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Contents Lesson 5-1Slope Lesson 5-2Slope and Direct Variation Lesson 5-3Slope-Intercept Form Lesson 5-4Writing Equations in Slope-Intercept Form Lesson 5-5Writing Equations in Point-Slope Form Lesson 5-6Geometry: Parallel and Perpendicular Lines Lesson 5-7Statistics: Scatter Plots and Lines of Fit

Lesson 1 Contents Example 1Positive Slope Example 2Negative Slope Example 3Zero Slope Example 4Undefined Slope Example 5Find Coordinates Given Slope Example 6Find a Rate of Change

Example 1-1a Find the slope of the line that passes through (–3, 2) and (5, 5). Let and Substitute.

Example 1-1b Simplify. Answer:The slope is

Example 1-1c Find the slope of the line that passes through (4, 5) and (7, 6). Answer:

Example 1-2a Find the slope of the line that passes through (–3, –4) and (–2, –8). Substitute. Let and

Example 1-2b Answer:The slope is –4. Simplify.

Example 1-2c Find the slope of the line that passes through (–3, –5) and (–2, –7). Answer: –2

Let and Example 1-3a Find the slope of the line that passes through (–3, 4) and (4, 4). Substitute.

Example 1-3b Answer:The slope is 0. Simplify.

Example 1-3c Find the slope of the line that passes through (–3, –1) and (5, –1). Answer: 0

Let and Example 1-4a Find the slope of the line that passes through (–2, –4) and (–2, 3). Answer:Since division by zero is undefined, the slope is undefined.

Example 1-4b Find the slope of the line that passes through (5, –1) and (5, –3). Answer:undefined

Example 1-5a Find the value of r so that the line through ( 6, 3 ) and ( r, 2 ) has a slope of Slope formula Substitute. Subtract.

Example 1-5b Find the cross products. Simplify. Add 6 to each side. Answer:Simplify.

Example 1-5c Find the value of p so that the line through ( p, 4 ) and ( 3, –1 ) has a slope of Answer: –5

Example 1-6a Travel The graph to the right shows the number of U.S. passports issued in 1991, 1995, and Find the rates of change for and Use the formula for slope. millions of passports years

Example 1-6b : Substitute. Simplify. Answer: The number of passports issued increased by 1.9 million in a 4-year period for a rate of change of 475,000 per year.

Example 1-6c : Substitute. Simplify. Answer: Over this 4-year period, the number of U.S. passports issued increased by 1.4 million for a rate of change of 350,000 per year.

Example 1-6d Explain the meaning of slope in each case. Answer: For , on average, 475,000 more passports were issued each year than the last. For , on average, 350,000 more passports were issued each year than the last.

Example 1-6e How are the different rates of change shown on the graph? Answer: There is a greater rate of change from than from Therefore, the section of the graph for has a steeper slope.

Example 1-6f Airlines The graph shows the number of airplane departures in the United States in recent years. a.Find the rates of change for and Answer:240,000 per year; 180,000 per year

Example 1-6g b.Explain the meaning of the slope in each case. Answer:For , the number of airplane departures increased by about 240,000 flights each year. For , the number of airplane departures increased by about 180,000 flights each year.

Example 1-6h c.How are the different rates of change shown on the graph? Answer:There is a greater vertical change for than for Therefore, the section of the graph for has a steeper slope.

End of Lesson 1

Lesson 2 Contents Example 1Slope and Constant of Variation Example 2Direct Variation with k > 0 Example 3Direct Variation with k < 0 Example 4Write and Solve a Direct Variation Equation Example 5Direct Variation Equation

Example 2-1a Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. Slope formula Simplify. Answer: The constant of variation is 2. The slope is 2.

Example 2-1b Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. Slope formula Simplify. Answer: The constant of variation is –4. The slope is –4.

Example 2-1c Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. a. Answer:constant of variation: 4 ; slope: 4

Example 2-1d Answer:constant of variation: –3 ; slope: –3 Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. b.

Example 2-2a Step 1Write the slope as a ratio. Step 2Graph (0, 0). Step 3From the point (0, 0), move up 1 unit and right 1 unit. Draw a dot. Step 4Draw a line containing the points.

Example 2-2b Answer:

Example 2-3a Step 1Write the slope as a ratio. Step 2Graph (0, 0). Step 3From the point (0, 0), move down 3 units and right 2 units. Draw a dot. Step 4Draw a line containing the points.

Example 2-3b Answer:

Example 2-4a Suppose y varies directly as x, andwhen Write a direct variation equation that relates x and y. Find the value of k. Direct variation formula Replace y with 9 and x with –3. Divide each side by –3.

Example 2-4b Simplify. Answer:Therefore,

Example 2-4c Use the direct variation equation to find x when Direct variation equation Answer:Therefore, when Replace y with 15. Divide each side by –3. Simplify.

Example 2-4d Suppose y varies directly as x, andwhen a.Write a direct variation equation that relates x and y. b.Use the direct variation equation to find x when Answer: –15 Answer:

Example 2-5a Travel The Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours. Write a direct variation equation to find the distance driven for any number of hours. WordsThe distance traveled is 330 miles, and the time is 5.5 hours. Variables Distance equals rate times time. Equation 330 mi r5.5h

Example 2-5b Solve for the rate. Answer:Therefore, the direct variation equation is Original equation Divide each side by 5.5. Simplify.

Example 2-5c Graph the equation. The graph ofpasses through the origin with a slope of 60. Answer:

Example 2-5d Estimate how many hours it would take to drive 600 miles. Original equation Replace d with 600. Divide each side by 60. Simplify. Answer:At this rate, it will take 10 hours to drive 600 miles.

Example 2-5e Dustin ran a 26-mile marathon in 3.25 hours. a.Write a direct variation equation to find the distance ran for any number of hours. b.Graph the equation. Answer:

Example 2-5f c. Estimate how many hours it would take to jog 16 miles. Answer: 2 hours

End of Lesson 2

Lesson 3 Contents Example 1Write an Equation Given Slope and y-Intercept Example 2Write an Equation Given Two Points Example 3Graph an Equation in Slope-Intercept Form Example 4Graph an Equation in Standard Form Example 5Write an Equation in Slope-Intercept Form

Example 3-1a Write an equation of the line whose slope is and whose y-intercept is –6. Slope-intercept form Replace m with and b with –6. Answer:

Example 3-1b Write an equation of the line whose slope is 4 and whose y-intercept is 3. Answer:

Example 3-2a Write an equation of the line shown in the graph. Step 1You know the coordinates of two points on the line. Find the slope. Let

Example 3-2b Simplify. The slope is 2. Step 2The line crosses the y-axis at (0, –3). So, the y-intercept is –3. Step 3Finally, write the equation. Slope-intercept form Replace m with 2 and b with –3. Answer:The equation of the line is

Example 3-2c Write an equation of the line shown in the graph. Answer:

Example 3-3a Graph Step 1The y-intercept is –7. So graph (0, –7). Step 2The slope is 0.5 or From (0, –7), move up 1 unit and right 2 units. Draw a dot. Step 3Draw a line connecting the points. y = 0.5x – 7

Example 3-3b Graph Answer:

Example 3-4a Graph Step 1Solve for y to find the slope-intercept form. Original equation Subtract 5x from each side. Simplify. Divide each side by 4.

Example 3-4b Divide each term in the numerator by 4. Answer: Step 2The y-intercept of is 2. So graph (0, 2).

Example 3-4c Step 3The slope is From (0, 2), move down 5 units and right 4 units. Draw a dot. Step 4Draw a line connecting the points. 5x + 4y = 8

Example 3-4d Graph Answer:

Example 3-5a Health The ideal maximum heart rate for a 25-year- old who is exercising to burn fat is 117 beats per minute. For every 5 years older than 25, that ideal rate drops 3 beats per minute. Write a linear equation to find the ideal maximum heart rate for anyone over 25 who is exercising to burn fat. WordsThe rate drops 3 beats per minute every 5 years, so the rate of change isbeats per minute each year. The ideal maximum heart rate for a 25-year-old is 117 beats per minute.

Example 3-5b VariablesLet R = the ideal heart rate. Let a = years older than 25. Equation ideal rate Ideal rate ofyears older for 25- rateequalschangetimes than 25plusyear-old. Ra117 Answer:

Example 3-5c Graph the equation. The graph passes through (0, 117) with a slope of Answer:

Example 3-5d Find the ideal maximum heart rate for a person exercising to burn fat who is 55 years old. The age 55 is 30 years older than 25. So, Ideal heart rate equation Replace a with 30. Simplify. Answer:The ideal heart rate for a 55-year- old person is 99 beats per minute.

Example 3-5e The amount of money spent on Christmas gifts has increased by an average of $150,000 ($0.15 million) per year since Consumers spent $3 million in a.Write a linear equation to find the average amount spent for any year since Answer:where D is the amount of money spent in millions of dollars, and n is the number of years since 1986

Example 3-5f The amount of money spent on Christmas gifts has increased by an average of $150,000 ($0.15 million) per year since Consumers spent $3 million in b.Graph the equation. Answer:

Example 3-5g The amount of money spent on Christmas gifts has increased by an average of $150,000 ($0.15 million) per year since Consumers spent $3 million in c.Find the amount spent by consumers in Answer:$4.95 million

End of Lesson 3

Lesson 4 Contents Example 1Write an Equation Given Slope and One Point Example 2Write an Equation Given Two Points Example 3Write an Equation to Solve a Problem Example 4Linear Extrapolation

Example 4-1a Write an equation of a line that passes through (2, –3) with slope Step 1The line has slope To find the y-intercept, replace m with and ( x, y ) with ( 2, –3 ) in the slope-intercept form. Then, solve for b.

Example 4-1b Slope-intercept form Replace m with, y with –3, and x with 2. Multiply. Subtract 1 from each side. Simplify.

Example 4-1c Step 2Write the slope-intercept form using Slope-intercept form Replace m with and b with –4. Answer:The equation is

Example 4-1d CheckYou can check your result by graphing on a graphing calculator. Use the CALC menu to verify that it passes through (2, –3).

Example 4-1e Write an equation of a line that passes through (1, 4) and has a slope of –3. Answer:

Example 4-2a Multiple-Choice Test Item The table of ordered pairs shows the coordinates of two points on the graph of a function. Which equation describes the function? AB CD xy –3–4 –2–8 Read the Test Item The table represents the ordered pairs (–3, –4) and (–2, –8).

Example 4-2b Solve the Test Item Step 1Find the slope of the line containing the points. Let and. Slope formula Simplify.

Example 4-2c Step 2You know the slope and two points. Choose one point and find the y-intercept. In this case, we chose (–3, –4). Slope-intercept form Replace m with –4, x with –3, and y with –4. Multiply. Subtract 12 from each side. Simplify.

Example 4-2d Step 3Write the slope-intercept form using Answer:The equation isThe answer is D. Slope-intercept form Replace m with –4 and b with –16.

Example 4-2e xy –13 26 Multiple-Choice Test Item The table of ordered pairs shows the coordinates of two points on the graph of a function. Which equation describes the function? AB CD Answer:B

Example 4-3a Economy In 2000, the cost of many items increased because of the increase in the cost of petroleum. In Chicago, a gallon of self-serve regular gasoline cost $1.76 in May and $2.13 in June. Write a linear equation to predict the cost of gasoline in any month in 2000, using 1 to represent January. Explore You know the cost of regular gasoline in May and June. PlanLet x represent the month and y represent the cost of gasoline that month. Write an equation of the line that passes through (5, 1.76) and (6, 2.13).

Example 4-3b SolveFind the slope. Let and. Slope formula Simplify.

Example 4-3c Choose (5, 1.76) and find the y-intercept of the line. Slope-intercept form Replace m with 0.37, x with 5, and y with Multiply. Subtract 1.85 from each side. Simplify.

Example 4-3d Slope-intercept form Write the slope-intercept form using and Replace m with 0.37 and b with –0.09. Answer:The equation is

Example 4-3e ExamineCheck your result by substituting the coordinates of the point not chosen, (6, 2.13), into the equation. Original equation Replace y with 2.13 and x with 6. Multiply. Simplify.

Example 4-3f The average cost of a college textbook in 1997 was $ In 2000, the average cost was $ Write a linear equation to estimate the average cost of a textbook in any given year since Let x represent years since Answer:

Example 4-4a Economy The Yellow Cab Company budgeted $7000 for the July gasoline supply. On average, they use 3000 gallons of gasoline per month. Use the prediction equation where x represents the month and y represents the cost of one gallon of gasoline, to determine if they will have to add to their budget. Explain. Original equation Replace x with 7. Simplify.

Example 4-4b Answer:If gas increases at the same rate, a gallon of gasoline will cost $2.50 in July gallons at this price is $7500, so they will have to add $500 to their budget.

Example 4-4c A student is starting college in 2004 and has saved $400 to use for textbooks. Use the prediction equation where x is the years since 1997 and y is the average cost of a college textbook, to determine whether he will have enough money for 5 textbooks. Answer:If the cost of textbooks increases at the same rate, the average cost will be $82.15 in Five textbooks at this price is $410.75, so he will not have enough money.

End of Lesson 4

Lesson 5 Contents Example 1Write an Equation Given Slope and a Point Example 2Write an Equation of a Horizontal Line Example 3Write an Equation in Standard Form Example 4Write an Equation in Slope-Intercept Form Example 5Write an Equation in Point-Slope Form

Example 5-1a Write the point-slope form of an equation for a line that passes through (–2, 0) with slope Point-slope form Simplify. Answer:The equation is

Example 5-1b Write the point-slope form of an equation for a line that passes through (4, –3) with slope –2. Answer:

Example 5-2a Write the point-slope form of an equation for a horizontal line that passes through (0, 5). Point-slope form Simplify. Answer:The equation is

Example 5-2b Write the point-slope form of an equation for a horizontal line that passes through (–3, –4). Answer:

Example 5-3a Writein standard form. In standard form, the variables are on the left side of the equation. A, B, and C are all integers. Original equation Multiply each side by 4 to eliminate the fraction. Distributive Property

Answer:The standard form of the equation is 3x – 4y = 20. Example 5-3b Subtract 3x from each side. Simplify.

Example 5-3c Writein standard form. Answer: 2x – y = –11

Example 5-4a Writein slope-intercept form. In slope-intercept form, y is on the left side of the equation. The constant and x are on the right side. Original equationDistributive Property Add 5 to each side.

Example 5-4b Simplify. Answer:The slope-intercept form of the equation is

Example 5-4c Writein slope-intercept form. Answer:

Example 5-5a The figure shows trapezoid ABCD with basesand Write the point-slope form of the lines containing the bases of the trapezoid.

Example 5-5b Step 1First find the slopes of and Slope formula

Example 5-5c Step 2You can use either point for ( x 1, y 1 ) in the point-slope form. Method 1 Use (–2, 3). Method 2 Use (4, 3).

Example 5-5d Method 1 Use (1, –2). Method 2 Use (6, –2). Answer:The point-slope form of the equation containing The point-slope form of the equation containing

Example 5-5e Write each equation in standard form. Original equation Add 3 to each side. Answer: Simplify. Original equation Subtract 2 from each side. Answer: Simplify.

Example 5-5f The figure shows right triangle ABC. a.Write the point-slope form of the line containing the hypotenuse b.Write the equation in standard form. Answer:

End of Lesson 5

Lesson 6 Contents Example 1Parallel Line Through a Given Point Example 2Determine Whether Lines are Perpendicular Example 3Perpendicular Line Through a Given Point Example 4Perpendicular Line Through a Given Point

Example 6-1a Write the slope-intercept form of an equation for the line that passes through (4, –2) and is parallel to the graph of The line parallel tohas the same slope, Replace m with and (x 1, y 1 ) with (4, -2) in the point-slope form.

Example 6-1b Point-slope form Replace m with y with – 2, and x with 4. Simplify. Distributive Property Subtract 2 from each side.

Example 6-1c Write the equation in slope- intercept form. Answer:The equation is

Example 6-1d CheckYou can check your result by graphing both equations. The lines appear to be parallel. The graph of passes through (4, –2).

Example 6-1e Write the slope-intercept form of an equation for the line that passes through (2, 3) and is parallel to the graph of Answer:

Example 6-2a Geometry The height of a trapezoid is measured on a segment that is perpendicular to a base. In trapezoid ARTP, and are bases. Can be used to measure the height of the trapezoid? Explain.

Example 6-2b Find the slope of each segment. Slope of

Example 6-2c Answer: The slope of and is 1 and the slope of is not perpendicular to and, so it cannot be used to measure height.

Example 6-2d The graph shows the diagonals of a rectangle. Determine whether is perpendicular to Answer: The slope of is and the slope of is Sinceis not perpendicular to

Example 6-3a Write the slope-intercept form for an equation of a line that passes through (4, –1) and is perpendicular to the graph of Step 1Find the slope of the given line. Original equation Subtract 7x from each side. Simplify.

Example 6-3b Divide each side by –2. Simplify.Step 2The slope of the given line isSo, the slope of the line perpendicular to this line is the opposite reciprocal ofor

Example 6-3c Step 3Use the point-slope form to find the equation. Point-slope form and Simplify. Distributive Property

Example 6-3d Subtract 1 from each side. Simplify. Answer: The equation of the line is

Example 6-3e CheckYou can check your result by graphing both equations on a graphing calculator. Use the CALC menu to verify that passes through (4, –1).

Example 6-3f Write the slope-intercept form for an equation of a line that passes through (–3, 6) and is perpendicular to the graph of Answer:

Example 6-4a Write the slope-intercept form for an equation of a line perpendicular to the graph of and passes through (0, 6). Step 1Find the slope of Original equation Subtract 5x from each side. Simplify.

Example 6-4b Divide each side by 2. Simplify.Step 2The slope of the given line isSo, the slope of the line perpendicular to this line is the opposite reciprocal ofor

Example 6-4c Step 3Substitute the slope and the given point into the point-slope form of a linear equation. Then write the equation in slope-intercept form. Point-slope form Replace x 1 with 0, y 1 with 6, and m with Distributive Property Answer: The equation of the line is

Example 6-4d Write the slope-intercept form for an equation of a line perpendicular to the graph of and passes through the x -intercept of that line. Answer:

End of Lesson 6

Lesson 7 Contents Example 1Analyze Scatter Plots Example 2Find a Line of Fit Example 3Linear Interpolation

Example 7-1a Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe it. The graph shows average personal income for U.S. citizens. Answer:The graph shows a positive correlation. With each year, the average personal income rose.

Example 7-1b Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe it. The graph shows the average students per computer in U.S. public schools. Answer: The graph shows a negative correlation. With each year, more computers are in the schools, making the students per computer rate smaller.

Example 7-1c Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe it. a.The graph shows the number of mail-order prescriptions. Answer:Positive correlation; with each year, the number of mail-order prescriptions has increased.

Example 7-1d Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe it. Answer:no correlation b.The graph shows the percentage of voter participation in Presidential Elections.

Example 7-2a The table shows the world population growing at a rapid rate. YearPopulation (millions)

Example 7-2b Draw a scatter plot and determine what relationship exists, if any, in the data. Let the independent variable x be the year and let the dependent variable y be the population (in millions). The scatter plot seems to indicate that as the year increases, the population increases. There is a positive correlation between the two variables.

Example 7-2c Draw a line of fit for the scatter plot. No one line will pass through all of the data points. Draw a line that passes close to the points. A line is shown in the scatter plot.

Example 7-2d Write the slope-intercept form of an equation for the line of fit. The line of fit shown passes through the data points ( 1850, 1000 ) and ( 1998, 5900 ). Step 1Find the slope. Slope formula Let and Simplify.

Example 7-2e Step 2Use m = 33.1 and either the point-slope form or the slope-intercept form to write the equation. You can use either data point. We chose (1850, 1000). Point-slope formSlope-intercept form Answer:The equation of the line is.

Example 7-2f CheckCheck your result by substituting (1998, 5900) into Line of fit equation Subtract. The solution checks. Replace x with 1998 and y with Multiply.

Example 7-2g The table shows the number of bachelor’s degrees received since Years Since Bachelor’s Degrees Received (thousands) Source: National Center for Education Statistics

Example 7-2h a.Draw a scatter plot and determine what relationship exists, if any, in the data. Answer: The scatter plot seems to indicate that as the number of years increases, the number of bachelor’s degrees received increases. There is a positive correlation between the two variables.

Example 7-2i b.Draw a line of best fit for the scatter plot. c.Write the slope-intercept form of an equation for the line of fit. Answer: Using (4, 1136) and (10, 1184),

Example 7-2a Use the prediction equation where x is the year and y is the population (in millions), to predict the world population in Original equation Replace x with Simplify. Answer: 6,296,000,000

Example 7-3b Use the equationwhere x is the years since 1988 and y is the number of bachelor’s degrees (in thousands), to predict the number of bachelor’s degrees that will be received in Answer: 1,240,000

End of Lesson 7

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