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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 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.

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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 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: Grapher?!?! This is an example of a relation that defines two separate functions implicitly. (the functions are “hidden” within the relation…)

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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??? 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.

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**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 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) (4, 15)

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**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 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!!!

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**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 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!!!

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**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

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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 odd

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