1. 4b Relations, Implicitly Defined Functions, and Parametric Equations

Consider this problem: Does this equation describe a function??? No way, Jose!!! But, it does describe a mathematical relation…

Definition: Relation always sometimes 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 function is ____________ a relation. sometimes A relation is ____________ 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? The points (2, –5) and (2, 1) are in the relation, but (1, 3) is not. Since the relation gives two different y-values (–5 and 1) to the same x-value (2), the relation is not a function!!!

Revisiting the “Do Now”… This relation is not a function itself, but it can be split into two equations that do define functions: Grapher?!?! This is an example of a relation that defines two separate functions implicitly. (the functions are “hidden” within the relation…)

More Examples Find two functions defined implicitly by the given relation. Graph the implicit functions, and describe the graph of the relation. This is a hyperbola!!! (recall the reciprocal function???)

More Examples Find two functions defined implicitly by the given relation. Graph the implicit functions, and describe the graph of the relation. This is an ellipse!!!

More Examples Find two functions defined implicitly by the given relation. Graph the implicit functions, and describe the graph of the relation. The terms on the left are a perfect square trinomial!!! Factor: This is a pair of parallel lines!

Now on to parametric equations…

What are they??? x = f (t ), y = g (t ) It is often useful to define both elements of a relation (x and y) in terms of another variable (often t ), called a parameter… The graph of the ordered pairs (x, y ) where x = f (t ), y = g (t ) are functions defined on an interval I of t -values is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parameter interval.

x = t + 1 y = t + 2t First Example: Defining a function parametrically Consider the set of all ordered pairs (x, y) defined by the equations 2 x = t + 1 y = t + 2t where t is any real number. 1. Find the points determined by t = –3, –2, –1, 0, 1, 2, and 3. t x y (x, y) –3 –2 3 (–2, 3) –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) 3 4 15 (4, 15)

x = t + 1 y = t + 2t First Example: Defining a function parametrically Consider the set of all ordered pairs (x, y) defined by the equations 2 x = t + 1 y = t + 2t where t is any real number. 2. Find an algebraic relationship between x and y. Is y a function of x? Substitute!!!  This is a function!!!

x = t + 1 y = t + 2t First Example: Defining a function parametrically Consider the set of all ordered pairs (x, y) defined by the equations 2 x = t + 1 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!!!

x = t + 2t y = t + 1 More Practice: Using the Graphulator?!?! NO!!! Consider the set of all ordered pairs (x, y) defined by the equations 2 x = t + 2t y = t + 1 where t is any real number. 1. Use a calculator to find the points determined by t = –3, –2, –1, 0, 1, 2, and 3. 2. Use a calculator to graph the relation in the (x, y) plane. NO!!! 3. Is y a function of x? 4. Find an algebraic relationship between x and y. 2 x = y – 1

Guided Practice: For the given parametric equations, find the points determined by the t-interval –3 to 3, find an algebraic relationship between x and y, and graph the relation. (–2, 15), (–1, 8), (0, 3), (1, 0), (2, –1), (3, 0), (4, 3) (this is a function)

Guided Practice: For the given parametric equations, find the points determined by the t-interval –3 to 3, find an algebraic relationship between x and y, and graph the relation. Not defined for t = –3, –2, or –1, (0, –5), (1, –3), ( 2, –1), ( 3, 1) (this is a function) Homework: p. 128 25-37 odd