Maintaining Mathematical Proficiency

Transcription

1 Name Date Chapter Maintaining Mathematical Proficienc Use the graph to answer the question. A H B F G E C D x 1. What ordered pair corresponds to point A?. What ordered pair corresponds to point H? 3. What ordered pair corresponds to point E?. Which point is located in Quadrant III? 5. Which point is located in Quadrant IV?. Which point is located on the negative x-axis? Solve the equation for. 7. x = x + = x = 1. = 3 x = 3x x = x 13. Rectangle ABCD has vertices A(, ), B(, 5 ), and C( ) coordinates of vertex D? 7, 5. What are the Copright Big Ideas Learning, LLC Algebra 1 95

2 Name Date.1 Writing Equations in Slope-Intercept Form For use with Exploration.1 Essential Question Given the graph of a linear function, how can ou write an equation of the line? 1 EXPLORATION: Writing Equations in Slope-Intercept Form Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Find the slope and -intercept of each line. Write an equation of each line in slope-intercept form. Use a graphing calculator to verif our equation. a. b. (, 3) (, ) 9 (, 1) 9 9 (, ) 9 c. d. 9 ( 3, 3) (3, 1) 9 9 (, ) (, 1) 9 9 Algebra 1 Copright Big Ideas Learning, LLC

3 Name Date.1 Writing Equations in Slope-Intercept Form (continued) EXPLORATION: Mathematical Modeling Work with a partner. The graph shows the cost of a smartphone plan. a. What is the -intercept of the line? Interpret the -intercept in the context of the problem. b. Approximate the slope of the line. Interpret the slope in the context of the problem. Cost per month (dollars) Smartphone Plan x Data usage (megabtes) c. Write an equation that represents the cost as a function of data usage. Communicate Your Answer 3. Given the graph of a linear function, how can ou write an equation of the line?. Give an example of a graph of a linear function that is different from those above. Then use the graph to write an equation of the line. Copright Big Ideas Learning, LLC Algebra 1 97

4 Name Date.1 Notetaking with Vocabular For use after Lesson.1 In our own words, write the meaning of each vocabular term. linear model Notes: 98 Algebra 1 Copright Big Ideas Learning, LLC

5 Name Date.1 Notetaking with Vocabular (continued) Extra Practice In Exercises 1, write an equation of the line with the given slope and -intercept. 1. slope:. slope: 1 3. slope: -intercept: 9 -intercept: -intercept: 3. slope: 3 5. slope:. slope: 1 3 -intercept: 7 -intercept: -intercept: In Exercises 7 1, write an equation of the line in slope-intercept form ( 1, 3) (, ) (, 3) (, 1) x (, ) x (, 1) x (, 5) x (3, ) (, 3) x (, ) x (, 3) (, ) Copright Big Ideas Learning, LLC Algebra 1 99

6 Name Date.1 Notetaking with Vocabular (continued) In Exercises 13 18, write an equation of the line that passes through the given points. 13. ( 3, 1 ), ( 8, ) 1. (,1 ), ( 3, 5 ) 15. (, ), (, 3 ) 1. ( 3, ), (, 1) 17. ( 8, ), (, 8 ) 18. ( 1, 7 ), (, 5) In Exercises 19, write a linear function f with the given values. 19. f( ) =, f( ) = 3. f( 5) = 5, f ( 5) = f() f() 8 = 3, 9 =. f ( ) =, f ( 7) = 3. f( ) =, f( ) = 1. f( ) f ( ) =, = 8 5. An electrician charges $1 after hours of work and $19 after hours of work. a. Write a linear model that represents the total cost as a function of the number of hours worked. b. What is the electrician s initial fee? c. How much does the electrician charge per hour? 1 Algebra 1 Copright Big Ideas Learning, LLC

7 Name Date. Writing Equations in Point-Slope Form For use with Exploration. Essential Question How can ou write an equation of a line when ou are given the slope and a point on the line? 1 EXPLORATION: Writing Equations of Lines Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Sketch the line that has the given slope and passes through the given point. Find the -intercept of the line. Write an equation of the line. a. m = 1 b. m = x x EXPLORATION: Writing a Formula Work with a partner. The point (x 1, 1 ) is a given point on a nonvertical line. The point (x, ) is an other point on the line. Write an equation that represents the slope m of the line. Then rewrite this equation b multipling each side b the difference of the x-coordinates to obtain the point-slope form of a linear equation. (x 1, 1 ) (x, ) x Copright Big Ideas Learning, LLC Algebra 1 11

8 Name Date. Writing Equations in Point-Slope Form (continued) 3 EXPLORATION: Writing an Equation Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. For four months, ou have saved $5 per month. You now have $175 in our savings account. a. Use our result from Exploration to write an equation that represents the balance A after t months. Account balance (dollars) Savings Account A 5 15 (, 175) Time (months) t b. Use a graphing calculator to verif our equation. Communicate Your Answer. How can ou write an equation of a line when ou are given the slope and a point on the line? 5. Give an example of how to write an equation of a line when ou are given the slope and a point on the line. Your example should be different from those above. 1 Algebra 1 Copright Big Ideas Learning, LLC

9 Name Date. Notetaking with Vocabular For use after Lesson. In our own words, write the meaning of each vocabular term. point-slope form Core Concepts Point-Slope Form Words A linear equation written in the form = m x x is in point-slope form. 1 ( 1) The line passes through the point ( x ) and the slope of the line is m. slope,, 1 1 (x, ) (x 1, 1 ) x x 1 1 x Algebra 1 = m(x x 1 ) passes through (x 1, 1 ) Notes: Copright Big Ideas Learning, LLC Algebra 1 13

10 Name Date. Notetaking with Vocabular (continued) Extra Practice In Exercises 1, write an equation in point-slope form of the line that passes through the given point and has the given slope. 1. (,1 ); m = 3. ( 3, 5 ); m = 3. ( ) 1, ; m = 1. ( 5, ); m = 5. (, ); m = 7. ( 1, 1 ); m = 3 In Exercises 7 1, write an equation in slope-intercept form of the line shown (3, ) (1, ) (1, ) x (, 1) x (, 3) x (, 1) (, 1) x (, 1) x (1, 3) ( 1, 9) (1, ) (, ) x 1 Algebra 1 Copright Big Ideas Learning, LLC

11 Name Date. Notetaking with Vocabular (continued) In Exercises 13 18, write a linear function f with the given values. 13. f( 3) = 1, f ( ) = 1. f( ) = 1, f( 1) = f ( ) f ( ) 1 =, 3 = 3 1. f( ) =, f( ) = f () 1 =, f ( ) = f() f() 3 = 5, = In Exercises 19 and, tell whether the data in the table can be modeled b a linear equation. Explain. If possible, write a linear equation that represents as a function of x. 19. x x Craig is driving at a constant speed of miles per hour. After driving 3 hours, his odometer reads 5 miles. Write a linear function D that represents the miles driven after h hours. What does the odometer read after 7 hours of continuous driving? Copright Big Ideas Learning, LLC Algebra 1 15

12 Name Date.3 Writing Equations of Parallel and Perpendicular Lines For use with Exploration.3 Essential Question How can ou recognize lines that are parallel or perpendicular? 1 EXPLORATION: Recognizing Parallel Lines Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Write each linear equation in slope-intercept form. Then use a graphing calculator to graph the three equations in the same square viewing window. (The graph of the first equation is shown.) Which two lines appear parallel? How can ou tell? a. 3x + = b. 5x + = 3x + = 1 x + = 3 x + 3 = 1.5x + = = x + 3 = x Algebra 1 Copright Big Ideas Learning, LLC

13 Name Date.3 Writing Equations of Parallel and Perpendicular Lines (continued) EXPLORATION: Recognizing Perpendicular Lines Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Write each linear equation in slope-intercept form. Then use a graphing calculator to graph the three equations in the same square viewing window. (The graph of the first equation is shown.) Which two lines appear perpendicular? How can ou tell? a. 3x + = b. x + 5 = 1 3x = 1 x + = 3 x 3 = 1.5x = = x + 3 = x + 5 Communicate Your Answer 3. How can ou recognize lines that are parallel or perpendicular?. Compare the slopes of the lines in Exploration 1. How can ou use slope to determine whether two lines are parallel? Explain our reasoning. 5. Compare the slopes of the lines in Exploration. How can ou use slope to determine whether two lines are perpendicular? Explain our reasoning. Copright Big Ideas Learning, LLC Algebra 1 17

14 Name Date.3 Notetaking with Vocabular For use after Lesson.3 In our own words, write the meaning of each vocabular term. parallel lines perpendicular lines Core Concepts Parallel Lines and Slopes Two lines in the same plane that never intersect are parallel lines. Two distinct nonvertical lines are parallel if and onl if the have the same slope. All vertical lines are parallel. Notes: Perpendicular Lines and Slopes Two lines in the same plane that intersect to form right angles are perpendicular lines. Nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals. Vertical lines are perpendicular to horizontal lines. Notes: = x + 1 = x 1 x 18 Algebra 1 Copright Big Ideas Learning, LLC

15 Name Date.3 Notetaking with Vocabular (continued) Extra Practice In Exercises 1, determine which of the lines, if an, are parallel. Explain. 1.. a b c ( 1, ) (, 1) (1, 1) x (, 1) ( 1, 3) (, 3) a b c ( 1, 1) 3 (1, ) (1, 3) (, 1) (, ) x (3, ) 3. Line a passes through (, 1) and (, ).. Line a passes through (, 5) and (,1 ). Line b passes through ( 5, 3) and ( 5,1 ). Line b passes through (, 3) and ( 3, ). Line c passes through (, 3) and (, 1 ). Line c passes through ( 3, ) and ( ),. 5. Line a: x = Line a: 5 x = Line b: 8 = x + 1 Line b: 5 = x + 7 Line c: = 3x + 9 Line c: 5 x = 5 In Exercises 7 and 8, write an equation of the line that passes through the given point and is parallel to the given line. 7. ( 3, 1 ); = 1 x 3 8. ( ) 3 1, ; = x + 1 Copright Big Ideas Learning, LLC Algebra 1 19

16 Name Date.3 Notetaking with Vocabular (continued) In Exercises 9 1, determine which of the lines, if an, are parallel or perpendicular. Explain ( 1, 3) (, 1) (1, 1) x ( 1, 1) (1, ) ( 1, ) b ( 3, 3) ( 1, ) c (1, ) (, ) (3, 1) x (, 1) b 5 c a a 11. Line a passes through (, ) and ( 1, 1 ). 1. Line a passes through (, ) and ( ) Line b passes through (,1) and (, ). Line b passes through ( 1, ) and ( 1, ). Line c passes through ( 1, ) and ( 1, ). Line c passes through (, 3) and (, ). 1, Line a: = 3 x Line a: 5 x = 1 Line b: 3 = x 3 Line b: = 5 x 1 Line c: = 3x + 9 Line c: = x In Exercises 15 and 1, write an equation of the line that passes through the given point and is perpendicular to the given line. 15. (, ); = x + 1. ( ) 3 3, 1 ; = x 3 11 Algebra 1 Copright Big Ideas Learning, LLC

17 Name Date. Scatter Plots and Lines of Fit For use with Exploration. Essential Question How can ou use a scatter plot and a line of fit to make conclusions about data? A scatter plot is a graph that shows the relationship between two data sets. The two data sets are graphed as ordered pairs in a coordinate plane. 1 EXPLORATION: Finding a Line of Fit Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. A surve was taken of 179 married couples. Each person was asked his or her age. The scatter plot shows the results. a. Draw a line that approximates the data. Write an equation of the line. Explain the method ou used. Wife s age Ages of Married Couples Husband s age b. What conclusions can ou make from the equation ou wrote? Explain our reasoning. Copright Big Ideas Learning, LLC Algebra 1 111

18 Name Date. Scatter Plots and Lines of Fit (continued) EXPLORATION: Finding a Line of Fit Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. The scatter plot shows the median ages of American women at their first marriage for selected ears from 19 through 1. a. Draw a line that approximates the data. Write an equation of the line. Let x represent the number of ears since 19. Explain the method ou used. Age Ages of American Women at First Marriage Year b. What conclusions can ou make from the equation ou wrote? c. Use our equation to predict the median age of American women at their first marriage in the ear. Communicate Your Answer 3. How can ou use a scatter plot and a line of fit to make conclusions about data?. Use the Internet or some other reference to find a scatter plot of real-life data that is different from those given above. Then draw a line that approximates the data and write an equation of the line. Explain the method ou used. 11 Algebra 1 Copright Big Ideas Learning, LLC

19 Name Date. Notetaking with Vocabular For use after Lesson. In our own words, write the meaning of each vocabular term. scatter plot correlation line of fit Core Concepts Scatter Plot A scatter plot is a graph that shows the relationship between two data sets. The two data sets are graphed as ordered pairs in a coordinate plane. Scatter plots can show trends in the data. Notes: Copright Big Ideas Learning, LLC Algebra 1 113

20 Name Date. Notetaking with Vocabular (continued) Using a Line of Fit to Model Data Step 1 Make a scatter plot of the data. Step Decide whether the data can be modeled b a line. Step 3 Draw a line that appears to fit the data closel. There should be approximatel as man points above the line as below it. Step Write an equation using two points on the line. The points do not have to represent actual data pairs, but the must lie on the line of fit. Notes: Extra Practice 1. The scatter plot shows the weights (in pounds) of a bab over time. Weight of a Bab Weight (in pounds) x Age (in months) a. What is the weight of the bab when the bab is four months old? b. What is the age of the bab when the bab weighs 17. pounds? c. What tends to happen to weight of the bab as the age increases? 11 Algebra 1 Copright Big Ideas Learning, LLC

21 Name Date. Notetaking with Vocabular (continued) In Exercises 5, tell whether x and show a positive, a negative, or no correlation.. 3. x x. 5. x x. The table shows the depth (in centimeters) of water filling a bathtub after x minutes. Time (minutes), x Depth (centimeters), a. Write an equation that models the depth of the water as a function of time. b. Interpret the slope and -intercept of the line of fit. Copright Big Ideas Learning, LLC Algebra 1 115

22 Name Date.5 Analzing Lines of Fit For use with Exploration.5 Essential Question How can ou analticall find a line of best fit for a scatter plot? 1 EXPLORATION: Finding a Line of Best Fit Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. The scatter plot shows the median ages of American women at their first marriage for selected ears from 19 through 1. In Exploration in Section., ou approximated a line of fit graphicall. To find the line of best fit, ou can use a computer, spreadsheet, or graphing calculator that has a linear regression feature. a. The data from the scatter plot is shown in the table. Note that, 5, 1, and so on represent the numbers of ears since 19. What does the ordered pair (5, 3.3) represent? b. Use the linear regression feature to find an equation of the line of best fit. You should obtain results such as those shown below. Age L Ages of American Women at First Marriage Year L1(55)= L L3 LinReg =ax+b a= b= r = r= c. Write an equation of the line of best fit. Compare our result with the equation ou obtained in Exploration in Section.. 11 Algebra 1 Copright Big Ideas Learning, LLC

23 Name Date.5 Analzing Lines of Fit (continued) Communicate Your Answer. How can ou analticall find a line of best fit for a scatter plot? 3. The data set relates the number of chirps per second for striped ground crickets and the outside temperature in degrees Fahrenheit. Make a scatter plot of the data. Then find an equation of the line of best fit. Use our result to estimate the outside temperature when there are 19 chirps per second. Chirps per second Temperature ( F) Chirps per second Temperature ( F) Copright Big Ideas Learning, LLC Algebra 1 117

24 Name Date.5 Notetaking with Vocabular For use after Lesson.5 In our own words, write the meaning of each vocabular term. residual linear regression line of best fit correlation coefficient interpolation extrapolation causation Notes: 118 Algebra 1 Copright Big Ideas Learning, LLC

25 Name Date.5 Notetaking with Vocabular (continued) Core Concepts Residuals A residual is the difference of the -value of a data point and the corresponding -value found using the line of fit. A residual can be positive, negative, or zero. positive residual data point line of fit negative residual data point A scatter plot of the residuals shows how well a model fits a data set. If the model is a good fit, then the absolute values of the residuals are relativel small, and the residual points will be more or less evenl dispersed about the horizontal axis. If the model is not a good fit, then the residual points will form some tpe of pattern that suggests the data are not linear. Wildl scattered residual points suggest that the data might have no correlation. Notes: Extra Practice In Exercises 1 and, use residuals to determine whether the model is a good fit for the data in the table. Explain. 1. = 3x + x Copright Big Ideas Learning, LLC Algebra 1 119

26 Name Date.5 Notetaking with Vocabular (continued). =.5x + 1 x The table shows the number of visitors to a particular beach for average dail temperatures x. a. Use a graphing calculator to find an equation of the line of best fit. Then plot the data and graph the equation in the same viewing window. Average Dail Temperature ( F) Number of Beach Visitors b. Identif and interpret the correlation coefficient. c. Interpret the slope and -intercept of the line of best fit. 1 Algebra 1 Copright Big Ideas Learning, LLC

27 Name Date. Arithmetic Sequences For use with Exploration. Essential Question How can ou use an arithmetic sequence to describe a pattern? An arithmetic sequence is an ordered list of numbers in which the difference between each pair of consecutive terms, or numbers in the list, is the same. 1 EXPLORATION: Describing a Pattern Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use the figures to complete the table. Plot the points given b our completed table. Describe the pattern of the -values. a. n = 1 n = n = 3 n = n = 5 Number of stars, n Number of sides, n b. n = 1 n = n = 3 n = n = 5 n Number of circles, n Copright Big Ideas Learning, LLC Algebra 1 11

28 Name Date. 1 Arithmetic Sequences (continued) EXPLORATION: Describing a Pattern (continued) c. n = 1 n= n=3 n= 1 n=5 1 8 Number of rows, n n 5 Number of dots, Communicate Your Answer. How can ou use an arithmetic sequence to describe a pattern? Give an example from real life. 3. In chemistr, water is called HO because each molecule of water has two hdrogen atoms and one oxgen atom. Describe the pattern shown below. Use the pattern to determine the number of atoms in 3 molecules. n=1 1 Algebra 1 n= n=3 n= n=5 Copright Big Ideas Learning, LLC

29 Name Date. Notetaking with Vocabular For use after Lesson. In our own words, write the meaning of each vocabular term. sequence term arithmetic sequence common difference Core Concepts Arithmetic Sequence In an arithmetic sequence, the difference between each pair of consecutive terms is the same. This difference is called the common difference. Each term is found b adding the common difference to the previous term common difference Terms of an arithmetic sequence Notes: Equation for an Arithmetic Sequence Let a n be the nth term of an arithmetic sequence with first term a 1 and common difference d. The nth term is given b Notes: ( ) an = a1 + n 1 d. Copright Big Ideas Learning, LLC Algebra 1 13

30 Name Date. Notetaking with Vocabular (continued) Extra Practice In Exercises 1, write the next three terms of the arithmetic sequence. 1. 1, 8, 15,,., 1, 8,, 3. 1, 1, 3, 39,. 5, 1, 19,, 5. 3, 7, 11, 15,., 1,, 38, In Exercises 7 1, graph the arithmetic sequence. 7. 1, 3, 5, 7, 8. 9,, 3,, 9. 15, 13, 11, 9, 1. 1,.5,, 5.5, 11. 1,, 7, 1, 1. 1, 5, 9, 13, 1 Algebra 1 Copright Big Ideas Learning, LLC

31 Name Date. Notetaking with Vocabular (continued) In Exercises 13 15, determine whether the graph represents an arithmetic sequence. Explain a n (, 1) (1, ) 3 (3, ) (, 9) 5 7 n a n (, 5) (3, ) (, 3) (1, ) n a n (1, 1) (, 18) (3, 15) (, 1) n In Exercises 1 1, write an equation for the nth term of the arithmetic sequence. Then find a ,., 7.8, 9., 17. 3, 38, 33, 8, 18., 1, 1, 18, , 9, 7, 5,. 3, 37,, 3, 1. 9, 7, 5, 3,. In an auditorium, the first row of seats has 3 seats. Each row behind the first row has more seats than the row in front of it. How man seats are in the 5th row? Copright Big Ideas Learning, LLC Algebra 1 15

32 Name Date.7 Piecewise Functions For use with Exploration.7 Essential Question How can ou describe a function that is represented b more than one equation? 1 EXPLORATION: Writing Equations for a Function Work with a partner. a. Does the graph represent as a function of x? Justif our conclusion. x b. What is the value of the function when x =? How can ou tell? c. Write an equation that represents the values of the function when x. f x = x ( ), if d. Write an equation that represents the values of the function when x >. f x = x > ( ), if e. Combine the results of parts (c) and (d) to write a single description of the function. ( ) f x =,ifx,ifx > 1 Algebra 1 Copright Big Ideas Learning, LLC

33 Name Date.7 Piecewise Functions (continued) EXPLORATION: Writing Equations for a Function Work with a partner. a. Does the graph represent as a function of x? Justif our conclusion. x b. Describe the values of the function for the following intervals. ( ) f x, if x < 3, if 3 x < =, if x < 3, if 3 x < Communicate Your Answer 3. How can ou describe a function that is represented b more than one equation? x. Use two equations to describe the function represented b the graph? Copright Big Ideas Learning, LLC Algebra 1 17

34 Name Date.7 Notetaking with Vocabular For use after Lesson.7 In our own words, write the meaning of each vocabular term. piecewise function step function Core Concepts Piecewise Function A piecewise function is a function defined b two or more equations. Each piece of the function applies to a different part of its domain. An example is shown below. ( ) f x x, if x = x + 1, if x > The expression x represents the value of f when x is less than or equal to. The expression x + 1 represents the value of f when x is greater than. Notes: f(x) = x, x f(x) = x + 1, x > x 18 Algebra 1 Copright Big Ideas Learning, LLC

35 Name Date.7 Notetaking with Vocabular (continued) Extra Practice In Exercise 1 9, evaluate the function. ( ) f x 3x 1, if x 1 = 1 x, if x > 1 ( ) gx 3x 1, if x 3 =, if 3 < x< 1 3, x ifx 1 1. ( ) f. f () 1 3. f () 5. f ( ) 5. g ( ). g( 3) 7. () 1 g 8. () 3 g 9. g( 5) In Exercise 1 13, graph the function. Describe the domain and range. 1. x, if x = 11., if x > x, if x < = x + 3, if x Copright Big Ideas Learning, LLC Algebra 1 19

36 Name Date.7 Notetaking with Vocabular (continued) 1. x, if x < =, if x < x, if x 13. 1, if x 1 =, if 1 < x < 1, if x In Exercise 1 and 15, write a piecewise function for the graph x x 1. A postal service charges $ for shipping an package weighing up to but not including 1 pound and $1 for each additional pound or portion of a pound up to but not including 5 pounds. Packages 5 pounds or over have different rates. Write and graph a step function that shows the relationship between the number x of pounds a package weighs and the total cost for postage. 13 Algebra 1 Copright Big Ideas Learning, LLC