1. 4b relations, implicitly defined functions, and parametric equations
TRANSCRIPT
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