1. 4b Relations, 1. 4b Relations, Implicitly Defined Implicitly Defined

Functions, and Functions, and Parametric EquationsParametric Equations

Consider this problem:

2 2 4x y Does this equation

describe a function???

No way, Jose!!!No way, Jose!!!

But, it does describea mathematical relation…

Definition: Relation

In Math-Land, a relation is the general term for aset of ordered pairs (x, y).

Fill in the blank with always, sometimes, or never.

A function is ____________ a relation.

A relation is ____________ a function.

alwaysalways

sometimessometimes

Verifying Pairs in a RelationDetermine 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 5x y y

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 functionthe relation is not a function!!!

Revisiting the “Do Now”…

2 2 4x y

This relation is not a function itself, but it can be split into twoequations that do define functions:

This is an example of a relation that defines two separatefunctions implicitly. (the functions are “hidden” within therelation…)

24y x 2 24y x

Grapher?!

?!

Grapher?!

?!

21 4y x 2

2 4y x

More Examples

2 22 5x y

Find two functions defined implicitly by the given relation. Graphthe implicit functions, and describe the graph of the relation.

22 2 5y x 2

1 2 5y x This is a hyperbola!!! (recall the reciprocal function???)

More Examples

2 24 8x y

Find two functions defined implicitly by the given relation. Graphthe implicit functions, and describe the graph of the relation.

2

2 24

xy

2

1 24

xy This is an ellipse!!!

More ExamplesFind two functions defined implicitly by the given relation. Graphthe implicit functions, and describe the graph of the relation.

2 22 1x xy y

The terms on the left are a perfect square trinomial!!!Factor:

21x y 1x y 1x y 1x y

1 1y x 2 1y x This is a pair of parallel lines!

Now on to parametric Now on to parametric equations…equations…

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 aparametric curve. The equations are parametricequations for the curve, the variable t is a parameter,and I is the parameter interval.

First Example: Defining a function parametricallyConsider 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.

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)

First Example: Defining a function parametricallyConsider 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?

Substitu

te!!!

Substitu

te!!!

1t x 2 2y t t

2 1x This is a function!!!This is a function!!!

First Example: Defining a function parametricallyConsider 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!!!

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!!!NO!!!

xx = = y y – 1 – 122

Guided Practice: For the given parametric equations, findthe points determined by the t-interval –3 to 3, find analgebraic relationship between x and y, and graph the relation.

2 2y t t

2 4 3y x x

1x t

(–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, findthe points determined by the t-interval –3 to 3, find analgebraic relationship between x and y, and graph the relation.

2 5y t

22 5y x

x tNot 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