Ready To Go On? Skills Intervention 4-1 Graphing Relationships

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1 Read To Go On? Skills Intervention -1 Graphing Relationships Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular continuous graph discrete graph Relating Graphs to Situations An object rises from the ground at a constant rate for several minutes. The object stas at that elevation for several minutes before dropping down, then rising to its previous altitude before slowl descending back to the ground. To relate a graph to a given situation, use ke words in the description. If a ke word is rose steadil the line segment showing that description should be slanting. If a ke word is constant the line segment should be a The table below lists the ke words, in order, from the situation above. Complete the table, using the graphs shown below. line. Ke Words Segment Description Graphs... Rises from the ground Slanting upward Graphs B and C Stas at that elevation Horizontal Graphs and Dropping down Slanting Graphs, B, and Rising Slanting Graphs,, and Slowl descending Slanting Graphs and C Graph A Graph B Graph C Elevation Elevation Elevation Time Time Time Which graph shows all the ke phrases in order? Copright b Holt, Rinehart and Winston. Holt Algebra 1

2 Read to Go On? Problem Solving Intervention -1 Graphing Relationships Graphs with connected curves or lines are called continuous graphs. Graphs that onl show distinct points are called discrete graphs. Dennis has 0 raffle tickets to sell for a school fund-raiser. Each booklet has raffle tickets. Sketch a graph to show how man raffle tickets he has left if he sells 1,, 3, or booklets of tickets. Understand the Problem 1. How man tickets does Dennis have to sell?. Each booklet has how man raffle tickets? 3. If he sells one booklet, he has sold tickets and 90 remain.. If he sells two booklets, he has sold 0 tickets and tickets remain. Make a Plan. What are ou being asked to do?. Decide the tpe of graph ou should draw. A graph has connected lines or curves. A graph has onl distinct points. Since ou can onl count whole numbers of tickets sold, construct a graph. Solve 7. What should ou title the graph? 8. Let the -ais represent the number of booklets sold. The -ais should be labeled from 0 to. 9. Let the -ais represent the number of tickets remaining. The -ais should be labeled from 0 to.. Complete the ordered pairs: (1, 90), (, 80), (3, ), (, ), (, ) 11. Plot the ordered pairs. Look Back 1. If Dennis sells 3 books, how man tickets remain? 13. Is the point shown on our graph? Tickets Remaining Booklets Sold Copright b Holt, Rinehart and Winston. Holt Algebra 1

3 Find these vocabular words in Lesson - and the Multilingual Glossar. Vocabular Read To Go On? Skills Intervention - Relations and Functions relation domain range function Finding the Domain and Range of a Relation Give the domain and range of each relation. A set of ordered pairs is called a. The of a relation is the set of first elements (or -coordinates) of the ordered pairs. The of a relation is the set of second elements (or -coordinates) of the ordered pairs. A. 3 B List the ordered pairs: ( 3, ); (0, ); List the ordered pairs of the endpoints: (0, ); (3, ) ( 3, ); (, ) The domain is { 3,, }. The domain is all -values from to, inclusive: 3 The range is {,, }. The range is all -values from 1 to, Identifing Functions Tell whether the relation is a function. inclusive: 1 A function is a special tpe of relation that pairs each domain 0 value with eactl range value. List the ordered pairs (, ); (, ); (, ); (, ); (, ) The domain is {,,,, }. The range is { }. Is a domain value (-coordinates) paired with more than one range value? Is the relation a function? Copright b Holt, Rinehart and Winston. Holt Algebra 1

4 Read To Go On? Skills Intervention -3 Writing Functions Find these vocabular words in Lesson -3 and the Multilingual Glossar. Vocabular independent variable dependent variable function rule function notation Using a Table to Write an Equation Determine a relationship between the - and -values. Write an equation If 1, what can ou subtract to get? 1 If 1, what can ou multipl b to get? 1( ) Determine which relationship works for the other - and -values. Subtracting: 3 Multipling: ( ) 3( ) ( ) Which relationship works so that when ou input the -values ou get the -values in the table? Write an equation: Identifing Independent and Dependent Variables Identif the independent and dependent variables given that 1. The input of a function is the variable. The of a function is the dependent variable. If 3,. If,. The value of depends on the. Independent variable: Dependent variable: Evaluating Functions Evaluate the function g () for 3. g( 3) ( ) ( 3) Substitute 3 for. g( 3) ( ) ( 3) Evaluate the eponent. g( 3) ( 3) Multipl. g( 3) Add the opposite. g( 3) Simplif. Copright b Holt, Rinehart and Winston. 7 Holt Algebra 1

5 Read to Go On? Problem Solving Intervention -3 Writing Functions A function describes the relationship between - and -values. A tanning salon charges a one-time maintenance fee of $1 plus $ for each tanning visit. Write a function to describe the situation. Find a reasonable domain and range for the function for up to visits. Understand the Problem 1. What is the tanning salon s maintenance fee?. How much does each tanning visit cost? 3. The problem is asking ou to do three different tasks. What are the three tasks? Make a Plan. Let v represent each visit. Write a function to represent this situation. Mone paid for tanning is $ per visit plus $1 maintenance fee f (v) v Solve. Write the function. f (v). What is the domain of the function? {0, 1,,,,, } 7. Substitute the domain values into the function rule to find the range of values f () (0) 1 1 (1) 1 1 () 1 (3) 1 () 1 () 1 () 1 8. What is the smallest value for f ()? ; largest value? 9. A reasonable range for this situation is {1, 1, 0,,,, }. Look Back. If ou pa the maintenance fee for the tanning salon and never have one tanning visit, how much will ou pa? If ou pa the maintenance fee for the tanning salon and have four tanning visits, how much will ou pa? Are these the same amounts given in the table in Eercise 7? Copright b Holt, Rinehart and Winston. 8 Holt Algebra 1

6 Read To Go On? Skills Intervention - Graphing Functions Graphing Solutions Given a Domain Graph the function for the domain D: {, 0,, }. Does the domain represent the - or -values of a function? Use the given values in the domain for and find values of. (, ) ( ) (, ) 0 (0) (0, ) ( ) (, ) ( ) (, ) If the first point in an ordered pair is negative, should ou move left or right from the origin? If the first point in an ordered pair is positive, should ou move left or right from the origin? If the second point in an ordered pair is negative, should ou move up or down? If the second point in an ordered pair is positive, should ou move up or down? Graph the ordered pairs (, ) from the table. Graphing Functions Graph the function. The absolute value of a number is alwas. Complete the table. (, ) ( 3, ) 0 (, ) ( 1, ) 0 0 (0, ) 1 (1, ) (, ) 3 (, ) Plot the ordered pairs. What shape graph do the ordered pairs appear to form? Copright b Holt, Rinehart and Winston. 9 Holt Algebra 1

7 Read to Go On? Problem Solving Intervention - Graphing Functions You can graph a function b finding ordered pairs that satisf the function. The function 9 represents how much mone Cameron earns in hours. Graph the function. Understand the Problem 1. What is the given function?. What variable represents hours? What variable represents mone? 3. If Cameron works 0 hours, how much mone will she earn? If she works 1 hour how much will she earn?. So, the more Cameron works, the more she will earn. Make a Plan. Can Cameron earn negative mone? So, the domain values for this function should onl be numbers.. The graph will onl be graphed in quadrant. Solve 7. Complete the table for the given function values. 8. The -coordinate of an ordered pair tells ou to move on the grid. 9. The -coordinate of an ordered pair tells ou to move on the grid.. Graph each ordered pair. 9 (, ) 0 9(0) 0 (0, ) 1 9(1) 9 (1, ) 9( ) 18 (, ) 3 9( ) (3, ) 9( ) (, ) Look Back 11. Use our graph to determine: What -value corresponds with an -value of? What -value corresponds with an -value of? What -value corresponds with an -value of? 1. If Cameron works more hours, does the graph show an increase or decrease in the amount of mone she earns? Income Hours Worked Does this correspond to our understanding of the problem in Eercise? Copright b Holt, Rinehart and Winston. 70 Holt Algebra 1

8 Read To Go On? Quiz -1 Graphing Relationships Choose the graph that best represents each situation. 1. Your distance from the ground as ou ride a Ferris wheel for three minutes. Graph A Graph B Height Height. The height of a o-o during a competition. Time Time 3. Julius goes to a carnival with $. Each ride ticket costs $. Sketch a graph to show his remaining amount of mone if he purchases 1,, 3,, or ride tickets. - Relations and Functions Give the domain and range of each relation. Tell whether the relation is a function. Eplain Amount Remaining, $ Tickets Purchased Domain: Range: Eplain: Domain: Range: Eplain: -3 Writing Functions Determine a relationship between the - and -values. Write an equation Copright b Holt, Rinehart and Winston. 71 Holt Algebra 1

9 Read to Go On? Quiz continued Identif the dependent and independent variables. Write a rule in function notation for each situation. 8. An administrative assistant can tpe words per minute. 9. An appliance repair compan charges a $ service fee plus $ per hour. Evaluate each function for the given input values.. For f () 3, find f () when For g() 3, find g() when A graphics design compan charges an initial $ set up fee and $1 per t-shirt printed. Write a function to describe the situation. Find a reasonable domain and range for the function for up to t-shirts. - Graphing Functions Graph each function for the given domain ; D: {, 0,, } D: {, 0, } D: { 1, 0, 1, } Graph each function. 1. 7; Copright b Holt, Rinehart and Winston. 7 Holt Algebra 1

10 Read to Go On? Enrichment Profit-Loss-Revenue An important concept in business is the abilit to make a profit. Profit is equal to the amount of sales minus the cost of production. If the sales are greater than the cost, the business makes a profit. If the sales are less than the cost, the business is losing mone. Use the information below to answer each question. A manufacturer of compact-disc plaers sells them to a retailer for $ each. It costs the manufacturer $00 plus $ each to produce the compact-disc plaer. 1. Write a function, s, to represent the total amount of sales of compact-disc plaers, n.. Write a function, c, to represent the total cost of producing the compact-disc plaers, n. 3. Graph the functions s and c on the same coordinate grid. Amount, $ Number of Compact-Disc Plaers. For what dollar amount is the sales and the cost equal?. For what value of n is the sales and the cost equal?. Write an inequalit that represents the value(s) of n for which the cost is more than the sales. 7. Write an inequalit that represents the value(s) of n for which the manufacturer makes a profit. Copright b Holt, Rinehart and Winston. 73 Holt Algebra 1

11 B Read To Go On? Skills Intervention - Scatter Plots and Trend Lines Find these vocabular words in Lesson - and the Multilingual Glossar. Vocabular scatter plot correlation positive correlation negative correlation no correlation trend line Graphing a Scatter Plot from Given Data Graph a scatter plot using the given data A is a graph with points plotted to show a possible relationship between two sets of data. List the sets of ordered pairs: (, 1); (, ); (, ); (, ); (, ); (, ). When plotting an ordered pair the -coordinate tells ou to move and the -coordinate tells ou to move or or To plot the point (, 1) ou move units right from the origin and units up. To plot the point (, 17) ou move units from the origin and units up. To plot the point (, 19) ou move units from the origin and units. Plot the ordered pairs on the grid. Describing Correlations from Scatter Plots Describe the correlation illustrated b the above scatter plot. A describes a relationship between two sets of data. In a positive correlation, both sets of data values. In a decreases. There is correlation, one set of data values increases as the other set correlation when the data values are scattered about. Look at the scatter plot ou just drew above. As the -value the -value also increases. Therefore, a correlation eists between the two data sets. Copright b Holt, Rinehart and Winston. 7 Holt Algebra 1

12 B Read to Go On? Problem Solving Intervention - Scatter Plots and Trend Lines You can graph a function on a scatter plot to show the relationship of data. Sometimes the function is a straight line. The line, called a trend line helps show the correlation between data sets more clearl. It can also be helpful when making predictions based on the data. The scatter plot shows the estimated annual sales for a plahouse franchise of stores for the ears Based on this relationship, predict the total annual sales in 01. Understand the Problem 1. What are ou being asked to predict? Sales (millions $) Year. What information are ou given? Make a Plan 3. What can be drawn on the scatter plot to help make a prediction?. Does our line have to go through all the data points?. When drawing a trend line, ou want about the same number of points above and the line. Solve. Draw the trend line on the scatter plot above. 7. To make the prediction find the point on the line whose -value is. The corresponding -value is the predicted annual sales. 8. The estimated sales for the plaground and plahouse franchise in 01 are. Look Back 9. The annual sales increase from ear to ear, based on the data. Is our answer a reasonable prediction of how much in sales the plaground and plahouse franchise will collect in the ear 01?. Plot the point (01, 1) on the grid. Does the point fall near our trend line? Copright b Holt, Rinehart and Winston. 7 Holt Algebra 1

13 B Find these vocabular words in Lesson - and the Multilingual Glossar. Vocabular Read To Go On? Skills Intervention - Arithmetic Sequences sequence term arithmetic sequence common difference Identifing Arithmetic Sequences Determine whether the sequence 0, 1,,, appears to be an arithmetic sequence. If so, find the common difference and the net three terms. A is a list of numbers that often forms a pattern. Each number in a sequence is called a. Find the difference between successive terms: 1 0 ; 1 ; ( ) Is the difference the same between each set of terms? If so, the common difference is. Use the common difference to find the net three terms. 0, 1,,,,, 8 Finding the n th Term of an Arithmetic Sequence Find the 1 th term of the arithmetic sequence 17, 8, 1,,. What is the common difference, d? 8 17 Write the rule to find the n th term: In the formula which variable represents the first term? What is a 1 in the above sequence? Solve for a n. a n a 1 (n 1)d a n (1 1)( 9) Substitute. a n 13( 9) Subtract. a n Multipl. The variable n represents the number of a n Subtract. term ou are looking for, so n. The 1 th term of the sequence is. Copright b Holt, Rinehart and Winston. 7 Holt Algebra 1

14 B Read to Go On? Problem Solving Intervention - Arithmetic Sequences The nth term of an arithmetic sequence with common difference d and first term a 1 is: a n a 1 (n 1)d. The marching band has 1 marchers in the front row, 1 in the second row, 18 in the third row, 0 in the fourth row, and so on. How man marchers are in the 1 th row? Understand the Problem 1. What are ou asked to find?. What are ou given? Make a Plan 3. How man marchers are in each of the first four rows? 1, 1,,. What is the difference between each two terms? 1 1 ; What tpe of sequence does this appear to be? Solve. What is the first term of the sequence, a 1? 7. What is the common difference, d? 8. What term in the sequence do ou need to find? n 9. What is the formula for finding the n th term?. Complete the formula for the number of marchers in the 1 th row. a n a 1 (n 1)d a n 1 (1 1) Substitute for a 1, n, and d. a n 1() Simplif the epression in parentheses. a n Multipl. a n Add. 11. The number of marchers in the 1 th row is. Look Back 1. Complete the table for the number of marchers in each row Does our answer from Eercise 11 match the table? Copright b Holt, Rinehart and Winston. 77 Holt Algebra 1

15 B Read to Go On? Quiz - Scatter Plots and Trend Lines The table shows the number of hits and runs scored in a softball game. 1. Graph a scatter plot using the given data. hits runs Runs Hits. Describe the correlation illustrated b the scatter plot. 3. Predict the number of runs out of 17 hits. Choose the scatter plot below that best represents the described relationship. Eplain.. age of a car and value of the car. age of a car and miles per gallon. age of a car and the annual cost to repair the car Graph A Graph B Graph C ,000,000,000 0,000 18,000 1,000 1,000 1,000, Copright b Holt, Rinehart and Winston. 78 Holt Algebra 1

16 B 7. The scatter plot shows the estimated annual ta returns for the ears 00 to 0. Based on this relationship, predict the number of ta returns in 013. Returns (million) Annual Estimated Ta Returns Read to Go On? Quiz continued Year - Arithmetic Sequences Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the net three terms. 8. 1, 13,, 3, 9., 8, 1, 3,..,, 1., 1, Find the indicated term of the arithmetic sequence. 11. th term: 9,,, 1, 1. 1 th term: a 1 8; d With no air resistance, a ball will roll down a ramp 9 feet during the first second, 1 feet during the net second, 3 feet during the third second, 30 feet during the fourth second, and so on. How man feet will the ball roll during the eighth second? Copright b Holt, Rinehart and Winston. 79 Holt Algebra 1

17 B Read to Go On? Enrichment Geometric Sequences A geometric sequence is a sequence in which each term is a product of the previous term and a common ratio, r. For eample,,, 8, 1, is a geometric sequence. The common ratio, r is. Each term is the product of the previous term and. The common ratio can be determined b finding the quotient of two consecutive terms. In the sequence 1,, 1,, the common ratio is because A geometric sequence has the general form a n a 1 r n 1, where n is the term number, and a 1 is the first term in the sequence. Determine whether each of the following is a geometric sequence. 1. 1, 1, 3, 9,.,,, 8, , 1,, 3,.,,,, Determine the common ratio for each of the geometric sequences.., 1,, 13,. 8,,, 1, 7.,, 8, 1, 8. 1, 1,,, Write the general form of the geometric sequence. 9.,, 0, 0,. 1, 3, 9, 7, 11. 0,, 1, 1, 1. 1,, 3, 3, Copright b Holt, Rinehart and Winston. 80 Holt Algebra 1