1. 4b Relations, Implicitly Defined Functions, and Parametric Equations Homework: p. 128 25-37 odd Consider this problem: 2 2 x y 4 Does this equation describe a function??? No way, Jose!!! But, it does describe a mathematical relation Definition: Relation
In Math-Land, a relation is the general term for a set of ordered pairs (x, y). Fill in the blank with always, sometimes, or never. always a relation. A function is ____________ A relation issometimes ____________ a function. Verifying Pairs in a Relation Determine which of the ordered pairs (2, 5), (1, 3) and (2, 1) are in the relation defined below. Is the relation a function? 2 2 x y y 5 he points (2, 5) and (2, 1) are in the relation, but (1, 3) is no
Since the relation gives two different y-values (5 and 1) to the same x-value (2), the relation is not a function!!! function Revisiting the first problem his relation is not a function itself, but it can be split into two quations that do define functions: 2 2 x y 4 ?! ! r? e h p a
r G 1 2 y 4 x y 4 x 2 2 y 4 x y2 4 x 2
2 This is an example of a relation that defines two separate unctions implicitly. (the functions are hidden within the elation) More Examples nd two functions defined implicitly by the given relation. Grap e implicit functions, and describe the graph of the relation. 2 2 2 x y 5 2 y1 2 x 5
2 y2 2 x 5 This is a hyperbola!!! (recall the reciprocal function???) More Examples nd two functions defined implicitly by the given relation. Grap e implicit functions, and describe the graph of the relation. 2 2 x 4 y 8 2 x y1 2 4
2 x This is an ellipse!!! y 2 2 4 More Examples nd two functions defined implicitly by the given relation. Gra e implicit functions, and describe the graph of the relation. 2 2 x 2 xy y 1 The terms on the left are a perfect square trinomial!! Factor: 2
1 x y 1 x y 1 x y 1 y1 x 1 y2 x 1 x y This is a pair of parallel lines! Now on to parametric equations What are they??? is often useful to define both elements of a relation (x and y n terms of another variable (often t ), called a parameter he graph of the ordered pairs (x, y ) where
x = f (t ), y = g (t ) e functions defined on an interval I of t -values is a arametric curve. The equations are parametric quations for the curve, the variable t is a paramete nd I is the parameter interval. irst Example: Defining a function parametrica Consider the set of all ordered pairs (x, y) defined by the equations 2 where t is any real number. x=t+1 y=t + 1. Find 2tthe points determined by t = 3, 2, 1, 0, 1, 2, and 3 t 3
2 1 0 1 2 3 x 2 1 0 1 2 3 4 y 3 0 1
0 3 8 15 (x, y) (2, 3) (1, 0) (0, 1) (1, 0) (2, 3) (3, 8) (4, 15) irst Example: Defining a function parametrica Consider the set of all ordered pairs (x, y) defined by the equations x=t+1
2 y = t + 2t where t is any real number. 2. Find an algebraic relationship between x and y. Is y a function of x? !! ! te 2 u y t 2t it t s b Su
t x 1 2 x 1 This is a function!!! rst Example: Defining a function parametrica Consider the set of all ordered pairs (x, y) defined by the equations x=t+1 2 y = t + 2t where t is any real number. 3. Graph the relation in the (x, y) plane. We can plot our original points, or just graph the function we found in step 2!!! More Practice: Using the Graphulator?!?!
Consider the set of all ordered pairs (x, y) defined by the equations 2 where t is any real number. x = t + 2t y=t+1 Use a calculator to find the points determined by t = 3, 2, 1, 0, 1, 2, and 3. Use a calculator to graph the relation in the (x, y) plane. 3. Is y a function of x? NO!!! Find an algebraic relationship between x and y. 2
x=y 1 Guided Practice: For the given parametric equations, find he points determined by the t-interval 3 to 3, find an algebraic relationship between x and y, and graph the relatio 2 x t 1 y t 2t (2, 15), (1, 8), (0, 3), (1, 0), (2, 1), (3, 0), (4, 3) 2 y x 4 x 3(this is a function) Guided Practice: For the given parametric equations, find he points determined by the t-interval 3 to 3, find an algebraic relationship between x and y, and graph the relatio
x t y 2t 5 ot defined for t = 3, 2, or 1, (0, 5), (1, 3), ( 2, 1), ( 3, 1 y 2 x 2 5 (this is a function)