions parametrically • The set of all ordered pairs (x, y) is defined by the equations x = t + 1 and y = t2 + 2t • Find the points determined by t =-2, -1, 0, 1, and 2

  4. Cont’d • 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 -1 2. Substitute your new equation into the y equation: y = t2 + 2t y = (x – 1)2 + 2(x – 1) now simplify y = x2 – 2x + 1 + 2x – 2 y = x2 - 1

  5. Cont’d • C. graph the relation in the (x, y) plane

  6. 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

  7. Examples • 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) • x = 3t and y = t2 + 5 • x = 5t – 7 and y = 17 – 3t • x = |t + 3| and y = 1/t

  8. Inverse functions • 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

  9. Finding an inverse • 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

  10. Examples • Find the inverse of each function • f(x) = x/(x + 1) • f(x) = 3x – 6 • f(x) = x - 3

  11. Inverse Reflection Principle • 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

  12. Verifying inverse functions • 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.

  13. Homework • p. 126 #5-6 all, 14-22 even, 28-32 even, 34- 36 all

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