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Another natural way to define relations is to define both
elements of the ordered pair (x, y), in terms of another
variable t, called a parameter
Parametric equations: equations in the form
x = f(t) and y = g(t) for all t in the interval I. The
variable t is the parameter and I .is the parameter
interval
The set of all ordered pairs (x, y) is defined by
the equations
x = t + 1 and y = t2 + 2t
a. Find the points determined by t =-2, -1, 0,
1, and 2

t
x=t+1
y = t2 + 2 t
(x, y)
-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)
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b. find an algebraic relationship between x
and y. (can be called “eliminating the
parameter”). Is y a function of x?
1. Solve the x equation for t
x = t + 1, so t = x -1
2. Substitute your new equation into the y equation:
y = t2 + 2t
y = (x – 1)2 + 2(x – 1) now simplify
y = x2 – 2x + 1 + 2x – 2
y = x2 - 1
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C. graph the relation in the (x, y) plane
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Mode: arrow down 3, change from FUNC to
PAR
Hit y =
Enter your x and y equations
Go to window: tmin: -4, tmax: 2, tstep:.1,
xmin: -5, xmax: 5, ymin:-5, ymax:5
Go to 2nd window: have TblStart = 0, indpnt:
auto, depend: auto
Hit graph
Hit 2nd graph to get your table of values
find a) find the points determined by t = -3,
-2, -1, 0, 1, 2, 3
b) Find the direct algebraic relationship (rewrite
the equation in terms of t)
c) Graph the relationship (this can be done
either by hand or on the calculator)
1. x = 3t and y = t2 + 5
2. x = 5t – 7 and y = 17 – 3t
3. x = |t + 3| and y = 1/t
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The ordered pair (a, b) is in a relation if and
only if the ordered pair (b, a) is in the inverse
relation
Inverse functions: if f is a one-to-one
function with domain D and range R, then the
inverse function of f, denoted f-1, is the
function with domain R and range D defined
by : f-1 (b) = a if and only if f (a) = b
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Change f(x) to y
Switch your x and y
Solve for y
Rewrite as f-1(x)
Determine if f-1(x) is a function
Find the inverse of each function
1. f(x) = x/(x + 1)
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2.
f(x) = 3x – 6
3.
f(x) = x - 3
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The points (a, b) and (b, a) in the coordinate
plane are symmetric with respect to the line y
= x. The points (a, b) and (b, a) are
reflections of each other across the line y = x.
Inverse Composition Rule: a function is oneto-one with inverse function g if and only if:
f(g(x)) = x for every x in the domain of g, and
g(f(x)) = x for every x in the domain of f
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Algebraically: use the Inverse Composition
Rule, find both f(g(x)) and g(f(x)) and if the
answers are the same, the functions are
inverses
Graphically: in parametric mode, graph and
compare the graphs of the 2 sets of
parametric equations.
 p.
126
#5-7 odd, 13-21 odd, 27-31
odd, 34- 36 all