first&name& &&&&&&&last&name& …...a function is any...

First Name _________________________ Last Name _________________________________ Period ___ Algebra 2/Trig Unit 2: Functions as the Cornerstones of Algebra II Classwork Packet for Lessons #1 – 7

Pages # Lesson Reflection (circle one) Notes 1 Introduction to Functions o Mastery

o Proficiency o Developing o Beginning

2 Function Notation o Mastery o Proficiency o Developing o Beginning

3 Function Composition o Mastery o Proficiency o Developing o Beginning

4 The Domain and Range of a Function o Mastery o Proficiency o Developing o Beginning

5 One to One Functions o Mastery o Proficiency o Developing o Beginning

6 Inverse Functions o Mastery o Proficiency o Developing o Beginning

7 Key Features of Functions o Mastery o Proficiency o Developing o Beginning

Lesson Reflection Key:

o Mastery: I can do this on my own and clearly teach someone else what I did. o Proficiency: I can do this on my own without any help. o Developing: I understood parts of the problem, but I need some more help or practice. o Emerging: I’m just starting to learn this and I don’t understand yet.

Reflection Questions:

1) What went well for you this unit, and why? Please explain.

2) What do you still need to practice or improve on?

3) Please use the space below to make any questions you still have, or comments you want us to read.

Unit 2: Functions as the Cornerstones of Algebra II Vocabulary List

Word Definition Example

Function

Independent Variable

Dependent Variable

Vertical Line Test

Function Notation

(write and label)

Composition of Functions

Domain (include a synonym for this

term)

Range (include a synonym for this

term)

One-‐to-‐one Function

Horizontal Line Test

Inverse Function Notation

Existence of Inverse Functions

Alternate Names for x-‐intercepts (List them)

Absolute Maximum of a Function

Absolute Minimum of a Function

First Name _________________________ Last Name _________________________________ Period ___ Date________________ Common Core Algebra 2 Lesson #2-‐1: Introduction to Functions 8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Aim: I can create an equation, table, and graph, given a scenario. I can determine whether a relationship is a function or not using the vertical line test.

WARM-‐UP

An internet music service offers a plan whereby users pay a flat monthly fee of $5 and can then download songs for 10 cents each. Use the vocabulary definitions below to try to determine the following… What are the independent and dependent variables in this scenario? Independent: Dependent:

VOCABULARY

A function is any “rule” that assigns exactly one output value (y-value) for each input value (x-value). These rules can be expressed in different ways, the most common being equations, graphs, and tables of values. We call the input variable independent and output variable dependent.

LESSON NOTES

Exercise #1: (a) Fill in the table below

(b) Let the number of downloads be represented by the variable x, and the amount charged

in dollars be represented by the variable y. Write an equation that models y as a function of x.

(c) Based on the equation you found in part (b), produce a graph of this function for all values of x on the interval. Use a calculator TABLE to generate additional coordinate pairs to the ones you found in part (b).

QUICK CHECK / GUIDED PRACTICE NOTES

Exercise #2: Evin is walking home from the museum. She starts 38 blocks from home and walks 2 blocks each minute. Evin’s distance from home is a function of the number of minutes she has been walking.

(a) Which variable is independent and which variable is dependent in this scenario?

(b) Fill in the table below for a variety of time values.

Time, t, in minutes 0 1 5 10

Distance from home, D, in blocks

(c) Determine an equation relating the distance, D, that Evin is from home as a function of the number of minutes, t, that she has been walking.

(d) Determine the number of minutes, t, that it takes for Evin to reach home.

VOCABULARY

The Vertical Line Test states that if you can draw a vertical line anywhere on a graph so that it hits the graph more than once, then the graph is NOT a function.

LESSON NOTES

Exercise #2: One of the following graphs shows a relationship where y is a function of x and one does not.

(a) draw the vertical line whose equation is x=3 on both graphs.

(b) Give all output values for each graph at an input of 3. Relationship A: Relationship B:

(c) Explain which of these relationships is a function and why.


Determine for each of the following graphed relationships whether y is a function of x using the Vertical Line Test.

(a) (b)

CLASSWORK NOTES

1 The graph of the function 𝑦 = 𝑥! − 4𝑥 + 1 is shown below

. (a) State this function’s y-‐intercept.

(b) Between what two consecutive integers does the larger x-intercept lie?

(c) Draw the horizontal line 𝑦 = −2 on this graph.

(d) Using these two graphs, find all values of x that solve the equation below: 𝑥! − 4𝑥 + 1 = −2

(e) Verify that these values of x are solutions by using STORE on your graphing calculator.

2 Determine for each of the following graphed relationships whether y is a function of x using the Vertical Line Test.

(a) (b) (b)

(c) (d)

3 What are the outputs for an input of given functions defined by the following formulas: (a) 𝑦 = 3𝑥 − 4 (b) 𝑦 = 50 − 2𝑥! (c)𝑦 = 2!

EXIT TICKET NOTES

In one of the following tables, the variable y is a function of the variable x. Explain which relationship is a function and why the other is not.

First Name _________________________ Last Name _________________________________ Period ___ Date________________ Common Core Algebra 2 Lesson #2-‐2: Function Notation

8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Aim: I can evaluate and analyze the outputs of a function given its equation, graph, or table.

WARM-‐UP

Evaluate each of the following given the function definitions and input values. (a) (b) (c)

VOCABULARY

LESSON NOTES

Exercise #1: Boiling water at 212 degrees Fahrenheit is left in a room that is at 65 degrees Fahrenheit and begins to cool. Temperature readings are taken each hour and are given in the table below. In this scenario, the temperature, T, is a function of the number of hours, h.

(a) Evaluate �(2) and �(6). (b) For what value of h is �(�) = 76? (c) Between what two consecutive hours will �(�) = 100?


Exercise #2: The function � = �(�) is defined by the graph shown below. Answer the following questions based on this graph.

(a) Evaluate �(−1),�(1),and �(5).

(b) Evaluate �(0). What special feature on a graph does �(0) always correspond to?

(c) What values of x solve the equation �(�) = 0? What special features on a graph does

the set of x-values that solve �(�) = 0 correspond to?

(d) Between what two consecutive integers does the largest solution to �(�) = 3 lie?

CLASSWORK NOTES

1 Without using your calculator, evaluate each of the following given the function definitions and input values.

2

3 For a function � = �(�) it is known that �(−2) = 7. Which of the following points must lie on the graph of �(�)?

5 Physics students drop a ball from the top of a 50 foot high building and model its height as a function of time with the equation �(�) = 50− 16�2. Using TABLES on your calculator, determine, to the nearest tenth of a second, when the ball hits the ground. Provide tabular outputs to support your answer.

EXIT TICKET NOTES

First Name _________________________ Last Name _________________________________ Period ___ Date________________ Common Core Algebra 2 Lesson #2-‐3: Function Composition HSF.BF.A.1: Write a function that describes a relationship between two quantities; HSF.BF.A.1.C: Compose functions. Aim: I can perform a composition of functions given graphs, equations, or scenarios.

WARM-‐UP

VOCABULARY

Composition of Functions: applying one function to the results of another. In the warm up we see that the output of an area function is used as the input to a cost function. This idea can be generalized to generic functions, f and g as shown in the diagram below:

This can be written in two ways:

In this situation, you apply f BEFORE g

LESSON NOTES


1.

2.

LESSON NOTES


CLASSWORK NOTES

1

2

3

EXIT TICKET NOTES

EXTENSION

First Name _________________________ Last Name _________________________________ Period ___ Date________________ Common Core Algebra 2 Lesson #2-‐4: The Domain and Range of a Function HSF.IF.A.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. Aim: I can determine the domain and range of a function given an equation, graph, or set of ordered pairs.

WARM-‐UP

VOCABULARY

LESSON NOTES


CLASSWORK NOTES

1

2

3

4

EXIT TICKET NOTES

First Name _________________________ Last Name _________________________________ Period ___ Date________________ Common Core Algebra 2 Lesson #2-‐5: One-‐to-‐one Functions

HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Aim: I can identify a function as one-‐to-‐one by examining inputs and outputs and by performing the horizontal line test.

WARM-‐UP

(HINT: Mapping means, writing the number in the range and then drawing an arrow from the domain to the range.)

VOCABULARY

LESSON NOTES


VOCABULARY

LESSON NOTES


CLASSWORK NOTES

1

2

3

4

EXIT TICKET NOTES

APPLICATIONS + EXTENSION

First Name _________________________ Last Name _________________________________ Period ___ Date________________ Common Core Algebra 2 Lesson #2-‐6: Inverse Functions HSF.BF.B.4: Find inverse functions; HSF.BF.B.4.B: (+) Verify by composition that one function is the inverse of another; HSF.BF.B.4.C: (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. Aim: I can find and verify inverse functions and read values of an inverse function from a graph or table.

WARM-‐UP

LESSON NOTES

VOCABULARY

LESSON


VOCABULARY

CLASSWORK NOTES

1

2

3

4

5

EXIT TICKET NOTES

APPLICATION/HOMEWORK NOTES

First Name _________________________ Last Name _________________________________ Period ___ Date________________ Common Core Algebra 2 Lesson #2-‐7: Key Features of Functions HSF.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Aim: I can understand and implement proper functon terminology.

WARM-‐UP / LESSON


LESSON NOTES

CLASSWORK NOTES

1

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3

EXIT TICKET NOTES

first&name& &&&&&&&last&name& …...a function is any...

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