F8-8 Finding the -intercept from Ordered Pairs Pages 5 Standards: 8.F.A., 8.F.B. Goals: Students will find the -intercept of a line from a set of ordered pairs. Prior Knowledge Required: Can add, subtract, and divide integers Can find the slope and the -intercept of a line from its graph, its equation, or its table Vocabular: ordered pair, quadrant, rise, run, slope, -coordinate, -coordinate, -intercept Review plotting ordered pairs. On a grid on the board, mark the point (, ) and ASK: What is the -coordinate of this point? () Write on the board: = Repeat with the -coordinate. ASK: How do we write this as an ordered pair? (as (, )) Repeat with a point at (, ), emphasizing the importance of the order of the coordinates. Mark a point in the second quadrant, such as (, ), and show students how to identif the signs of its coordinates. SAY: The point is on the negative side of the -ais and the positive side of the -ais, so the signs for the coordinates of the ordered pair will be (, +). Repeat with a point in the fourth quadrant. Eercises: Plot the points on a grid. A (5, ) B (, ) C (, 5) D (, ) E (, ) F (, ) G (, ) H (, ) Answers: 5 H F E A 5 0 5 D G B C 5 L- Teacher s Guide for AP Book 8. Unit Functions
(MP.7) Finding the -intercept from ordered pairs b graphing. Eplain to students that, when the plot two ordered pairs on a grid, the can join the points to make a line and then find the -intercept b etending the line. As an eample, plot two points, (, ) and (, 5), on a grid on the board. Ask a volunteer to join them to make a line and then etend the line to find the -intercept. (-intercept: ; see completed graph below) 0 Eercises: Graph the two ordered pairs and join them to make a line. Etend the line to find the -intercept. a) (, ), (, ) b) (, ), (, ) c) (, ), (, ) Answers: a) b) c) 0 -intercept: Finding the -intercept from ordered pairs without graphing. SAY: You just learned how to find the -intercept from ordered pairs b graphing. But the graphing method onl works well when two points are close to the -ais. Write on the board: (, 5) (8, ) SAY: You would need a big grid to plot these ordered pairs, draw the line that goes through the points, and etend it to cross the -ais. However, as in the previous lesson, ou can use a table to find the -intercept. Draw on the board: slope 5 8 0 -intercept: 0 -intercept: Teacher s Guide for AP Book 8. Unit Functions L-5
ASK: What is the run? PROMPT: The run is the change in. ( ) ASK: What is the rise? (the change in, +) Write the run and rise in the circles, as shown below: slope 5 8 + ASK: What is the slope? (/( )= ) Multipl the -coordinate in each row b the slope and complete the table. Draw an arrow underneath the table, from the second column to the third column. The table should look like this: slope 5 + 8 8 ASK: What must ou add to the second column to get? (9) Write Add 9 underneath the arrow and SAY: The -intercept is 9. (MP.7) Eercises: For the line that goes through the given points in the previous eercise, find the -intercept without graphing. a) (, ), (, ) b) (, ), (, ) c) (, ), (, ) Solutions: a) slope b) slope c) slope + + + Add Subtract rise + rise - slope = = = slope = = = run + run - -intercept: -intercept: Subtract rise - slope = = =- run + -intercept: Summarize finding the -intercept. Remind students that the can find the -intercept using the following methods:. From a graph: Students can plot two points on a graph, join the points, and etend the line to cross the -ais. The intersection point is the -intercept.. From an equation: B substituting 0 for, students can find the -intercept. Point out this trick: in an equation with the form = m + b, b is the constant term and is the -intercept.. From a table: Students can write the coordinates in a table and find the run, rise, and slope. For each value, multipl b the slope in the second column. The number that students must add (or subtract) to the second column to get, in the third column, is the -intercept.. From ordered pairs: Students can write the ordered pairs in a table and find the -intercept from the table. Where = 0, is the -intercept. L- Teacher s Guide for AP Book 8. Unit Functions
Note to students that the graphing method is the general method for finding the -intercept. Even if students have the equation or table or ordered pairs the might choose to graph to find the -intercept. The other methods are options for students who do not want to graph. (MP., MP., MP.5) Eercises: a) Four linear functions are represented in different was below. Find the -intercept for each. A. B. (, ), (, ), (, ) C. D. = + 7 7 5 5 b) Which has the highest -intercept? c) Which has a negative -intercept? d) Which goes through the origin? Answers: a) A:, B: 0, C: +.5, D: +; b) C; c) A; d) B 0 Etension Without graphing, find the -intercept of the line that goes through the given points. a) (, ), (, ) b) (, ), (, ) c) (, ), (, ) Answers: a), b) /, c) Teacher s Guide for AP Book 8. Unit Functions L-7
F8-9 Writing an Equation of a Line Using the Slope and -intercept Pages 7 9 Standards: 8.F.A., 8.F.B. Goals: Students will write the equation of a line in slope-intercept form. Prior Knowledge Required: Can add, subtract, and divide integers Can find the slope and the -intercept from a graph, table of values, or ordered pairs Vocabular: coefficient, constant term, rise, run, slope, slope-intercept form, -intercept Review slopes between an two points on a straight line are equal. Draw on the board: 5 C 5 0 5 B A 5 ASK: What is the run from A to B? (+) What is the rise from A to B? (+) What is the slope of AB? (/ or ) Have a volunteer find the slope of BC. (run = +, rise = +8, slope = 8/ or ) Then have another volunteer find the slope of AC. (run = +5, rise = +0, slope = 0/5 or ) ASK: Is the slope the same between an two points on a straight line? (es) Point to the line and SAY: The slope is equal between ever set of two points on a straight line, and that means that, if we found the slope using a table of values for that linear function, we would get the same answer. Draw on the board: a) b) c) 5 5 5 5 L-8 Teacher s Guide for AP Book 8. Unit Functions
Ask a volunteer to find the slope for each linear function. (a) run = +, rise = +, slope = / = ; b) run = +, rise = +8, slope = 8/ = ; c) run = +5, rise = +0, slope = 0/5 = ) ASK: For the graph we did earlier, what is the -intercept? () SAY: The coordinates of the -intercept are (0, ). Ask a volunteer to find the slope from a table with two points, the -intercept (0, ) and C (, 5). (see solution below) 0 + + 5 rise run = +, rise = +, slope = = = run ASK: Among all the points we have used to find the slope on the graph, which two -coordinates were the easiest to use to find the slope? ( equals 0 and equals ) SAY: When ou use = 0 and =, the run is and the rise is the slope of the line. Eercises: Make a table with = 0 and = for the given equation. Find the slope and -intercept. a) = + b) =.5 c) = +.5 Answers: a) 0 5 -intercept: run = +, rise = + slope = / = b) 0 0.5 -intercept: run = +, rise = +.5 slope =.5/ =.5 c) 0.5 0.5 -intercept:.5 run = +, rise = slope = / = Remind students that, in an equation such as = +, the coefficient of the (in this case, ) is the slope of the line and the constant term (in this case, +) is the -intercept of the line. Write on the board: + = + SAY: I had a student who said that the -intercept of this equation is +. ASK: How can we check if that is right or wrong? (b replacing with 0 and finding ) Write on the board: + = (0) + + = = = Teacher s Guide for AP Book 8. Unit Functions L-9
ASK: What is the -intercept? (+) Eplain to students that, to find the -intercept in an equation, first the need to rearrange the equation to isolate. The constant term in the rearranged equation is the -intercept. Write on the board: + = + = + = + So the -intercept is +. Eercises: Find the slope and the -intercept of the line from the equation. a) = + 0.5 b) = c) = d) = 5 + Sample solution: d) = 5 +, so = + 9, so the slope is + and the -intercept is +9 Answers: a) slope:, -intercept: +0.5; b) slope:, -intercept: ; c) slope:, -intercept: (MP., MP., MP.) Writing an equation in slope-intercept form. SAY: To write an equation for a line, multipl b the slope, add the -intercept, and write the result equal to. Write on the board: slope =, -intercept = 5, so the equation is = + 5 Write on the board: = m + b SAY: If m is the slope of a line and b is the -intercept, then = m + b. Mathematicians call this wa of writing the equation the slope-intercept form of the line. Eercises: For a line with the given slope and -intercept, write the equation of the line in slope-intercept form. a) slope =, -intercept = b) slope = 0.5, -intercept = c) slope =, -intercept =.5 d) slope =, -intercept = e) slope =, -intercept = - f) slope = -, -intercept = - 5 Answers: a) = +, b) = 0.5, c) = +.5, d) = +, e) =--, f) =- - 5 Writing the equation of a line using the -intercept and a point. Eplain to students that the can write the equation of a line using the -intercept and another point. SAY: Whenever there are two points, ou can find the slope of the line that goes through the points. Write on the board: -intercept =, P (, ) L-0 Teacher s Guide for AP Book 8. Unit Functions
SAY: The -intercept is, which means the line crosses the -ais at (0, ). Let s name the -intercept point A. Write on the board: A (0, ), P (,) SAY: With two points A and P, ou can find the slope. Then ou can use the slope and the -intercept to write the equation of the line. Ask a volunteer to find the slope and write the equation of the line, as shown below. run = +, rise = +, equation: = + rise slope = = = run Eercises:. Find the slope and the -intercept. Write the equation of the line. a) A (0, ), B (, ) b) A (, ), B (0, ) c) A (, ), B (0, 0) Answers: a) run = +, rise = +, slope = / =, -intercept =, equation: = b) run =, rise =, slope = / =, -intercept =, equation: = c) run = +, rise = +, slope = / =.5, -intercept = 0, equation: =.5.Check our answers to Eercise b substituting one point in each equation. Sample answer: a) =, = (0), = Writing the equation of a line using the graph of the line. Remind students that, to find the slope of a line from a graph, it is easier to use two points with integer coordinates. Then, remind students that the can also etend the line to cross the -ais to find the -intercept. Draw on the board: 0 B A ASK: What is the run from A to B? (+) What is the rise from A to B? (+) What is the slope of AB? (/ or ) Etend the line to find the -intercept (), then write the equation of the line on the board in slope-intercept form, as shown below: = + Teacher s Guide for AP Book 8. Unit Functions L-
Eercises: (MP., MP.). Etend the line to find the -intercept. Mark two points with integer coordinates and find the slope of the line. a) b) c) 0 0 0 Answers: a) -intercept =, run = +, rise = +, slope = / = b) -intercept =, run = +, rise = +, slope = / = c) -intercept = 0.5, run = +, rise =, slope = / = 0.5. Write the equation for each line in Eercise in slope-intercept form. Answers: a) = +, b) =, c) = 0.5 + 0.5 Etensions. a) Find the -intercept of the line = + b replacing with 0. b) Find a formula for finding the -intercept of the line = m + b b replacing with 0. Solutions: a) = +, so 0 = +, =, / =, so = b) = m + b, so 0 = m + b, b = m, b/m =. Determine m and b, then use the formula that ou found in Etension.b) to find the -intercept of the line. a) = + b) =.5 c) = + d) = Answers: a) m =, b =, -intercept = / = ; b) m =.5, b = 0, -intercept = 0/.5 = 0; c) m =, b =, -intercept = / = ; d) m =, b =, -intercept = / = 0.5 L- Teacher s Guide for AP Book 8. Unit Functions
F8-0 Comparing Linear Functions Pages 0 Standards: 8.F.A. Goals: Students will compare linear functions represented in different was b graphing. Prior Knowledge Required: Can graph a line using its equation Can find the absolute value of a number Vocabular: absolute value, greater slope, slope-intercept form, steeper, steepness (MP., MP.) Comparing lines b graphing. Draw on the board: A. = + B. = + 0 0 Ask a volunteer to find the slope of each line. (, ) ASK: Which line has the greater slope? (B) Draw on the board to the right of graphs A and B: C. = + 7 0 ASK: What is the slope of this line? ( / = ) Point to the three graphs on the board and SAY: I would like to compare these three graphs. Ask a volunteer to order all the lines from smallest to largest slope. (C, B, A) ASK: Which has the greatest slope? (B) SAY: Suppose that ou want to compare steepness, or how steep the are. Point out that comparing the steepness of different lines is like comparing the slopes of different mountains; students will need to look at two points on each graph and think about which line would be the most difficult to walk up. Teacher s Guide for AP Book 8. Unit Functions L-
ASK: Which graph has the steepest slope? (C) Does a greater slope alwas mean a steeper slope? (no) SAY: The sign (positive or negative) for a slope indicates which direction the line goes in, but the absolute value indicates whether a slope will be more or less steep than another one. Remind students that the absolute value of a number is its distance from zero, regardless of whether it is to the left or right of zero; for eample, = and + =. Ask a volunteer to order the three slopes graphed earlier b absolute value, from smallest to largest. (A, B, C) SAY: A steeper slope means a higher absolute value for the slope. Eercises: Graph the two functions on the same grid. Determine which function has the greater slope and which is steeper. a) = +.5 b) = + c) = = = + = + Answers: a) b) c) 0 = +.5 = greater slope: = steeper: = 0 = + = + greater slope: = + steeper: = + = + 0 = greater slope: = steeper: = You can ask some higher-level questions if all students answered the previous eercises. ASK: If two lines are parallel, what do ou know about their slopes? (their slopes are equal) What do ou know about how steep the are? (the are equall steep) If two lines intersect, what do ou know about their slopes? (the have different slopes) If two lines are equall steep, do the have the same slope? (it depends for eample, two lines each with slope have equal slopes and are equall steep, but a line with slope and another line with slope are equall steep, but have different slopes.) Eercises: a) The information needed to find a slope is given in four different was. Find the slopes of each of the given linear functions. A. B. C. = D. (, 0), (0, ), (, ) 5 0 7 0 b) Which two have the same slope? L- Teacher s Guide for AP Book 8. Unit Functions
c) Which has the greatest slope? d) Which has the steepest slope? e) Write the equation in slope-intercept form for A, B, and D. Answers: a) slopes: A:, B:.5, C:, D: ; b) C and D; c) A; d) B; e) A: = + 7, B: =.5 +, D: = + Etensions. You can find the intersection of two lines b graphing. Graph both functions on the same grid and find the coordinates of the intersection point. a) = + b) = + c) = = = + = + Answers: a) b) c) 0 = + = intersection point: (, ) intersection point: (, ) intersection point: (, ). A line passes through the points A (, ) and B (, 9) and has slope. Find the value of. 9 - Solution: run = = 5, rise = 9, so slope = =. After cross multipling, -5 9 = 0 and = 9 + 0 = 9. A line passes through A (, ) and has slope. Will the line pass through point B (7, )? Eplain. Answer: No, the line will not pass through point B. As the slope is, the run from point A to point B would be 7 ( ) = 0 and the rise would be 0 (because 0/0 = ), which means the line would pass through (7, + 0) or (7, ) rather than B (7, ). NOTE: The following etension is for advanced students onl. 0 = + = + 0 = = +. A line passes through A (, ) and B (k, k). a) Can ou determine a line for k =? b) What will the slope be for all other values of k? Solutions: a) No, k = gives point B (, ), which has the same coordinates as point A, and ou cannot use a single point to determine a line. b) rise = k ( ) = k + = (k ), run = k = (k ), so the slope for all other -( k -) - values of k will be = =-.5 ( k -) Teacher s Guide for AP Book 8. Unit Functions L-5
F8- Using the Equation of a Line to Solve Word Problems Pages Standards: 8.F.B. Goals: Students will write the equation of a function to model a linear relationship between two quantities. Students will understand that the -intercept is the initial value or flat fee. Prior Knowledge Required: Can add, subtract, multipl, and divide integers Can graph a linear function on grid paper Can find the slope and the -intercept of a line from a graph, equation, or table of values Vocabular: flat fee, ordered pairs, rise, run, slope, -intercept (MP.) Solving word problems using linear functions. Eplain to students that man realworld problems involve linear relationships. Draw on the board: 5 0 55 50 5 0 5 0 5 0 5 0 5 0 5 7 8 9 SAY: The graph shows the cost of renting an e-bike (electric bike) for a $0 flat fee plus $5 per hour. ASK: How much do ou have to pa to rent the e-bike for hour? ($5) For 5 hours? ($5) Ask a volunteer to find the slope of the line. (5) ASK: How are the slope and the cost related? (the slope is the cost per hour) What is the -intercept of the line? (0) How are the -intercept and the cost related? (the -intercept is the flat fee or initial cost) If Beth paid $50 for renting an e-bike, how man hours did she rent the e-bike for? (8) Eplain to students that the can use the graph to find the answer, or the can use the equation of the line. SAY: The slope is 5 and the -intercept is 0. Write the equation of the line on the board: = 5 + 0 L- Teacher s Guide for AP Book 8. Unit Functions
SAY: This is the equation for the straight line. Ask a volunteer to find the value for when = 50. ASK: In other words, how man hours can be paid for with $50? (see solution below) 50 = 5 + 0 50 0 = 5 0 = 5, so = 8 Eplain to students that using algebra seems to take longer for small numbers, but for large numbers, it might be the best or onl solution. Solve the word problems in the following eercises with the class. Eercises: (MP.). A phone compan charges a $5 flat fee, plus $0.0 for each international tet message. a) Create a table of values for the cost of sending,, and international tet messages. b) Plot the ordered pairs from the table of values in part a) on a grid paper. c) Etend the line to find the -intercept. d) Find the run, rise, and slope of the line. e) Write an equation to represent the cost of sending international tet messages. f) Substitute = in the equation to find the cost of sending international tet messages. g) Ben has $0. What is the maimum number of international tet messages he can send? Solutions: a) Number of Cost ($) Messages 5.0 5.0 5.0 b) Cost of International Tet Messages Cost ($) 5.0 5.0 5.0 5.0 5.00 0 Number of Messages c) -intercept = 5.00 d) run =, rise = 0.0, slope = 0. e) = 0.0 + 5.00 f) = 0.0() + 5.00 = 5.0 g) 0 = 0.0 + 5.00, 0 5 = 0.0, 5 = 0.0, so = 50. Ben can send 50 international tet messages with $0. Teacher s Guide for AP Book 8. Unit Functions L-7
(MP.). A balloon is released from the top of a tower. The graph shows the height of the balloon over time.,000 900 800 700 00 B 500 00 00 A 00 00 0 0 0 0 50 0 Time (seconds) a) What is the -intercept of the line? b) What is the height of the tower? c) What is the slope of the line? d) What is the speed of the balloon? e) Write the equation of the line. f) What is the height of the balloon after minute? g) How long does it take for the balloon to reach a height of 000 m? Answers: a) 00 b) 00 m c) 0 d) 0 m/s e) = 0 + 00, stands for seconds and shows the height of the balloon after seconds f),00 m g) 95 seconds Height (m) See Questions on AP Book 8., pp. for more practice. Etension (MP.) A sports arena will host a new event soon. The arena manager estimates that if tickets are priced at $7.50 each, then 500 tickets will be sold per da, and if tickets are priced at $0.00 each, then 50 tickets will be sold per da. a) Fill in the table, then plot the ordered pairs from the table of values on a grid. Ticket Price ($) Estimated Ticket Sales b) What is the rate of change relating estimated ticket sales to ticket price? c) Write an equation for the line. d) Use the equation to estimate how man tickets will sell in a da if each ticket costs $9.00. e) Use the equation to estimate the ticket price needed to sell 0 tickets per da. L-8 Teacher s Guide for AP Book 8. Unit Functions
Answers: a) Ticket Price ($) Estimated Ticket Sales 7.50 500 0 50 Estimated Tickets 500 50 00 50 00 7.5 8.0 8.5 9.0 9.5 0.0 Ticket Price ($) b) run =.5, rise = 50, slope = 50/.5 = 0 c) = 0 + 950 d) = 0(9) + 950 = 0 e) 0 = 0 + 950, so = 8.5 Teacher s Guide for AP Book 8. Unit Functions L-9
F8- Describing Graphs Pages Standards: 8.F.B.5 Goals: Students will describe the relationship between two quantities b analzing a graph. Students will sketch a graph to represent a verball described linear function. Prior Knowledge Required: Can write an equation of a line from its graph Vocabular: decreasing lines, increasing lines, speed, velocit Materials: BLM Immigration to the United States, 900 000 (p. L-57) BLM Mountain Climbers (p. L-58) Review increasing and decreasing lines. Eplain to students that the graph of a line represents important and useful information about the relationship between two quantities. Remind students of the terms increasing line and decreasing line and what the mean. SAY: For an increasing line, both run and rise are positive, so the slope of an increasing line is positive. When the slope of a line is positive, then the line is increasing and goes from bottom left to top right on a grid. SAY: For eample, Kell is saving for a new bike; she has $0 alread and will save $5 per week; describe the graph that shows the mone Kell saves and time. (sample answer: the line has a positive slope and goes from bottom left to top right) ASK: When will she have enough mone for a bike that costs $50? (after 5. weeks of saving, or after about weeks) SAY: In contrast, a decreasing line has a negative slope and goes from top left to bottom right on a grid. Draw on the board: Graph A Graph B Graph C 7 5 0 7 5 0 7 5 0 ASK: Is graph A increasing or decreasing? (increasing) What about graph B? (decreasing) Can ou sa graph C is increasing? (no, because it decreases after = ) Eplain to students L-50 Teacher s Guide for AP Book 8. Unit Functions
that graph C is neither increasing nor decreasing, but can still give useful information about the relationship between two quantities. SAY: Graph C, for eample, might show the gallons of water in a bathtub over four minutes. In the first minute after turning on the tap, we could run three gallons of water into the tub, but then we turn off the tap and open the drain, at which point the tub takes three minutes to empt. Describing graphs. Solve the following word problem as an eample with the class. Draw on the board: Distance from Home (km) 5 B A C D O E 0 0 0 0 0 50 0 Time (min) SAY: Kim starts at her home and eercises for 0 minutes. The graph shows the elapsed time in 0-minute intervals and Kim s distance from home. Kim starts b running. ASK: How far did Kim run in the first 0 minutes? ( km) In the net 0 minutes? ( km) What is the run from O to A? (0 min) What is the rise from O to A? ( km) Write on the board: rise slope = = = 0. km/min run 0 Point to the unit (km/min) and SAY: Speed is the rate of distance traveled in a certain time. Kim runs km in 0 minutes, so her speed is 0. km/min. As a class, find Kim s speed in each 0 minute interval. (OA: 0. km/min, AB: 0. km/min, BC: 0. km/min, CD: 0 km/min, DE: 0. km/min) Eplain to students the difference between the speed in AB and BC. SAY: The AB line segment shows that Kim runs at a speed of 0. km/min awa from home. But at point B, she turns toward home and runs at a speed of 0. km/min toward home. Both speeds are equal, but the are in different directions. Later, in high school, ou will learn that velocit is the speed with the direction considered; in this case, ou ma sa the velocit during AB is 0. km/min and the velocit during BC is 0. km/min. ASK: Which line segment has a slope equal to 0? (CD) What is Kim s speed in the CD line segment? (0) Point to the graph and SAY: You can guess Kim had stopped for 0 minutes to rest. (MP.) Eercise: Complete BLM Immigration to the United States, 900 000. Answers:. a) decade ending at 90 and decade ending at 000; b) decade ending at 90 c) decade ending at 980 d) i), ii), iii), iv) +, v) +, vi) +, vii) 0, viii) +, i) + e) viii (decade ending at 990) f) iii (decade ending at 90) Teacher s Guide for AP Book 8. Unit Functions L-5
g) vii (decade ending at 980) Bonus: possible answers could mention the Great Depression in the United States and economic depressions in other areas around the world (MP.) Sketching graphs. Eplain to students that, in the previous section, the described the relationship between two quantities when the graph is given, but, in this section, the have to do the reverse. SAY: Now ou will sketch a graph to show the relationship between two quantities. Distribute BLM Mountain Climbers and have students complete the BLM as an eample. (see answers below). a),500 m b) Time p.m. p.m. p.m. :0 p.m. 5 p.m. p.m. Height (m),000,500,500,700,00,00,700 c) 5,000,800 Height (meters),00,00,00,000 0 5 Time (hours from noon) d) i) slope: 50, speed: 50 m/h, ii) slope: 0, speed: 0 m/h, iii) slope: 00, speed: 00 m/h, iv) slope: 00, speed: 00 m/h downward, v) slope: 0, speed, 0 m/h, vi) slope: 00, speed: 00 m/h e) part i) from noon to p.m. Etension Julie gets on her bike for a ride that includes a hill. First, Julie accelerates for 0 seconds until she reaches a constant speed of 0 m/s. Julie maintains her speed for 0 seconds. She then starts to ride uphill, at which point her speed slows considerabl for 0 seconds. When she gets to the top of the hill, she stops for 0 seconds. Julie then starts downhill and reaches a speed of m/s after 5 seconds. Julie slows down over 5 seconds until she stops on the other side of the hill. Sketch a graph that shows the relationship between Julie s speed and the time, from the time she gets on her bike until she stops on the other side of the hill. Label each ais with the appropriate variable. L-5 Teacher s Guide for AP Book 8. Unit Functions
Answer: Speed (meters per second) 0 8 0 0 0 0 0 50 0 70 80 90 00 Elapsed Time (seconds) Teacher s Guide for AP Book 8. Unit Functions L-5
NAME Immigration to the United States, 900 000 DATE. The graph shows the number of new immigrant arrivals in the United States in each ten-ear interval from 90 000 (90 90, 9 90, and so on). For eample, approimatel 9 million people immigrated to the US in the decade from 90 90. The numbers are rounded to the nearest million. a) In what decades did the highest number of people immigrate to the US? b) In what decade did the lowest number of people immigrate to the US? c) In what decade did the number of immigrants not change from the previous decade? New Immigrant Arrivals (millions) 0 9 8 7 5 900 0 0 0 0 50 0 70 80 90 000 Decade Ending d) Label the line segments with Roman numerals. Find the slope of each line segment in millions per decade. i) ii) iii) iv) v) vi) vii) viii) i) COPYRIGHT 05 JUMP MATH: TO BE COPIED. CC EDITION e) Which line segment has the greatest slope? f) Which line segment has the steepest slope? g) Which line segment has slope equal to 0? Bonus Wh do ou think there was a low number of immigrants from 9 to 90? Blackline Master Functions Teacher s Guide for AP Book 8. L-57
NAME Mountain Climbers DATE. A group of mountain climbers starts to climb a mountain at :00 p.m. (noon). The climbers start from a height of,000 meters above sea level. The climb 500 meters in hours and then rest for one hour. Then the climb another 00 meters in an hour, but because of poor weather conditions the then climb downhill 00 meters in half an hour and rest for another half an hour. Finall, the climbers take another path and climb 00 meters in hour and set up their camp at that height. a) What is the height of the mountain climbers at :00 pm? b) Fill the table with the heights of the mountain climbers. Time p.m. p.m. p.m. p.m. :0 p.m. 5 p.m. p.m. Height (m),000 c) Sketch a graph using the table in part b). 5,000,800 Height (m),00,00,00,000 0 5 Time (hours from noon) d) Label the line segments with Roman numerals. Find the slope of each line segment to find the climbers speed during each part of their climb. i) ii) iii) iv) v) vi) e) In what period of time does the graph have the steepest slope? COPYRIGHT 05 JUMP MATH: TO BE COPIED. CC EDITION L-58 Blackline Master Functions Teacher s Guide for AP Book 8.