1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Start-Up Create an equation for each translated parent graph

2 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Sections 1.5 & 1.7 Day 6/7 Objective: Student will graph a parametric function. Students will use the Vertical Line and Horizontal Line Test to determine the nature of the relation. Students will determine and verify an inverse function both algebraically and graphically. Student will create a model function from a given formula, graph, or verbal description, including conversion between units of measure. Essential Questions: What is a parameter and why is it relevant in the context of real world problem? How do we determine if a given function will have an inverse that is also a function? How do you determine the best model for the given function? Home Learning: p. 126 #9-12, 16, 17, 19, 28, 30 & 34 + Pg. 148 #13, 33, 43 ‐ 46

3 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 1.5 Parametric Relations and Inverses Demana, Waits, Foley, Kennedy

4 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. What you’ll learn about Relations Parametrically Inverse Relations and Inverse Functions … and why Some functions and graphs can best be defined parametrically, while some others can be best understood as inverses of functions we already know.

5 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Relations Defined Parametrically Another natural way to define functions or, more generally, relations, is to define both elements of the ordered pair (x, y) in terms of another variable t, called a parameter.

6 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Defining a Function Parametrically

7 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution tx = t – 1y = t 2 + 2(x, y) –3–411(–4, 11) –2–36(–3, 6) –1–23(–2, 3) 0–12(–1, 2) 103(0, 3) 216(1, 6) 3211(2, 11)

8 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Defining a Function Parametrically

9 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

10 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn…p. 126 #1-4 Can you graph #1? Can you eliminate the parameter & re-write in terms of x and y?

11 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Inverse Relation The ordered pair (a,b) is in a relation if and only if the pair (b,a) is in the inverse relation.

12 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Horizontal Line Test The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.

13 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Inverse Function

14 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Is it a function? Is the inverse a function?

15 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Finding an Inverse Function Algebraically

16 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

17 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. The Inverse Reflection Principle 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.

18 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Finding an Inverse Function Graphically The graph of a function y = f (x) is shown. Sketch a graph of the function y = f –1 (x). Is f a one-to-one function?

19 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution All we need to do is to find the reflection of the given graph across the line y = x.

20 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution All we need to do is to find the reflection of the given graph across the line y = x.

21 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn…determine #14, 15 & 20

22 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. The Inverse Composition Rule

23 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Verifying Inverse Functions

24 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

25 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn…Confirm Algebraically!

26 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. How to Find an Inverse Function Algebraically

27 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Let’s try and apply!