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

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

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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. A function is ____________ a relation. A relation is ____________ a function. always sometimes

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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 relation is not a function the same x-value (2), the relation is not a function!!!

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

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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???)

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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!!!

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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!

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Now on to parametric equations…

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What are they??? 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.

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First Example: Defining a function parametrically Consider the set of all ordered pairs (x, y) defined by the equations where t is any real number. x = t + 1 y = t + 2t 2 1. Find the points determined by t = –3, –2, –1, 0, 1, 2, and 3. txy(x, y) –3–23(–2, 3) –2–10(–1, 0) –10–1(0, –1) 010(1, 0) 123(2, 3) 238(3, 8) 3415(4, 15)

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

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

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

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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)

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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

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