Maintaining Mathematical Proficiency

Transcription

1 Name Date Chapter 3 Maintaining Mathematical Proficienc Plot the point in a coordinate plane. Describe the location of the point. 1. A( 3, 1). B (, ) 3. C ( 1, 0). D ( 5, ) 5. Plot the point that is on the -ais and 5 units down from the origin. Evaluate the epression for the given value of.. + 1; = ; = ; = ; = The length of a side of a square is represented b ( ) length of the side of the square when =? 3 feet. What is the 58 Algebra 1 Copright Big Ideas Learning, LLC

2 Name Date 3.1 Functions For use with Eploration 3.1 Essential Question What is a function? 1 EXPLORATION: Describing a Function Work with a partner. Functions can be described in man was. b an equation b an input-output table using words b a graph as a set of ordered pairs a. Eplain wh the graph shown represents a function b. Describe the function in two other was. EXPLORATION: Identifing Functions Work with a partner. Determine whether each relation represents a function. Eplain our reasoning. a. Input, Output, b. Input, Output, Copright Big Ideas Learning, LLC Algebra 1 59

3 Name Date 3.1 Functions (continued) EXPLORATION: Identifing Functions (continued) c. Input, Output, d e. (, 5 ), ( 1, 8 ), ( 0, ), ( 1, ), (, 7) f. ( ) ( ) ( ) ( ) ( ), 0, 1, 0, 1, 1, 0, 1, 1,, (, ) g. Each radio frequenc in a listening area has eactl one radio station. h. The same television station can be found on more than one channel. i. = j. = + 3 Communicate Your Answer 3. What is a function? Give eamples of relations, other than those in Eplorations 1 and, that (a) are functions and (b) are not functions. 0 Algebra 1 Copright Big Ideas Learning, LLC

4 Name Date 3.1 Notetaking with Vocabular For use after Lesson 3.1 In our own words, write the meaning of each vocabular term. relation function domain range independent variable dependent variable Copright Big Ideas Learning, LLC Algebra 1 1

5 Name Date 3.1 Notetaking with Vocabular (continued) Core Concepts Vertical Line Test Words A graph represent a function when no vertical line passes through more than one point on the graph. Eamples Function Not a function The Domain and Range of a Function The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values. input output Algebra 1 Copright Big Ideas Learning, LLC

6 Name Date 3.1 Notetaking with Vocabular (continued) Etra Practice In Eercises 1 and, determine whether the relation is a function. Eplain. 1.. ( 0, 3 ), ( 1, 1 ), (, 1 ), ( 3, 0 ) Input, 0 1 Output, 5 5 In Eercises 3 and, determine whether the graph represents a function. Eplain In Eercises 5 and, find the domain and range of the function represented b the graph The function = 1 represents the number of pages of tet a computer printer can print in minutes. a. Identif the independent and dependent variables. b. The domain is 1,, 3, and. What is the range? Copright Big Ideas Learning, LLC Algebra 1 3

7 Name Date 3. Linear Functions For use with Eploration 3. Essential Question How can ou determine whether a function is linear or nonlinear? 1 EXPLORATION: Finding Patterns for Similar Figures Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Complete each table for the sequence of similar figures. (In parts (a) and (b), use the rectangle shown.) Graph the data in each table. Decide whether each pattern is linear or nonlinear. Justif our conclusion. a. perimeters of similar rectangles b. areas of similar rectangles P A P 0 A Algebra 1 Copright Big Ideas Learning, LLC

8 Name Date 3. Linear Functions (continued) 1 EXPLORATION: Finding Patterns for Similar Figures (continued) c. circumferences of circles of radius r d. areas of circles of radius r r C r A C 0 A r 0 8 r Communicate Your Answer. How do ou know that the patterns ou found in Eploration 1 represent functions? 3. How can ou determine whether a function is linear or nonlinear?. Describe two real-life patterns: one that is linear and one that is nonlinear. Use patterns that are different from those described in Eploration 1. Copright Big Ideas Learning, LLC Algebra 1 5

9 Name Date 3. Notetaking with Vocabular For use after Lesson 3. In our own words, write the meaning of each vocabular term. linear equation in two variables linear function nonlinear function solution of a linear equation in two variables discrete domain continuous domain Algebra 1 Copright Big Ideas Learning, LLC

10 Name Date 3. Notetaking with Vocabular (continued) Core Concepts Representations of Functions Words An output is 3 more than the input. Equation = + 3 Input-Output Table Mapping Diagram Graph Input, Output, Input, Output, Discrete and Continuous Domains A discrete domain is a set of input values that consists of onl certain numbers in an interval. Eample: Integers from 1 to A continuous domain is a set of input values that consists of all numbers in an interval. Eample: All numbers from 1 to Copright Big Ideas Learning, LLC Algebra 1 7

11 Name Date 3. Notetaking with Vocabular (continued) Etra Practice In Eercises 1 and, determine whether the graph represents a linear or nonlinear function. Eplain. 1.. In Eercises 3 and, determine whether the table represents a linear or nonlinear function. Eplain In Eercises 5 and, determine whether the equation represents a linear or nonlinear function. Eplain. 5. = 3. = 3 3 In Eercises 7 and 8, find the domain of the function represented b the graph. Determine whether the domain is discrete or continuous. Eplain Algebra 1 Copright Big Ideas Learning, LLC

12 Name Date 3.3 Function Notation For use with Eploration 3.3 Essential Question How can ou use function notation to represent a function? 1 EXPLORATION: Matching Functions with Their Graphs Work with a partner. Match each function with its graph. a. f( ) = 3 b. g ( ) = + c. h ( ) = 1 d. j ( ) = 3 A. B. C. D. Copright Big Ideas Learning, LLC Algebra 1 9

13 Name Date 3.3 Function Notation (continued) EXPLORATION: Evaluating a Function Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Consider the function ( ) = + 3. f Locate the points (, f ( ) ) on the graph. Eplain how ou found each point. a. ( 1, f ( 1) ) 5 3 f() = + 3 b. ( 0, f ( 0) ) ( ) c. 1, f () 1 d. (, f ( ) ) Communicate Your Answer 3. How can ou use function notation to represent a function? How are standard notation and function notation similar? How are the different? Standard Notation Function Notation = + 5 f( ) = Algebra 1 Copright Big Ideas Learning, LLC

14 Name Date 3.3 Notetaking with Vocabular For use after Lesson 3.3 In our own words, write the meaning of each vocabular term. function notation Copright Big Ideas Learning, LLC Algebra 1 71

15 Name Date 3.3 Notetaking with Vocabular (continued) Etra Practice In Eercises 1, evaluate the function when =, 0, and. 1. f( ) = +. g( ) = 5 3. h ( ) = 7. s( ) = t ( ) = + 3. u ( ) = Let nt () be the number of DVDs ou have in our collection after t trips to the video store. Eplain the meaning of each statement. a. ( 0) 8 n = b. n () 3 = 1 c. n() 5 > n() 3 d. n( ) n( ) 7 = 10 In Eercises 8 11, find the value of so that the function has the given value. 8. b ( ) = 3+ 1; b ( ) = 0 9. r ( ) r ( ) = 3; = m ( ) = 3 ; m ( ) = 11. w 5 ( ) w ( ) 5 = 3; = 18 7 Algebra 1 Copright Big Ideas Learning, LLC

16 Name Date 3.3 Notetaking with Vocabular (continued) In Eercises 1 and 13, graph the linear function. 1. s 1 ( ) = 13. t ( ) = 1 0 s() t() 1. The function Bm ( ) 50m 150 = + represents the balance (in dollars) in our savings account after m months. The table shows the balance in our friend's savings account. Who has the better savings plan? Eplain. Month Balance $330 $10 $90 Copright Big Ideas Learning, LLC Algebra 1 73

17 Name Date 3. Graphing Linear Equations in Standard Form For use with Eploration 3. Essential Question How can ou describe the graph of the equation A + B = C? 1 EXPLORATION: Using a Table to Plot Points Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. You sold a total of $1 worth of tickets to a fundraiser. You lost track of how man of each tpe of ticket ou sold. Adult tickets are $ each. Child tickets are $ each. adult Number of Number of adult tickets + child child tickets = a. Let represent the number of adult tickets. Let represent the number of child tickets. Use the verbal model to write an equation that relates and. b. Complete the table to show the different combinations of tickets ou might have sold. c. Plot the points from the table. Describe the pattern formed b the points. 8 8 d. If ou remember how man adult tickets ou sold, can ou determine how man child tickets ou sold? Eplain our reasoning. 7 Algebra 1 Copright Big Ideas Learning, LLC

18 Name Date 3. Graphing Linear Equations in Standard Form (continued) EXPLORATION: Rewriting and Graphing an Equation Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. You sold a total of $8 worth of cheese. You forgot how man pounds of each tpe of cheese ou sold. Swiss cheese costs $8 per pound. Cheddar cheese costs $ per pound. pound Pounds of Pounds of Swiss + pound cheddar = a. Let represent the number of pounds of Swiss cheese. Let represent the number of pounds of cheddar cheese. Use the verbal model to write an equation that relates and. b. Solve the equation for. Then use a graphing calculator to graph the equation. Given the real-life contet of the problem, find the domain and range of the function. c. The -intercept of a graph is the -coordinate of a point where the graph crosses the -ais. The -intercept of a graph is the -coordinate of a point where the graph crosses the -ais. Use the graph to determine the - and -intercepts. d. How could ou use the equation ou found in part (a) to determine the - and -intercepts? Eplain our reasoning. e. Eplain the meaning of the intercepts in the contet of the problem. Communicate Your Answer 3. How can ou describe the graph of the equation A + B = C?. Write a real-life problem that is similar to those shown in Eplorations 1 and. Copright Big Ideas Learning, LLC Algebra 1 75

19 Name Date 3. Notetaking with Vocabular For use after Lesson 3. In our own words, write the meaning of each vocabular term. standard form -intercept -intercept Core Concepts Horizontal and Vertical Lines = b = a (0, b) (a, 0) The graph of = b is a horizontal line. The graph of = a is a vertical line. The line passes through the point ( 0, ). b The line passes through the point ( a ), 0. 7 Algebra 1 Copright Big Ideas Learning, LLC

20 Name Date 3. Notetaking with Vocabular (continued) Using Intercepts to Graph Equations The -intercept of a graph is the -coordinate of a point where the graph crosses the -ais. It occurs when = 0. (0, b) -intercept = b -intercept = a The -intercept of a graph is the -coordinate of a point where the graph crosses the -ais. It occurs when = 0. O (a, 0) To graph the linear equation A + B = C, find the intercepts and draw the line that passes through the two intercepts. To find the -intercept, let = 0 and solve for. To find the -intercept, let = 0 and solve for. Etra Practice In Eercises 1 and, graph the linear equation. 1. = 3. = Copright Big Ideas Learning, LLC Algebra 1 77

21 Name Date 3. Notetaking with Vocabular (continued) In Eercises 3 5, find the - and -intercepts of the graph of the linear equation = 1. = = 30 In Eercises and 7, use intercepts to graph the linear equation. Label the points corresponding to the intercepts = 7. + = 8. The school band is selling sweatshirts and baseball caps to raise $9000 to attend a band competition. Sweatshirts cost $5 each and baseball caps cost $10 each. The equation = 9000 models this situation, where is the number of sweatshirts sold and is the number of baseball caps sold. a. Find and interpret the intercepts. b. If 58 sweatshirts are sold, how man baseball caps are sold? c. Graph the equation. Find two more possible solutions in the contet of the problem. 78 Algebra 1 Copright Big Ideas Learning, LLC

22 Name Date 3.5 Graphing Linear Equations in Slope-Intercept Form For use with Eploration 3.5 Essential Question How can ou describe the graph of the equation = m + b? 1 EXPLORATION: Finding Slopes and -Intercepts Work with a partner. Find the slope and -intercept of each line. a. b. = + 3 = 1 c EXPLORATION: Writing a Conjecture Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Graph each equation. Then complete the table. Use the completed table to write a conjecture about the relationship between the graph of = m + b and the values of m and b. Equation Description of graph Slope of graph -Intercept a. = + 3 Line 3 b. = c. = + 1 d. = 3 3 a. b. Copright Big Ideas Learning, LLC Algebra 1 79

23 Name Date 3.5 Graphing Linear Equation in Slope-Intercept Form (continued) c EXPLORATION: Writing a Conjecture (continued) c. d. Communicate Your Answer 3. How can ou describe the graph of the equation = m + b? a. How does the value of m affect the graph of the equation? b. How does the value of b affect the graph of the equation? c. Check our answers to parts (a) and (b) b choosing one equation from Eploration and (1) varing onl m and () varing onl b. 80 Algebra 1 Copright Big Ideas Learning, LLC

24 Name Date 3.5 Notetaking with Vocabular For use after Lesson 3.5 In our own words, write the meaning of each vocabular term. slope rise run slope-intercept form constant function Core Concepts Slope The slope m of a nonvertical line passing through,, is the ratio of two points ( ) and ( ) 1 1 the rise (change in ) to the run (change in ). rise change in slope = m = = = run change in 1 1 (, ) ( 1, 1 ) rise = 1 run = 1 O When the line rises from left to right, the slope is positive. When the line falls from left to right, the slope is negative. Copright Big Ideas Learning, LLC Algebra 1 81

25 Name Date 3.5 Notetaking with Vocabular (continued) Slope Positive slope Negative slope Slope of 0 Undefined slope O O O O The line rises The line falls The line is horizontal. The line is vertical. from left to right. from left to right. Slope-Intercept Form Words A linear equation written in the form = m + b is in slope-intercept form. The slope of the line is m, and the -intercept of the line is b. (0, b) = m + b Algebra = m + b slope -intercept 8 Algebra 1 Copright Big Ideas Learning, LLC

26 Name Date 3.5 Notetaking with Vocabular (continued) Etra Practice In Eercise 1 3, describe the slope of the line. Then find the slope (3, ) (3, ) (, 3) (3, ) ( 1, ) (3, ) In Eercise and 5, the points represented b the table lie on a line. Find the slope of the line In Eercise 8, find the slope and the -intercept of the graph of the linear equation.. + = 7. = = 5 9. A linear function f models a relationship in which the dependent variable decreases units for ever 3 units the independent variable decreases. The value of the function at 0 is. Graph the function. Identif the slope, -intercept, and -intercept of the graph. Copright Big Ideas Learning, LLC Algebra 1 83

27 Name Date 3. Transformations of Graphs of Linear Functions For use with Eploration 3. Essential Question How does the graph of the linear function f( ) = compare to the graphs of g( ) = f( ) + c and h ( ) = fc ( )? 1 EXPLORATION: Comparing Graphs of Functions Work with a partner. The graph of f ( ) = is shown. Sketch the graph of each function, along with f, on the same set of coordinate aes. Use a graphing calculator to check our results. What can ou conclude? a. g ( ) = + b. g ( ) = + c. g ( ) = d. g ( ) = EXPLORATION: Comparing Graphs of Functions Work with a partner. Sketch the graph of each function, along with f ( ), = on the same set of coordinate aes. Use a graphing calculator to check our results. What can ou conclude? a. h ( ) = 1 b. h ( ) = c. h ( ) = 1 d. h ( ) = 8 Algebra 1 Copright Big Ideas Learning, LLC

28 Name Date 3. Transformations of Graphs of Linear Functions (continued) 3 EXPLORATION: Matching Functions with Their Graphs Work with a partner. Match each function with its graph. Use a graphing calculator to check our results. Then use the results of Eplorations 1 and to compare the graph of k to the graph of f ( ) =. a. k( ) = b. k( ) = + c. k ( ) = 1 + d. ( ) 1 k = A. B. C. D. 8 8 Communicate Your Answer. How does the graph of the linear function f ( ) g( ) = f( ) + cand h ( ) = f( c)? = compare to the graphs of Copright Big Ideas Learning, LLC Algebra 1 85

29 Name Date 3. Notetaking with Vocabular For use after Lesson 3. In our own words, write the meaning of each vocabular term. famil of functions parent function transformation translation reflection horizontal shrink horizontal stretch vertical stretch vertical shrink 8 Algebra 1 Copright Big Ideas Learning, LLC

30 Name Date 3. Notetaking with Vocabular (continued) Core Concepts A translation is a transformation that shifts a graph horizontall or verticall but does not change the size, shape, or orientation of the graph. Horizontal Translations Vertical Translations The graph of = f( h) is a horizontal The graph of ( ) translation of the graph of f( ) = f + k is a vertical =, where translation of the graph of = f( ), where h 0. k 0. = f( h), h < 0 = f() = f( h), h > 0 = f() + k, k > 0 = f() + k, k < 0 = f() Subtracting h from the inputs before evaluating Adding k to the outputs shifts the graph down the function shifts the graph left when h < 0 when k < 0 and up when k > 0. and right when h > 0. A reflection is a transformation that flips a graph over a line called the line of reflection. Reflections in the -ais Reflections in the -ais The graph of = f ( ) is a reflection in The graph of f( ) = is a reflection in the -ais of the graph of = f ( ). the -ais of the graph of = f( ). = f() = f( ) = f() = f() Multipling the outputs b 1 changes their signs. Multipling the inputs b 1 changes their signs. Copright Big Ideas Learning, LLC Algebra 1 87

31 Name Date 3. Notetaking with Vocabular (continued) Horizontal Stretches and Shrinks Vertical Stretches and Shrinks The graph of = f ( a) is a horizontal stretch The graph of a f( ) = is a vertical stretch or shrink b a factor of 1 a of the graph of or shrink b a factor of a of the graph of = f ( ), where a > 0 and a 1. = f ( ), where a > 0 and a 1. = f(a), a > 1 = f() = f(a), 0 < a < 1 The -intercept stas the same. = a f(), a > 1 = f() = a f(), 0 < a < 1 The -intercept stas the same. Transformations of Graphs The graph of = a f ( h) + k or the graph of ( ) obtained from the graph of = f ( ) b performing these steps. Step 1 Translate the graph of f( ) = horizontall h units. Step Use a to stretch or shrink the resulting graph from Step 1. Step 3 Reflect the resulting graph from Step when a < 0. = f a h + k can be Step Translate the resulting graph from Step 3 verticall k units. 88 Algebra 1 Copright Big Ideas Learning, LLC

32 Name Date 3. Notetaking with Vocabular (continued) Etra Practice In Eercises 1, use the graphs of f and g to describe the transformation from the graph of f to the graph of g f() = g() = f( ) f() = + 1 g() = f() 3.. g() = f( ) f() = 3 f() = 1 1 g() = f( ) 5.. f() = + f() = g() = f() 1 g() = f() 7. Graph f ( ) = and g( ) = 3. Describe the transformations from the graph of f to the graph of g. Copright Big Ideas Learning, LLC Algebra 1 89

33 Name Date 3.7 Graphing Absolute Value Functions For use with Eploration 3.7 Essential Question How do the values of a, h, and k affect the graph of the absolute value function g( ) = a h + k? 1 EXPLORATION: Identifing Graphs of Absolute Value Functions Work with a partner. Match each absolute value function with its graph. Then use a graphing calculator to verif our answers. a. g( ) = b. g( ) = + c. g( ) = + d. g( ) = e. g( ) = f. g( ) = + + A. B. C. D. E. F. 90 Algebra 1 Copright Big Ideas Learning, LLC

34 Name Date 3.7 Graphing Absolute Value Functions (continued) Communicate Your Answer. How do the values of a, h, and k affect the graph of the absolute value function g = a h + k ( )? 3. Write the equation of the absolute value function whose graph is shown. Use a graphing calculator to verif our equation. Copright Big Ideas Learning, LLC Algebra 1 91

35 Name Date 3.7 Notetaking with Vocabular For use after Lesson 3.7 In our own words, write the meaning of each vocabular term. absolute value function verte verte form 9 Algebra 1 Copright Big Ideas Learning, LLC

36 Name Date 3.7 Notetaking with Vocabular (continued) Core Concepts Absolute Value Function An absolute value function is a function that contains an absolute value epression. The parent absolute value function is f ( ) =. The graph of f ( ) = is V-shaped and smmetric about the -ais. The verte is the point where the graph changes direction. The verte of the graph f 0, 0. of ( ) = is ( ) The domain of f ( ) The range is 0. = is all real numbers. verte f() = Verte Form of an Absolute Value Function An absolute value function written in the form g( ) = a h + k, where a 0, is in verte form. The verte of the graph of g is ( h, k ). An absolute value function can be written in verte form, and its graph is smmetric about the line = h. Copright Big Ideas Learning, LLC Algebra 1 93

37 Name Date 3.7 Notetaking with Vocabular (continued) Etra Practice In Eercises 1, graph the function. Compare the graph to the graph of f =. Describe the domain and range. ( ) 1. t 1 ( ) =. u ( ) = 0 t() u() 3. p ( ) = 3. r ( ) = p() r() 9 Algebra 1 Copright Big Ideas Learning, LLC