Algebra I: Unit 4 Revised 10/16 Unit 4 Linear Functions Name: 1 P a g e
CONTENTS 3.4 Direct Variation 3.5 Arithmetic Sequences 2.3 Consecutive Numbers Unit 4 Assessment #1 (3.4, 3.5, 2.3) 4.1 Graphing Linear Functions Review 4.2 Writing Equations in Slope-Intercept Form Test Review #1 Unit 4 Evaluation #1 (3.4, 3.5, 2.3, 4.1, 4.2) 4.3 Writing Equations in Point Slope and Standard Form 3.1 Graphing Equations by Finding Intercepts 4.4 Parallel and Perpendicular Lines Unit 4 Assessment #2 (4.3, 3.1, 4.4) 4.5 Scatter Plots 4.6 Linear Regression Test Review #2 Unit 4 Evaluation #2 (4.3, 3.1, 4.4, 4.5, 4.6) 2 P a g e
Direct Variation ~ Section 3.4 A. Slope and Constant of Variation 3 1. Name the constant of variation for y = x. 4 2. Name the constant of variation for y = -x. B. Write and solve a direct variation equation. 3. Suppose y varies directly as x, and y = 51 when x = 3. i. Write a direct variation equation that relates x and y. ii. Use the direct variation equation that you got above (i) to find the value of x when y = 63. 3 P a g e
C. Direct Variation Word Problem 4. The distance a jet travels varies directly as the number of hours it flies. A jet traveled 3420 miles in 6 hours. Recall that distance = rate x time or d = rt. i. Write a direct variation equation for the distance d flown in time t. ii. Graph the equation. iii. How long will it take the jet to fly 6500 miles? 4 P a g e
5. A hot- air balloon s height varies directly as the balloon s ascent time in minutes. It rises 350 feet in 5 minutes. i. Write a direct variation for the distance d ascended in time t. ii. Use your equation to find how many minutes it would take to ascend 2100 ft. iii. About how many minutes would it take to ascend 3500 feet? iv. Graph the equation. 5 P a g e
3-4 Practice Direct Variation Name the constant of variation for each equation. Graph each equation. 1. y = 2x 2. y = x 3. y = x Suppose y varies directly as x. Write a direct variation equation that relates x and y. Then solve. 4. If y = 7.5 when x = 0.5, find y when x = 0.3. 5. If y = 80 when x = 32, find x when y = 100. 6. If y = when x = 24, find y when x = 12. Write a direct variation equation that relates the variables. Then graph the equation. 7. MEASURE The width W of a 8. TICKETS The total cost C of tickets is rectangle is two thirds of the length l. $4.50 times the number of tickets t. 9. PRODUCE The cost of bananas varies directly with their weight. Miguel bought 3 pounds of bananas for $1.12. Write an equation that relates the cost of the bananas to their weight. Then find the cost of 4 pounds of bananas. 6 P a g e
Arithmetic Sequences as Linear Functions ~ Section 3.5 Ex 1. -4, -2, 0, 2,. is an example of an arithmetic sequence because the difference between the terms is constant. That constant is called the common difference. The common difference here is 2. A. Determine whether each sequence is an arithmetic sequence. Write yes or no and state the common difference. 1. -10, -7, -4, 1,. Common Difference: 2. -3, 1, 5, 9, Common Difference: 7 P a g e
B. Find the next three terms of each arithmetic sequence. 3. 0.02, 1.08, 2.14, 3.2,,, 4. -1/2, 0, ½, 1,,, Determining the nth Term of an Arithmetic Sequence Ex. 2 The arithmetic sequence 8, 11, 14, 17,...has a constant difference of +3. Each term in an arithmetic sequence can be expressed in terms of the first term called a 1 and the common difference d. Term Symbol Expression Numbers 1 st term a 1 a 1 8 2 nd term a 2 a 1 + d 11 or 8 + 1(3) 3 rd term a 3 a 1 + 2d 14 or 8 + 2(3) 4 th term a 4 a 1 + 3d 17 or 8 + 3(3) 5 th term a 5 a 1 + 4d 20 or 8 + 4(3)... Nth term a n a 1 + (n 1)d or 8 + (n-1)(3) Using the previous example and the formula: 8 + (n-1)(3). What is the value of the: 20 th term? 52 nd term? 75 th term? Formula for the nth term: a n = a 1 + (n 1)d where d is a positive integer. 8 P a g e
Find the stated term in the arithmetic sequences. Also write an equation for the nth term of the arithmetic sequence. 1. The 15 th term in 8, -10, -28, -46. a 1 = d = Equation= 15 th term = 2. The 20 th term in 16, 13, 10, 7. a 1 = d = Equation= 20 th term = 3. The 33 rd term in -12, -14, -16, -18. a 1 = d = Equation= 33 rd term = 4. Consider the arithmetic sequence: -12, -8, -4, 0. a. Write an equation for the nth term of the arithmetic sequence. Step 1: Write the first term: a 1 = Step 2: Write the common difference: d= Step 3: Write an equation following this form: a n = a 1 + (n 1)d b. Complete the following table. n (term #) 1 2 3 4 5 a n Coordinate point c. Find the 15 th term of the sequence above. d. Which term, n, of the sequence is equal to 32? [ a n = 32 ] 9 P a g e
5. Consider the arithmetic sequence: 3, -10, -23, -36. a. Write an equation for the nth term of the arithmetic sequence. Step 1: Write the first term: a 1 = Step 2: Write the common difference: d= Step 3: Write an equation following this form: a n = a 1 + (n 1)d c. Complete the following table. n (term #) 1 2 3 4 5 a n Coordinate point b. Find the 15 th term of the sequence above. c. Which term, n, of the sequence is equal to -114? [ a n = -114 ] Arithmetic Sequences as Functions Marisol is mailing invitations to her party. The arithmetic sequence $.42, $.84, $1.26, $1.68 represents the cost of postage. a. Write a function to represent this sequence and express it as a table. b. What is the domain? b. What is the range? 10 P a g e
3-5 Skills Practice Arithmetic Sequences as Linear Functions Determine whether each sequence is an arithmetic sequence. Write yes or no. Explain. 1. 4, 7, 9, 12,... 2. 15, 13, 11, 9,... 3. 7, 10, 13, 16,... 4. 6, 5, 3, 1,... 5. 5, 3, 1, 1,... 6. 9, 12, 15, 18,... 7. 10, 15, 25, 40,... 8. 10, 5, 0, 5,... Find the next three terms of each arithmetic sequence. 9. 3, 7, 11, 15,... 10. 22, 20, 18, 16,... 11. 13, 11, 9, 7... 12. 2, 5, 8, 11,... 13. 19, 24, 29, 34,... 14. 16, 7, 2, 11,... 15. 2.5, 5, 7.5, 10,... 16. 3.1, 4.1, 5.1, 6.1,... Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence. 17. 7, 13, 19, 25,... 18. 30, 26, 22, 18,... 19. 7, 4, 1, 2,... 11 P a g e
3-5 Practice Arithmetic Sequences as Linear Functions Determine whether each sequence is an arithmetic sequence. Write yes or no. Explain. 1. 21, 13, 5, 3,... 2. 5, 12, 29, 46,... 3. 2.2, 1.1, 0.1, 1.3,... 4. 1, 4, 9, 16,... 5. 9, 16, 23, 30,... 6. 1.2, 0.6, 1.8, 3.0,... Find the next three terms of each arithmetic sequence. 7. 82, 76, 70, 64,... 8. 49, 35, 21, 7,... 9.,,, 0,... 10. 10, 3, 4, 11... 11. 12, 10, 8, 6,... 12. 12, 7, 2, 3,... Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence. 13. 9, 13, 17, 21,... 14. 5, 2, 1, 4,... 15. 19, 31, 43, 55,... 16. BANKING Madison deposited $115.00 in a savings account. Each week thereafter, she deposits $35.00 into the account. a. Write a function to represent the total amount Madison has deposited for any particular number of weeks after his initial deposit. a. How much has Madison deposited 30 weeks after her initial deposit. 17. STORE DISPLAYS Tamika is stacking boxes of tissue for a store display. Each row of tissues has 2 fewer boxes than the row below. The first row has 23 boxes of tissues. a. Write a function to represent the arithmetic sequence. b. How many boxes will there be in the tenth row? 12 P a g e
Consecutive Numbers ~ Section 2.3 13 P a g e
Exercises. Write an equation for each problem and solve for every solution. 1. Find 4 consecutive integers that add to 378. Equation: Solutions: 2. Find 3 consecutive integers that add to 201. Equation: Solutions: 3. Find three consecutive EVEN integers whose sum is 132. Equation: Solutions: 4. Find four consecutive ODD integers whose sum is 176. Equation: Solutions: 5. The greater of two consecutive even integers is 6 less than three times the lesser. Find the integers. Equation: Solutions: 6. The lesser of two consecutive EVEN integers is 10 more than one-half the greater. Find the integers. Equation: Solutions: 7. Find a set of four consecutive positive integers such that the greatest integer in the set is twice the least integer in the set. Equation: Solutions: 8. Find four consecutive odd integers whose sum is -80. Equation: Solutions: 9. Determine each of three consecutive integers such that their sum is 90. Equation: Solutions: 10. Determine each of five consecutive integers such that their sum is 820. Equation: Solutions: 14 P a g e
Graphing Linear Functions Review ~ Section 4.1 1. Write an equation in slope-intercept form of the line with slope of ¼ and a y-intercept of -1. 2. Graph 3x + 4y = 8. (Solve for y first). Horizontal and Vertical Lines 3. Graph y = 5. 4. Graph x = -3 on the same coordinate plane as question 3. 5. Write an equation in slope-intercept form (y = mx + b) for the given graphs. i. ii. 15 P a g e
6. The ideal maximum heart rate for a 25-year-old who is exercising to burn fat is 117 beats per minute. For every five years older than 25, that ideal rate drops three beats per minute. i. Write a linear equation in the form y = mx + b to find the maximum heart rate for anyone over 25 who is exercising to burn fat. Let Y = ideal maximum heart rate for a 25-yr-old and X is the number of years older than 25. ii. Graph the equation and find the ideal maximum heart rate for a 55-year-old person exercising to burn fat. 16 P a g e
4-1 Skills Practice ~ Graphing Equations in Slope-Intercept Form Write an equation of a line in slope-intercept form with the given slope and y-intercept. 1. slope: 5, y-intercept: 3 2. slope: 2, y-intercept: 7 3. slope: 3, y-intercept: 2 4. slope: 4, y-intercept: 9 5. slope: 1, y-intercept: 12 6. slope: 0, y-intercept: 8 Write an equation in slope-intercept form for each graph shown. 7. 8. 9. Graph each equation. 10. y = x + 4 11. y = 2x 1 12. x + y = 3 13. VIDEO RENTALS A video store charges $10 for a rental card plus $2 per rental. a. Write an equation in slope-intercept form for the total cost c of buying a rental card and renting m movies. Video Store Rental Costs b. Graph the equation. c. Find the cost of buying a rental card and renting 6 movies. 17 P a g e
4-1 Practice ~Graphing Equations in Slope-Intercept Form Write an equation of a line in slope-intercept form with the given slope and y-intercept. 1. slope:, y-intercept: 3 2. slope:, y-intercept: 4 3. slope: 1.5, y-intercept: 1 4. slope: 2.5, y-intercept: 3.5 Write an equation in slope-intercept form for each graph shown. 5. 6. 7. Graph each equation. 8. y = x + 2 9. 3y = 2x 6 10. 6x + 3y = 6 11. WRITING Carla has already written 10 pages of a novel. She plans to write 15 additional pages per month until she is finished. a. Write an equation to find the total number of pages P written after any number of months m. Carla s Novel b. Graph the equation on the grid at the right. c. Find the total number of pages written after 5 months. 18 P a g e
Activity 1: Changing b in y = mx + b 4.1 Tech Lab ~ The Family of Linear Graphs Graph: y = x, y = x + 4, and y = x 2 in the standard viewing window. Enter these equations into Y 1, Y 2, and Y 3 after hitting the Y= button. 1A. Compare the graph of y = x + 4 and the graph of y = x - 2. Describe specifically how they are different and how they are the same. Different: Same: 1B. Compare the graph of y = x 2 and the graph of y = x. Describe specifically how they are different and how they are the same. Different: Same: Activity 2: Changing m in y = mx + b, positive values. Graph y = x + 4, y = 2x = 4, and y = 3 1 x + 4 in the standard viewing window. 2A. Compare the graph of y = x + 4 and the graph of y = 2x + 4. Describe specifically how they are different and how they are the same. Different: Same: 2B. Which is steeper, the graph of y = 3 1 x + 4 or the graph of y = x + 4? 19 P a g e
Writing Equations in Slope-Intercept Form ~ Section 4.2 20 P a g e
Slope-intercept form of a linear equation: y = mx + b m = slope b = y-intercept Given a point and the slope Example 1: Find the equation of a line which passes through (-2, 4) with a slope of 3. Step 1: Find the y-intercept b. Step 2: Now write the equation using b from step 1. Try these. Find the equation of the line which passes through the given point with the given slope. 1A. (-1, 2), slope = 5 1B. (4, -7), slope = -1 1C. (8, 1), slope = 1D. (0, -2), slope = -5 21 P a g e
Given two points on the line Example 2: Find the equation of a line which passes through (2,1) and (5, -8). Step 1: Find the slope. Step 2: Use either point to find the y-intercept Step 3: Write equation of the line. Try these. Write the equation of the line which goes through these points. 2A. (-1, 12) and (4, -8) 2B. (5, -8) and (-7, 0) 2C. (6, -8) and (0, 4) 2D. (4, -4) and (0, 0) 22 P a g e
Real-World Problem The table shows the number of domestic flights in the USA from 2012 to 2016. Write an equation that could be used to predict the number of flights if it continues to decrease at the same rate (use 2012 as the base year). Base year: This is the starting point of the data (think y-intercept). Year Flights (millions) 2012 11.19 2013 10.88 2014 10.57 2015 10.26 2016 9.95 Estimate how many flights will occur in 2017 if this pattern continues. Estimate how many flights will occur in 2025 if this pattern continues. Estimate how many flights will occur in 2050 if this pattern continues. Why is this estimate unrealistic? Why is the data itself unrealistic? 23 P a g e
4-2 Skills Practice ~ Writing Equations in Slope-Intercept Form Write an equation of the line that passes through the given point with the given slope. 1. 2. 3. 4. (1, 9); slope 4 5. (4, 2); slope 2 6. (2, 2); slope 3 7. (3, 0); slope 5 8. ( 3, 2); slope 2 9. ( 5, 4); slope 4 Write an equation of the line that passes through each pair of points. 10. 11. 12. 24 P a g e
13. (1, 3), ( 3, 5) 14. (1, 4), (6, 1) 15. (1, 1), (3, 5) 16. ( 2, 4), (0, 6) 17. (3, 3), (1, 3) 18. ( 1, 6), (3, 2) 19. INVESTING The price of a share of stock in XYZ Corporation was $74 two weeks ago. Seven weeks ago, the price was $59 a share. a. Write a linear equation to find the price p of a share of XYZ Corporation stock w weeks from now. b. Estimate the price of a share of stock five weeks ago. 20. WRITING Carla has already written 10 pages of a novel. She plans to write 15 additional pages per month until she is finished. a. Write an equation to find the total number of pages P written after any number of months m. Carla s Novel b. Graph the equation on the grid at the right. c. Find the total number of pages written after 5 months. 25 P a g e
4-2 Practice ~ Writing Equations in Slope-Intercept Form Write an equation of the line that passes through the given point and has the given slope. 1. 2. 3. 4. ( 5, 4); slope 3 5. (4, 3); slope 6. (1, 5); slope 7. (3, 7); slope 8. ; slope 9. (5, 0); slope 0 Write an equation of the line that passes through each pair of points. 10. 11. 12. 26 P a g e
13. (0, 4), (5, 4) 14. ( 4, 2), (4, 0) 15. ( 2, 3), (4, 5) 16. (0, 1), (5, 3) 17. ( 3, 0), (1, 6) 18. (1, 0), (5, 1) 19. DANCE LESSONS The cost for 7 dance lessons is $82. The cost for 11 lessons is $122. Write a linear equation to find the total cost C for l lessons. Then use the equation to find the cost of 4 lessons. 20. WEATHER It is 76 F at the 6000-foot level of a mountain, and 49 F at the 12,000-foot level of the mountain. Write a linear equation to find the temperature T at an elevation x on the mountain, where x is in thousands of feet. 27 P a g e
Writing Equations in Point-Slope and Standard ~ Section 4.3 28 P a g e
Point-Slope Form of a Linear Equation: y y 1 = m(x x 1 ) where m is the slope of the line and (x 1, y 1 ) is a point on the line. Example 1: Given a point and a slope Write an equation in point-slope form for the line that passes through (1, -6) with a slope of 7. Then graph the equation. 29 P a g e
Try these! Write the equation for a line which goes through the following point and has the given slope. Then graph the equation. 1A (-2, 1) and slope = -6 1B (3, 5) and slope = 1/3 30 P a g e
Example 2: Given two points Write the equation of the line in point-slope form if the line goes through (4, 8) and (-2, 9). Example 3: This equation is in point-slope form. Convert it to standard form: Ax + By = C. 2 y 1 ( x 5) 3 Example 4: 3 y 3 ( x 1) 2 This equation is in point-slope form. Convert it to slope-intercept form: y = mx + b. 4-3 Practice ~ Writing Equations in Point-Slope Form Write an equation in point-slope form for the line that passes through each point with the given slope. 1. (2, 2), m = 3 2. (1, 6), m = 1 3. ( 3, 4), m = 0 4. (1, 3), m = 5. ( 8, 5), m = 6. (3, 3), m = 31 P a g e
Write each equation in standard form. Ax + By = C 7. y 11 = 3(x 2) 8. y 10 = (x 2) 9. y + 7 = 2(x + 5) 10. y 5 = (x + 4) 11. y + 2 = (x + 1) 12. y 6 = (x 3) 13. y + 4 = 1.5(x + 2) 14. y 3 = 2.4(x 5) 15. y 4 = 2.5(x + 3) Write each equation in slope-intercept form. Y = mx + b 16. y + 2 = 4(x + 2) 17. y + 1 = 7(x + 1) 18. y 3 = 5(x + 12) 19. y 5 = (x + 4) 20. y = 3(x + ) 21. y = 2(x ) 32 P a g e
22. CONSTRUCTION A construction company charges $15 per hour for debris removal, plus a onetime fee for the use of a trash dumpster. The total fee for 9 hours of service is $195. a. Write the point-slope form of an equation to find the total fee y for any number of hours x. b. Write the equation in slope-intercept form. c. What is the fee for the use of a trash dumpster? 23. MOVING There is a daily fee for renting a moving truck, plus a charge of $0.50 per mile driven. It costs $64 to rent the truck on a day when it is driven 48 miles. a. Write the point-slope form of an equation to find the total charge y for a one-day rental with x miles driven. b. Write the equation in slope-intercept form. c. What is the daily fee? 33 P a g e
Graphing Equations by Finding Intercepts ~ Section 3.1 34 P a g e
A small pond had a leak and was losing 4 gallons of water per hour. The table shows the function relating the volume of water in the pond and the time in hours that the pond has been draining. Hours (x) Gallons or volume (y) 0 2000 125 1500 250 1000 375 500 500 0 1. Graph and label the data on the grid. You can see that the graph is linear. 2. The x-intercept is where the line crosses the x-axis. What is the x-intercept on the graph? Notice where the x-intercept appears in the chart. 3. The y-intercept is where the line crosses the y-axis. What is the y-intercept on the graph? Notice where the y-intercept appears in the chart. 4. The equation which models this problem is y = 2000 4x. You can check this by plugging in coordinate points from the table and confirm that the points make the equation true. a. Show that (0, 2000) from the table is a point on the line y = 2000 4x. b. Show that (250, 1000) from the table is a point on the line y = 2000 4x. c. Show that (2, 2300) is NOT a point on the line y = 2000 4x. 35 P a g e
X and Y Intercepts: There is another way to find the x- and y-intercepts if you don t have a table of values. Follow these steps. *TO FIND THE X-INTERCEPT, let y = 0 and solve for x. Try it with the equation y = 2000 4x. *TO FIND THE Y-INTERCEPT, let x = 0 and solve for y. Try it with the equation y = 2000 4x. Now graph: 36 P a g e
Practice: Now find the x- and y-intercepts for the equations below and use them to graph the equations on the grids. Ex. 1 y = 4 + 2x Ex. 2 x = 5y + 5 Ex. 3 x + y = 4 Ex. 4 x y = -3 37 P a g e
Practice: Now take each of these equations and write them in Standard Form: Ax + By = C. Identify A, B, and C. Then find the x- and y-intercepts. f. y = -3x + 5 g. 9y + 3x = 27 2 h. y = 8 + 2x i. y = x 1 3 4 5 j. y = x 2 k. y = x 3 5 7 38 P a g e
3-1 Skills Practice Graphing Linear Equations Determine whether each equation is a linear equation. Write yes or no. If yes, write the equation in standard form. 1. xy = 6 2. y = 2 3x 3. 5x = y 4 4. y = 2x + 5 5. y = 7 + 6x 6. y = 3 + 1 7. y 4 = 0 8. 5x + 6y = 3x + 2 9. = 1 Find the x- and y-intercepts of each linear function. 10. 11. 12. Graph each equation by making a table. 13. y = 4 14. y = 3x 15. y = x + 4 Graph each equation by using the x and y-intercepts. 16. x y = 3 17. 10x = 5y 18. 4x = 2y + 6 39 P a g e
3-1 Practice Graphing Linear Equations Determine whether each equation is a linear equation. Write yes or no. If yes, write the equation in standard form and determine the x- and y-intercepts. 1. 4xy + 2y = 9 2. 8x 3y = 6 4x 3. 7x + y + 3 = y 4. 5 2y = 3x 5. = 1 6. = 7 Graph each equation. 7. y = 2 8. 5x 2y = 7 9. 1.5x + 3y = 9 40 P a g e
10. COMMUNICATIONS A telephone company charges $4.95 per month for long distance calls plus $0.05 per minute. The monthly cost c of long distance calls can be described by the equation c = 0.05m + 4.95, where m is the number of minutes. a. Find the y-intercept of the graph of the equation. b. Graph the equation. c. If you talk 140 minutes, what is the monthly cost? 11. MARINE BIOLOGY Killer whales usually swim at a rate of 3.2-9.7 kilometers per hour, though they can travel up to 48.4 kilometers per hour. Suppose a migrating killer whale is swimming at an average rate of 4.5 kilometers per hour. The distance d the whale has traveled in t hours can be predicted by the equation d = 4.5t. d. Graph the equation. b. Use the graph to predict the time it takes the killer whale to travel 30 kilometers. 41 P a g e
Parallel and Perpendicular Lines ~ Section 4.4 42 P a g e
On graph the coordinate plane below, graph the following lines. Try to graph 5 points for each line. Use the concept of rise/run to find more points. 2 3 Line 1: y x 1 Line 2: y 6 x 3 2 2 3 Line 3: y x 4 Line 4: y x 2 3 2 2 Line 5: y x 5 3 43 P a g e
#1 Make some observations about how these lines look. What is the relationship between how the lines look and the equations? #2 Re-write each equation below in slope-intercept form. Then write an equation which is parallel to this line and also an equation which is perpendicular to this line. LINE PARALLEL LINE PERPENDICULAR LINE a. 2x + 4y = 6 b. 2x + 7y = 14 c. -4x 2y = 16 d. x + 10y = 5 e. 6x + 2y = 18 f. 12x 2y = 15 44 P a g e
4.4 ~ Parallel Lines --------------------------------------------------------------------------------------------------------------------------------- Parallel Lines Lines in the same plane that do not intersect are called parallel lines. Parallel lines have the same slope. --------------------------------------------------------------------------------------------------------------------------------- Example. 1 Write an equation in slope-intercept form (y = mx + b) for the line that passes through (1, 3) and is parallel to the graph of y= 2x 4. Graph both lines. Example. 2 Write an equation in slope-intercept form (y = mx + b) for the line that passes through (4, -1) and is parallel to the graph of y = 1 4 x + 5. Graph both lines. 45 P a g e
4-4 Skills Practice ~ Parallel and Perpendicular Lines Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the graph of the given equation. 1. 2. 3. 4. (3, 2), y = 3x + 4 5. ( 1, 2), y = 3x + 5 6. ( 1, 1), y = x 4 7. (1, 3), y = 4x 1 8. ( 4, 2), y = x + 3 9. ( 4, 3), y = 6 46 P a g e
4.4 ~ Perpendicular Lines Lines that intersect at right angles are called perpendicular lines. The slopes of perpendicular lines are opposite reciprocals. If the slope of the line is 3/4, then the slope of the line which is perpendicular to it is -4/3. Example. 3 Write an equation in slope intercept form for the line that passes through (1, 3) and is perpendicular to the graph of y = -2x + 4. Then graph it. Step 1: Find the slope. Step 2: Find the new slope for the perpendicular line. Step 3: Now find the equation of the line which is perpendicular to the original line. This new line should go through (1, 3). Example. 4 Write an equation in slope-intercept form for the line that passes through (-2, -3) and is perpendicular to the graph of 2x + 3y = 12. Then graph it. Step 1: Solve 2x + 3y = 12 for y. Then find its slope. Step 2: Find the new slope for the perpendicular line. Step 3: Now find the equation of the line which is perpendicular to the original line. This new line should go through (-2, -3). 47 P a g e
4-4 Skills Practice ~ continued Determine whether the graphs of the following equations are parallel or perpendicular. Explain. 11. y = x + 3, y = x, 2x 3y = 8 12. y = 4x, x + 4 y = 12, 4x + y = 1 Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the given equation. 13. ( 3, 2), y = x + 2 14. (4, 1), y = 2x 4 15. ( 1, 6), x + 3y = 6 16. ( 4, 5), y = 4x 1 17. ( 2, 3), y = x 4 18. (0, 0), y = x 1 48 P a g e
4-4 Practice Parallel and Perpendicular Lines Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the graph of the given equation. 1. (3, 2), y = x + 5 2. ( 2, 5), y = 4x + 2 3. (4, 6), y = x + 1 4. (5, 4), y = x 2 5. (12, 3), y = x + 5 6. (3, 1), 2x + y = 5 7. ( 3, 4), 3y = 2x 3 8. ( 1, 2), 3x y = 5 9. ( 8, 2), 5x 4y = 1 10. ( 1, 4), 9x + 3y = 8 11. ( 5, 6), 4x + 3y = 1 12. (3, 1), 2x + 5y = 7 49 P a g e
Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the given equation. 13. ( 2, 2), y = x + 9 14. ( 6, 5), x y = 5 15. ( 4, 3), 4x + y = 7 16. (0, 1), x + 5y = 15 17. (2, 4), x 6y = 2 18. ( 1, 7), 3x + 12y = 6 19. ( 4, 1), 4x + 7y = 6 20. (10, 5), 5x + 4y = 8 21. (4, 5), 2x 5y = 10 22. (1, 1), 3x + 2y = 7 23. ( 6, 5), 4x + 3y = 6 24. ( 3, 5), 5x 6y = 9 50 P a g e
Scatter Plots ~ Section 4.5 Data with two variables are called bivariate data. A scatter plot shows the relationship between a set of data with two variables, graphed as ordered pairs on a coordinate plane. New Vocabulary Bivariate data: Line of fit: Scatter plot: Linear interpolation: A set of data which contains two variables A line which closely approximates the scatter plot for a set of data A set of bivariate data graphed as ordered pairs on a coordinate plane The process of using a linear equation to predict values inside the range of a set of data Example 1: The table shows the number of hours spent exercising per week and the age of a random sample of seven people. Age 18 26 32 38 52 59 Hours 10 5 2 3 1.5 1 Step 1: Identify the independent and the dependent variables. 51 P a g e
Step 2: Make the scatter plot and describe the correlation. Step 3: Draw a line of fit that passes close to the points. It will not go through all the points but will show the trend or general direction. Step 4: Determine a line of fit for the data. Write the slope-intercept form of the equation for the line of fit. Choose two points to find your slope. Then interpret your slope s meaning. Step 5: Use the line of fit to predict the number of hours exercised per week by a 15-year-old. 52 P a g e
Example 2: The table shows the largest vertical drops of nine roller coasters in the USA and the number of years after 1988 that they were opened. Yrs since 1988 Vertical drop (ft) 1 3 5 8 12 12 12 13 15 151 155 225 230 306 300 255 255 400 Step 1: Identify the independent and the dependent variables. Step 2: Make the scatter plot and describe the correlation. Step 3: Draw a line of fit that passes close to the points. It will not go through all the points but will show the trend or general direction. Step 4: Determine a line of fit for the data. Write the slope-intercept form of the equation for the line of fit. Choose two points to find your slope. Then interpret your slope s meaning. Step 5: Use the line of fit to predict the vertical drop in a roller coaster built 30 years after 1988. 53 P a g e
EXAMPLE 3: Use the scatter plot. a. Use points (5, 71205) and (9, 68611) to write the slope-intercept form of an equation for the line of fit shown in the scatter plot. b. Predict the attendance at a game in 2020. c. Can you use the equation to make a decision about the average attendance in any given year in the future? Explain. 4-5 Skills Practice ~ Scatter Plots and Lines of Fit Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. 1. 2. 3. 4. 54 P a g e
5. BASEBALL The scatter plot shows the average price of a major-league baseball ticket from 1997 to 2006. a. Determine what relationship, if any, exists in the data. Explain. b. Use the points (1998, 13.60) and (2003, 19.00) to write the slope-intercept form of an equation for the line of fit shown in the scatter plot. c. Predict the price of a ticket in 2009. 55 P a g e
4.6 Linear Regression for Scatter Plots Calculator Instructions LINEAR REGRESSION This is a process done on a graphing calculator that will find the line of best fit for a scatter plot. Calculator Steps: To graph your data: 1) 2 nd, MEM, 7, 1, 2 2) to enter data: STAT, ENTER, then enter data into L 1 and L 2. 3) to plot the scatter plot: 2 nd, STAT PLOT, ENTER, ENTER, GRAPH 4) adjust window settings: WINDOW, ENTER 5) press GRAPH To find the list of best fit 1) STAT, CALC, LinReg, calculate or ENTER 2) round your a and b values 3) write equation in form y = mx + b 4) type your equation into Y=, then press GRAPH Example 1: The table shows the money made by movies in the USA. Graph and find the line of best fit. Let the year 2000 be represented as 0. year 2000 2001 2002 2003 2004 2005 2006 $ billions 7.48 8.13 9.19 9.35 9.27 8.95 9.25 A. What is the line of best fit? B. Using this equation and assuming that the trend continues, predict how much money will be made in the year 2020. Calculator Steps: To graph your data: 1) 2 nd, MEM, 7, 1, 2 2) to enter data: STAT, ENTER, then enter data into L 1 and L 2. 3) to plot the scatter plot: 2 nd, STAT PLOT, ENTER, ENTER, GRAPH 4) adjust window settings: WINDOW, ENTER 5) press GRAPH To find the line of best fit 1) STAT, CALC, LinReg (which is probably choice 4 ), choose calculate or ENTER 2) Round your a and b values to nearest hundreth 3) Write equation in form y = mx + b 4) Type your equation into Y=, then press GRAPH 56 P a g e
Example 2: A campground keeps a record of the number of campsites rented the week of July 4 for several years. Let x be the number of years since 2000. year 2002 2003 2004 2005 2006 2007 2008 2009 2010 Sites rented 34 45 42 53 58 47 57 65 59 Hint: When entering data, you may enter 2002 data as 2, 2003 data as 3, etc. A. What is the line of best fit? B. Using this equation and assuming that the trend continues, predict how many campsites will be rented in 2020. 4-6 Skills Practice ~ Regression and Median-Fit Lines Write an equation of the regression line for the data in each table below. Then find the correlation coefficient. 1. SOCCER The table shows the number of goals a soccer team scored each season since 2005. Year 2005 2006 2007 2008 2009 2010 Goals Scored 42 48 46 50 52 48 2. PHYSICAL FITNESS The table shows the percentage of seventh grade students in public school who met all six of California s physical fitness standards each year since 2002. Year 2002 2003 2004 2005 2006 Percentage 24.0% 36.4% 38.0% 40.8% 37.5% Source: California Department of Education 57 P a g e
3. TAXES The table shows the estimated sales tax revenues, in billions of dollars, for Massachusetts each year since 2004. Year 2004 2005 2006 2007 2008 Tax Revenue 3.75 3.89 4.00 4.17 4.47 Source: Beacon Hill Institute a. Find an equation for the median-fit line. b. How many diapers should SureSave anticipate selling in 2011? 4. PURCHASING The SureSave supermarket chain closely monitors how many diapers are sold each year so that they can reasonably predict how many diapers will be sold in the following year. Year 2006 2007 2008 2009 2010 Diapers Sold 60,200 65,000 66,300 65,200 70,600 a. Find an equation for the median-fit line. b. How many diapers should SureSave anticipate selling in 2011? 5. FARMING Some crops, such as barley, are very sensitive to how acidic the soil is. To determine the ideal level of acidity, a farmer measured how many bushels of barley he harvests in different fields with varying acidity levels. Soil Acidity (ph) 5.7 6.2 6.6 6.8 7.1 Bushels Harvested 3 20 48 61 73 a. Find an equation for the regression line. b. According to the equation, how many bushels would the farmer harvest if the soil had a ph of 10? c. Is this a reasonable prediction? Explain. 58 P a g e
Chapter 4 Test Review 1. Write an equation in slope-intercept form for each graph shown. (Lesson 4.1) a. b. 2. Graph each equation. (Lesson 4.1) a. y = 3x -1 b. y = x 2 3. The population of the United States is projected to be 320 million by the year 2010. Between 2010 and 2050, the population is expected to increase by about 2.5 million per year. (Lesson 4.1) a. Write an equation to find the population P in any year x between 2010 and 2050. b. Graph the equation on the grid at the right. Projected United States Population c. Use the equation to find the population in 2050. 59 P a g e
4. Write an equation in slope- intercept form of a line that passes through (4, 2) and has a slope of. (Lesson 4.2) 5. Write an equation in slope- intercept form of a line that passes through the points ( 3, 4) and (5, 8). (Lesson 4.2) 6. Write an equation in point-slope form for the line that passes through the point (-2, 7) with a slope of m = (Lesson 4.3) 7. Write the equation y 6 = (x + 2) in standard form. (Lesson 4.3) 8. Write the equation y + 1 = 5(x + 3) in slope-intercept form. (Lesson 4.3) 9. Find an equation of the line that has a y-intercept of 4 that is parallel to the line 2x + 4y = -8. (Lesson 4.4) 10. Write an equation in slope-intercept form for the line that passes through the (1, 3) and is perpendicular to the line y = x + 4 (Lesson 4.4) 60 P a g e
11. The table shows the number of students per computer in Easton High School for certain school years from 1996 to 2008. (Lesson 4.5-4.6) Year 1996 1998 2000 2002 2004 2006 2008 Students per Computer 26 20 16 11 7.5 4.3 3.8 a. Draw a scatter plot and determine what relationship exists, if any. b. Draw a line of fit for the scatter plot. c. Write the slope-intercept form of an equation for the line of fit. d. Use your graphing calculator to find the line regression for the data. 61 P a g e