3.03 Define and distinguish between relations and functions, dependent and independent variables, domain and range.

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1 3.03 Define and distinguish between relations and functions, dependent and independent variables, domain and range. A. These sports utility vehicles were listed in the classified section of the newspaper recently $21, $19, , , , , , , , , , , , , , , ,980 Which is the independent quantity, age or price? Are there other independent variables associated with the price of used SUVs? What are the realistic domain and range of the data? Does this set of data represent a function? Why? (4.02) Is there a general pattern to the data? Does the data exhibit a linear relationship? Explain. B. In July 2004 the Credit Union set the rate for the certificates of deposit (CD). A certificate of deposit is a savings plan that guarantees a fixed rate of return for a specified length of time, provided there are no withdrawals. Certificates of Deposit (CD) Interest Rate 6 Month Certificate Month Certificate Month Certificate Month Certificate Month Certificate Month Certificate Month Certificate Month Certificate 4.50 Does the CD information provided represent a function? Why? Identify the independent and dependent quantities. Are there other independent variables affecting the data? What are they? What are the domain and range of the relation? Are there realistic boundaries to the domain and range that are not shown? Algebra I Indicators p. 9

2 C. According to the internet ratings report for July 2000 from Nielsen/NetRatings, 52% of the home population have internet access and 32% of the home population surfed the Web in July. Nearly 144 million people in the U.S. had access to the internet from home, compared to million a year ago in July Lower prices for personal computers, inexpensive internet access, and competitive rates for high-speed internet access are reasons cited for the expanding access. Identify the independent and dependent variables associated with the internet ratings report. Are there other independent variables? What are they? What are the domain and range of the relation? Are there realistic boundaries to the domain and range that are not identified? Algebra I Indicators p. 10

3 3.04 Graph and interpret in the context of the problem, relations and functions on the coordinate plane. Include linear equations and inequalities, quadratics, and exponentials $ A. These compact cars were listed in the classified section of the newspaper recently. What are the independent variables associated with the price of used compact cars? What are the realistic domain and range of the data? Does this set of data represent a function? Why? Describe the data with respect to its graph. Is there a general pattern to the data? Does the data exhibit a linear relationship? Explain. B. One of the more important natural resources we have in the United States is petroleum. The equation P = -0.68Y is a linear model for the size of the petroleum reserves in the US. P is proven petroleum reserves in billions of barrels and Y is the number of years since the first commercial oil well was drilled (1859). Graph the relationship for the years since your birth. C. The function f(x) = (0.7475) (x-1) describes the winnings per player in the 2000 PGA Championship Golf Tournament according to a player's place (x) in the final standings. Describe the function as it would appear graphed. (3.12) If the minimum payout is $1000, how many places will receive money? Algebra I Indicators p. 11

4 3.05 Determine and use slopes of linear relationships to solve problems. a) Find the slope of a line given the graph of the line, an equation of the line, or two points on the line. b) Describe the slope of the line in the context of a problem situation. A. Suppose the value of a new car declines linearly over a ten-year period from the original value of $16,000 to the value of $1,500. What is the value of the car after six years? B. The number of licensed pilots in the United States has grown from 622,000 in 1996 to 640,000 in Assuming a constant annual increase in the number of pilots, how many pilots could we expect to have by 2002? Give the algebraic model for this growth. C. In 1998 there were 429,316 people employed in the United States as computer support specialists. According the Bureau of Labor Statistics, that number is expected to grow to 868,674 by Assuming a constant annual increase in the number of specialists, what is the number in 2006? Give the algebraic model for this growth. D. Nationally, the unemployment rate for teenagers (ages 16-19) seeking work dropped from a high of 16.1% in 1979 to a rate of 13.9% in For the same young people in North Carolina, the unemployment rate dropped from 18.3% in 1993 to 9.6% in How does the change in unemployment rates for teenagers compare between North Carolina and the nation? E. Students at a local university will see their student fees increase from $814 in 1997 to $842 in How much will students be paying in 2001? If this is a linear trend, what is the slope of an equation which describes the growth in fees? F. The percent of Americans with at least four years of high school education is shown in the chart. (4.03) Determine a linear best-fit equation (let 1940 be x = 0). Define the slope of the line with respect to the data. What are the variables that affect the number of people completing high school? How have those variables changed or modified since 1940? Algebra I Indicators p. 12

5 3.06 Write the equation of and graph linear relationships given relevant information: a) Slope and y-intercept b) Slope and one point on the line c) Two points on the line A. The natural gas company computes a monthly statement based on a rate of $ per hundred cubic feet of gas and a $7.74 facilities charge. The February statement is $ How much gas was used? B. According to an internet ratings report from Nielsen/NetRatings, nearly 144 million people in the U.S. had access to the internet from home in July 2000, compared to million a year earlier. Assuming the growth in internet access is linear, when (month and year) should every American have access to the internet? C. The Euro, the common European currency, converted to $ at the beginning of the year. Two months later the currency was worth $ Create a linear equation and estimate the value of a Euro at the end of the year. (1.01d, 3.06) How many more Euros can you buy for $500 at the end of the year? D. Between 1980 and 2000, the price of a pound of apples increased $0.019 per year. The price of a pound of apples in 1980 was $ Create a realistic algebraic model of the situation and estimate prices for the next five years. How realistic are your predictions? Explain. Algebra I Indicators p. 13

6 3.07 Investigate and determine the effects of changes in slope and intercepts on the graph and equation of a line. a) Change only slope. b) Change only the x- or y-intercept. c) Change the slope and an intercept. A. For the line y = 3x + 5: If the y-intercept moves to 7 and the slope remains unchanged, how does the x-intercept change? B. For the line y = ax + b where a>0 and b>0: If b increases and a remains constant, how does the x-intercept change? What happens to the line? If a increases and b remains constant, how does the x-intercept change? If b is multiplied by 1 and a remains constant, how does the line change? If a decreases, getting closer to 0, and b remains constant what happens to the line? C. Consider the lines y 1 = 1.5x and y 2 = (1.5x + 7.3) 6. How is y 2 different from y 1? How are the two lines similar? D. Consider the line y = ax + b. What changes when y = (ax + b) + 3? Identify similarities and differences between the two lines. E. Consider the lines y 1 = 0.95x 2 and y 2 = 0.95(x + 3) 2. How is y 2 different from y 1? How are the two lines similar? F. Consider the line y = ax + b. What changes when y = a(x 2.5) + b? Identify similarities and differences between the two lines. G. Consider the lines y 1 = 4x + 11 and y 2 = -4x How is y 2 different from y 1? How are the two lines similar? H. Consider the line y = ax + b. What changes when y = -ax + b? Identify similarities and differences between the two lines. I. Consider the lines y 1 = 0.5x and y 2 = 2(0.5x) How is y 2 different from y 1? How are the two lines similar? J. Consider the line y = ax + b. What changes when y = 3.1(ax) + b? Identify similarities and differences between the two lines. K. Consider the lines y 1 = 1.25x and y 2 = 3(1.25x ). How is y 2 different from y 1? How are the two lines similar? L. Consider the line y = ax + b. What changes when y = 1.66(ax + b)? Identify similarities and differences between the two lines. Algebra I Indicators p. 14

7 M. For y = -2.5x + 8: If the slope increases and the x-intercept remains constant, how does the y-intercept change? N. For the line y = ax + b where a<0 and b>0: If the slope increases and the x-intercept remains constant, how does the y-intercept change? If b increases and the x-intercept remains constant, how does the slope change? If a is multiplied by 1 and b remains constant, how does the x-intercept change? If a and b are multiplied by 2, how does the x-intercept change? O. For the years , the equation y = 6.25x represents the number of VCRs owned (in millions). For years , y = 3.2x represents the number of VCRs owned (in millions). In both equations, x represents the number of years since Discuss the changes in growth of VCR ownership described in the linear models. P. United Gameware is a company that makes games for PCs. For the last five years the equation y = 3.25x modeled the growth in value of the company s stock. $10.75 was the initial offering price, $3.25 is the average annual change in value of the stock, x is the number of years since the initial offering, and y is the value of the stock. A competitor, FedGames, issued its stock with the same initial value but only grew $1.95 a year in value. What would FedGames linear model look like? After five years which company s stock is worth more? by how much? After the fifth year suppose United Gameware began losing $0.85 per year in value. What would the linear equation look like then? Algebra I Indicators p. 15

8 3.08 Use linear equations or inequalities to solve problems. Solve by: a) Graphing. b) Using properties of equality; justify steps used. A. In 1990, US exports by North Carolina agriculture and industries were worth $8.01 billion. By 1997, exports from North Carolina had increased to $ billion. Assuming the growth in exports has been linear, (3.06) create an algebraic model of this growth. (3.05) What is the average annual growth in exports for the period? If North Carolina s exports continue to grow according to the model, when will the value reach $21 billion? (3.04) According to the model, prior to what year were there no exports? B. The power company uses two different rates to calculate a monthly power bill: For July-October, the basic customer charge is $6.75 plus $ per kilowatt-hour. For November-June, the basic customer charge is $6.75 plus $ per kilowatt-hour. 3% North Carolina sales tax is added for the final charge. If the May and September bills are both $127, what is the difference in the amount of power (kilowatt-hours) used each month? C. In football, the place kicker can score points two different ways. He can score 3 points with a field goal or 1 point with a point-after-touchdown (PAT). The coach expects his kicker to score at least 50 points during the season. (3.04) Construct the algebraic expression that represents the situation. Illustrate graphically how many field goals and PATs the kicker could score. D. A basketball player scored 20 points, including 3 free throws. How many 3-point field goals did she score? E. At the mid-season break, the Red Sox had won 45 out of the 85 games they had played. If the Red Sox win 55% of their remaining games (162 games in a season), how many games will they have won by the end of the season? If the Red Sox are to win 100 games, what is the smallest winning percentage they can have to accomplish this? Algebra I Indicators p. 16

9 3.09 Use systems of linear equations or inequalities in two variables to solve problems. Determine the solution by: a) Graphing. b) Substitution. c) Elimination. A. During the band s fruit sale, five dozen oranges cost as much as four dozen grapefruits. Terry bought two dozen oranges and a dozen grapefruit, spending $ What was the cost of a dozen oranges? B. The bill for a lunch of three hamburgers and two drinks is $9.67. The bill for a lunch of four hamburgers and three drinks is $ What is the total cost of one hamburger and one drink? C. For a special order, the Coverup Company manufactured 1200 shirts. Sweatshirts were priced at $14 each and T-shirts at $8 each. The company received a total of $11,400 for the shirts. How many of each type of shirt did the Coverup Company manufacture for this order? D. A movie theater charges $7 for an adult s ticket and $4.50 for a child s ticket. On a recent night, the sale of child s tickets was three times the sale of adult s tickets. If the total amount collected for ticket sales was $2,009, how many adults purchased tickets? E. A certificate of deposit (CD) is a savings plan that guarantees a fixed rate of return for a specified length of time, provided there are no withdrawals. Rutherford Central Bank is currently using the function R = 0.018m (R is the interest rate and m is the length of the CD in months) to set the rates for its CDs. Matthews Federal Credit Union uses the function R =.009m to set its interest rates. For which CDs does each banking institution offer the better rate? F. Nationally, the unemployment rate for teenagers (ages 16-19) seeking work dropped from a high of 16.1% in 1979 to a rate of 13.9% in For the same young people in North Carolina, the unemployment rate dropped from 18.3% in 1993 to 9.6% in How long has the unemployment rate for teenagers in North Carolina been better than the national rates? Algebra I Indicators p. 17

10 3.10 Graph quadratic functions. a) Locate the intercepts and the vertex. b) Recognize the x-intercepts of the function as the solutions of the equation. A. The function f(x) = x x describes the monthly price of gasoline for a recent 18-month period. At what month did the prices reach their peak? Assuming the function continues to model gasoline prices, how long will it be until the price returns to its initial value of $1.179 per gallon? B. The function f(x) = 0.194x x + 18 models the value of a share of stock in a computer camera company for a recent 12-month period. What was the lowest price of the camera stock? What was the greatest price for the 12-month period? If the function continues to accurately model the value of the stock, will the stock ever double its initial value of $18? When? C. The space shuttle uses solid rocket boosters (SRB) during the launch phase of its flight from Cape Canaveral. The SRBs burn for about two minutes, shut down, detach from the main rocket assembly, and fall back to Earth 140 miles downrange. Parachutes assist the ocean landing, beginning at an altitude of 20,000 feet. The SRBs are recovered and used again for a later launch. The function f(x) = x x 2, for x 115.5, describes the altitude (in miles) of the shuttle s SRBs since the launch (x is elapsed time in seconds). How long after launch do the SRBs splashdown? The SRBs separate from the shuttle after seconds. (3.11) How much longer do the SRBs continue to gain altitude? D. The United States annual trade balance with Mexico is modeled by the function f(x) = 2.75x x for the period (x = 0 for 1994). When f(x) > 0, the value of American exports exceeds imports from Mexico. According to the model, when is trade with Mexico in balance (f(x) = 0)? Algebra I Indicators p. 18

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