Chapter 1 Functions & Graphs Mr. J. Focht PreCalculus OHHS

1.5 Parametric Relations & Inverses Defined Relations Parametrically Inverse Relations and Inverse Functions

Relations Defined Parametrically Let’s look at all ordered pairs (x, y ) where x = t + 1 and y = t 2 + 2t Let’s look at all ordered pairs (x, y ) where x = t + 1 and y = t 2 + 2t When x and y are defined by a third variable, they are defined parametrically. t is called a parameter. Parametric relations are most commonly used to track positions (x, y) at certain times t.

Find 6 points in the relation Let’s look at all ordered pairs (x, y ) where x = t + 1 and y = t 2 + 2t tx= t +1y = t 2 +2t(x, y) (-2,3) -20(-1,0) 0 (0,-1) 010(1,0) 123(2,3) 238(3,8)

Find an algebraic relation between x and y This is also known as “eliminating the parameter”. x = t + 1 and y = t 2 + 2t t = x – 1 y = (x-1) 2 + 2(x-1) y= x 2 – 2x x – 2 y = x 2 -1

Class Work P. 135, #5

Using a Graphing Calculator in Parametric Mode x = t + 1 and y = t 2 + 2t Change your calculator from function mode to parametric mode.

Graphing x = t + 1 and y = t 2 + 2t Enter the two parametric equations

Graphing Change the range of the graph.

Graph

View the Table View the table to find a list of (x, y)’s.

Class Work P. 135, # 7

Inverse Relations An inverse relation can be found by reversing the order of the pairings.

Vertical Line Test A relation is a function if it can pass the vertical line test.

Horizontal Line Test The inverse of a relation is a function if it can pass the horizontal line test. The inverse of this relation is not a function.

Class Work P. 135, #9

Vocabulary If a relation can pass both the vertical and horizontal line tests, both the relation and its inverse are functions. The relation is called a one-to-one function.

Inverse Notation The inverse of a function f(x) is f -1 (x) Be careful. The -1 is not a power. It is part of the name and does not indicate any action. It means : if f(a) = b, then f -1 (b) = a

Finding the Inverse Algebraically If f(x) = 2x – 4, find f -1 (x). First rewrite the function as an equation. y = 2x – 4 Reverse the x and y. x = 2y – 4 Solve for y y = ½x + 2 f -1 (x) = ½x + 2

Class Work P. 135, #15

The Inverse Reflection Rule The graph of the inverse of a function can be found by reversing or inverting the x’s and y’s of the relation. For every (x,y) in the relation, graph (y,x). This is a reflection over the line y = x.

Graph the Inverse. Invert each point. (-2, 4) (4,0) (2,-1) (-1, 2) (4, -2) (0, 4) (-2, 4) (-1, 2) (0, 4)(4, 0) (2,-1) (4,-2) Graph. We don’t need to know the name of the relation.

Class Work P. 135, #23

The Inverse Composition Rule g(x) is the inverse function of f(x) if f(g(x)) = x and g(f(x)) = x

Example Verify that f(x) and g(x) are inverses of each other. Testing: f (g(x)) g(f(x))

Class Work P. 136 #27

Homework P #4, 6, 12, 14, 18, 25, 28, 39-44, 47