Precalculus Lesson 2.1.1 PDF

Summary

This document is a lesson on precalculus, focusing on characteristics of functions. It includes examples and questions related to increasing functions, concavity, and Riley's fictional drive, represented by a graph. It is designed as part of a math course.

Full Transcript

10/8/23, 12:21 PM CPM eBooks - Precalculus Third Edition Lesson 2.1.1 2.1.1 What is your best description? Characteristics of Functions Imagine how you might describe a basic idea, like addition, to someone who has never heard of it. Sometimes, the to describe succinctly and precisely. Mathematic...

10/8/23, 12:21 PM CPM eBooks - Precalculus Third Edition Lesson 2.1.1 2.1.1 What is your best description? Characteristics of Functions Imagine how you might describe a basic idea, like addition, to someone who has never heard of it. Sometimes, the to describe succinctly and precisely. Mathematicians use notation to describe terms, concepts, and procedures in v lesson, you will see familiar concepts written in mathematical notation. At first, the formal definitions may seem c clearer when you understand the concepts. Mastery takes time. Remember, everyone can learn this! 2-1. One day Riley decides to take a scenic drive. The graph below represents Riley’s trip. Obtain a Lesson 2.1.1 Resource Page. With a partner, write a story that matches the graph. Be sure to i 2-2. Consider Riley’s distance from home, using the graph from problem 2-1 as you answer the following q the nearest . a. When is Riley’s distance from home increasing? How is this represented on the graph? b. When is Riley’s distance from home decreasing? How is this represented on the graph? c. Is Riley’s distance from home ever constant? If yes, explain how you know. If no, describe what distance was constant over a period of time. 2-3. As mentioned in the introduction, mathematical notation can sometimes make concepts seem more com problems you described where the graph of a function was increasing. By definition, a function is incr if for all . This simply means that when you trace along a graph from left to r the function is increasing. a. Apply the definition of an increasing function to Riley’s drive. In Riley’s situation, let ? Is ? b. If you choose any two points for . https://ebooks.cpm.org/bookdb.php?title=pc3&name=2.1.1&type=lesson and between and , is for all 1/2 10/8/23, 12:21 PM CPM eBooks - Precalculus Third Edition Lesson 2.1.1 Another important property of functions is concavity. You may have heard the term “concave” before. studied optics, or maybe you wear contact lenses, which are concave, not flat. In mathematics, functions are described as being either “concave up” or “concave down” over given in function is concave up wherever you can trace a “cup” or a u-shaped curve that is right side up. Take a a. On the interval approximate answer. , the graph is concave up. What is another interval on which the b. On the interval , the graph is concave down. What is another interval on which t c. A point where the concavity of a graph changes is called a point of inflection. Identify the point d. Discuss with your team and come up with a definition for concavity. What does it mean for a fun down” over a given interval? Note that the formal definition of concavity is explored in a Calcul 2-6. WHAT ELSE CAN THE GRAPH TELL ME? Review the story you wrote and the graph of Riley’s drive from problem 2-1. After examining the grap 5, what other information can you add to the story? Work with your team to decide if you can answer t question if possible and explain what this looks like on the graph using the vocabulary you learned in t be used. If the question cannot be answered, explain why not. a. How fast was Riley driving when she was b. When was Riley’s speed minutes into her drive? mph? c. Was Riley ever speeding? d. When did Riley go from speeding up to slowing down? e. How far did Riley travel during her -hour drive? 2-7. Sketch a curve that is: a. increasing and concave down b. increasing and concave c. decreasing and concave down d. decreasing and concave https://ebooks.cpm.org/bookdb.php?title=pc3&name=2.1.1&type=lesson 2/2

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