1. 4b Relations, Implicitly
Defined Functions, and
Parametric Equations
Consider this problem:
x y 4
2
2
Does this equation
describe a function???
No way, Jose!!!
But, it does describe
a mathematical relation…
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.
always a relation.
A function is ____________
sometimes a function.
A relation is ____________
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?
2
2
x y y 5
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:
2
2
x y 4
y  4 x
2
y1  4  x
2
2
y   4 x
y2   4  x
2
2
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.
2x  y  5
2
2
y1  2 x  5
2
y2   2 x  5
2
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.
x  4y  8
2
2
2
x
y1  2 
4
2
This is an ellipse!!!
x
y2   2 
4
More Examples
Find two functions defined implicitly by the given relation. Graph
the implicit functions, and describe the graph of the relation.
2
2
x  2 xy  y  1
The terms on the left are a perfect square trinomial!!!
Factor:
2
 x  y
 1  x  y  1
x  y  1 x  y  1
y1   x  1 y2   x  1
This is a pair of parallel lines!
Now on to parametric
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 a
parametric curve. The equations are parametric
equations for the curve, the variable t is a parameter,
and I is the parameter interval.
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
–3
–2
–1
0
1
2
3
–2
–1
0
1
2
3
4
3
0
–1
0
3
8
15
(x, y)
(–2, 3)
(–1, 0)
(0, –1)
(1, 0)
(2, 3)
(3, 8)
(4, 15)
First Example: Defining a function parametrically
Consider the set of all ordered pairs (x, y) defined by the equations
x=t+1
2
y = t + 2t
where t is any real number.
2. Find an algebraic relationship between x and y. Is y a
function of x?
t  x 1
 x 1
2
y  t 2  2t
 This is a function!!!
First Example: Defining a function parametrically
Consider the set of all ordered pairs (x, y) defined by the equations
x=t+1
2
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!!!
More Practice: Using the Graphulator?!?!
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.
3. Is y a function of x?
NO!!!
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.
x  t 1 y  t  2t
2
(–2, 15), (–1, 8), (0, 3), (1, 0), (2, –1), (3, 0), (4, 3)
y  x  4 x  3 (this is a function)
2
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.
x t
y  2t  5
Not defined for t = –3, –2, or –1, (0, –5), (1, –3), ( 2, –1), ( 3, 1)
y  2x2  5
Homework: p. 128 25-37 odd
(this is a function)