Chapter 2 – Linear RelationshipsVocabulary
1) Linear Equation
2) Steepness
3) Variable
4) Coefficient
5) Standard Form
6) Slope-Intercept Form
7) Rate of Change
8) Slope
9) Starting Value
10) ∆ x
11) ∆ y
12) Unit Rate
13) Constant
2.1.1 & 2.1.2: Seeing Growth in Linear Representations / Slope
I Can ____________________________________________________________________.
2-1 a. Draw figures 0 & 4 Color in the new tiles in each figure with a colored pencil.
Figure 0 Figure 4
b. By how much is tile Pattern A growing?
Where are the tiles being added with each new figure?
c. What would Figure 100 look like for Pattern A? Describe it in words.
How many tiles would be in the 100th figure?
d. Write an equation that relates the figure number, x, to the number of tiles, y.
2-2 a. Draw figures 0 & 4 Color in the new tiles in each figure with a colored pencil.
Figure 0 Figure 4
b. By how much is tile Pattern B growing?
Where are the tiles being added with each new figure?
c. What would Figure 100 look like for Pattern B? Describe it in words.
How many tiles would be in the 100th figure?
d. Write an equation that relates the figure number, x, to the number of tiles, y.
2-3. The growth of a tile Pattern C is represented by the equation y = 3x + 1.
a. Fill in the table for Pattern C.
Figure #x 0 1 2 3 4
# of Tilesy
b. By how many tiles is Pattern C growing? What is the starting value?
c. Where do you look in the table to see the growth and starting value? d. Where do you look in the equation to see the growth and starting value?
2-5. a. Draw Figures 0 and 4 for this pattern.
Figure 0 Figure 4
b. Write an equation for the number of tiles in this pattern. Use color to show where the numbers in your equation appear in the tile pattern. Use x for the figure number, and y for the number of tiles in the figure.
c. Make a table for the equation you wrote in part (b).
Figure #x
# of Tilesy
d. What is the same about this pattern and Pattern C? What is different? What would those similarities and differences look like in a tile pattern?
e. What do the similarities and differences in part (d) look like in the equations?
f. What do the similarities and differences look like in the table?
2-11. Write an equation that represents the tile pattern in the table below. Figure # 0 1 2 3 4# of Tiles 2 7 12 17 22
Equation: _____________________2-12. Does the relation in the table above appear to be a function? Function Not a
Function
If so, write the equation in function notation ___________________________
2-13. Answer the questions for each of the graphs below:
2-14. The graph below shows a line for a tile pattern. How is the line growing? That is, how many tiles are added each time the figure number is increased by 1? Explain how you found your answer.
PRACTICE – Tell the slope and y intercept of each line and then write the equation in slope intercept form. (y = mx+b)
Slope: ____________ Slope: ____________ Slope: ____________Y-Int: ____________ Y-Int: ____________ Y-Int: ____________Equation: _______________ Equation: _______________ Equation: _____________
Growth: ________________Tiles in Fig. 0 : ______________Equation: _________________Function: yes or noFunction notation: ___________
Growth: _________________Tiles in Fig. 0 : ______________Equation: ________________Function: yes or noFunction notation: ____________
Growth: _________________Tiles in Fig. 0 : ______________Equation: __________________Function: yes or noFunction notation: ____________
Slope: ____________ Slope: ____________ Slope: ____________Y-Int: ____________ Y-Int: ____________ Y-Int: ____________Equation: _______________ Equation: _______________ Equation: _____________
Slope: ____________ Slope: __________ Y-Int: ____________ Y-Int: ____________Equation: _______________ Equation: _______________
2.1.3 & 2.1.4: Steepness / y=mx+b
I Can _____________________________________________________________________.
2-25 a. Which line from the graph is the steepest?
Which is the least steep? b. Does line D have a positive or negative slope?
What is the ∆y for line D? c. Label the slope triangles for each line and then
tell the slope.
Line A: _________ Line B: ________Line C: _________ Line D: ________
d. How is slope related to steepness of a graph?
2-26.
a. Which is the steepest line?
Which is steeper, line B or line C?
b. Draw slope triangles for the lines
using the points on each line.
Label Δx and Δy for each.
c. Match each line with its slope using the list below. Note: There are more slopes than lines.
m = 6 m = 2 m = m = m = 0 m = m = −5 m =
d. Viewed left to right, in what direction would a line with slope point? e. Viewed left to right, in what direction would a line with slope point?
How would it be different from the line in part (d)?
2-27. Graph a line to match each description below. Label each line.
a. y=35x−1
b. A line with Δx = 4 and Δy = −6.
c. f(x) = x − 7
d. A line that has Δy = 3 and Δx = 0.
2-36. Equations for linear patterns can all be written in the form y = mx + b.
a. When writing equations for tile patterns, what did x represent? What did y
represent?
b. What do m and b represent in a linear pattern like the tile patterns?
c. What effect does m have on a graph of the line? What effect does b
have?
2-37. Graph the following lines based on the info you have been given. If you do not have enough information to graph the line, write N/I next to the problem
a.Line A goes through the point (2, 5).
b.Line B has a slope of –3 and goes through the origin. c. Line C goes through points (–3, –2) and (3, 10). d.Line D has the following table.
x 2 3 4 5y 1 3.5 6 8.5
e.Line E grows by 4. f. Line F goes through the point (8, –1) and has a slope of –. g.Line G has the following table.
x −2 −1 0 1 2 3y 1 3 9 27
2-37 Graph
2-38. FINDING THE SLOPE OF A LINE WITHOUT GRAPHING (given 2 points or ∆x & ∆y) a. ∆y = 27 ∆x = -8b. ∆y = 3 ∆x = 15c. ∆y = 7 ∆x = 0d. Horizontal ∆= 6 Vertical ∆= 0e. Between (5,28) and (64,12)
f. Between (-3,2) and (5,-7)
2.2.2: Rate of Change (The Big Race)
I Can _____________________________________________________________________.
In the first heat, Leslie, Kristin, and Evie rode tricycles toward the finish line. Leslie began at the starting line and rode at a constant rate of 2 meters every
second. Kristin got an 8-meter head start and rode 2 meters every 5 seconds. Evie rode 5 meters every 4 seconds and got a 6-meter head start.
a. Graph and then write an equation in terms of x and y for the distance each rider travels. Let x represent time in seconds and y represent distance in meters. (scale by 2)
Leslie:
Kristen:
______________
Evie:
______________
b. After how many seconds did Leslie catch up to Evie?
How far were they from the starting line when Leslie caught up to Evie?
Confirm your answer algebraically and explain how to use your graph to justify your answer.
Slope Formula: Given 2 points
c. The winner of this heat will race in the final Big Race. If the race is 20 meters long, who won? Use both the graph and the equations to justify your answer.
d. How long did it take each participant to finish the race?
e. How fast was Kristin riding? Write your answer as a unit rate.
2-54. THE BIG RACE − HEAT 2
a. When the line representing Kaye’s race is graphed, the equation is .
What was her speed (in meters per second)? Did she get a head start?
b. Elizabeth’s race is given by the equation . Who is riding faster, Elizabeth or Kaye? How do you know?
c. At what unit rate was Hannah riding? Write your answer as a unit rate.
d. To entertain the crowd, a clown rode a tricycle in the race described by the equation
f(x) = 20 − x. Without graphing or making a table, fully describe the clown’s ride.
2-55. OTHER RATES OF CHANGE
A. For each graph below, o Explain what real-world quantities the slope and y-intercept represent. o Find the rate of change for each situation.
B. In each of the situations, would it make sense to draw a different line with a negative y-intercept?
2.2.3: Equations of Lines in Situations
I Can _____________________________________________________________________.
2-68 he Big Race – Finalsa. Draw a graph (on graph paper) showing all of the racers’ progress over time.
Identify the independent and dependent variables.b. Write an equation for each participant.c. Figure out who will win the race!
Rider A: You ride your tricycle 3 meters every 2 seconds and pass Rider B ten seconds after the race begins. The race is 25 meters long.
Rider B: You get the same head start as Elizabeth and ride 1 meter every 2 seconds
Rider C: You ride half as fast as Leslie but get a 4-meter head start.
Rider D: You get a 1 meter head start and catch up to Elizabeth in 4 seconds.
Elizabeth: is 11 meters into the race after 4 seconds but is only 13 meters in after 12 sec.
Leslie: rides 2 meters per second and starts 2 seconds after the start of the race
2-69. Use your results from “The Big Race – Finals ” to answer the following questions.
a. Who won the finals of The Big Race? Who came in last place?
b. How fast was Rider D traveling? How fast was Elizabeth
traveling?
c. At one point in the race, four different participants were the same distance from the starting line. Who were they and when did this happen?
Practice: Y = mx + b Word Problems1. Suppose that the water level of a river is 34 feet and that it is receding at a rate of 0.5 foot per day. Write an equation for the water level, L, after d days. In how many days will the water level be 26 feet?
2. Seth’s father is thinking of buying his son a six-month movie pass for $40. With the pass, matinees cost $1.00. If matinees are normally $3.50 each, how many times must Seth attend in order for it to benefit his father to buy the pass?
3. For babysitting, Nicole charges a flat fee of $3, plus $5 per hour. Write an equation for the cost, C, after h hours of babysitting. What do you think the slope and the y-intercept represent? How much money will she make if she baby-sits 5 hours?
6. A plumber charges $25 for a service call plus $50 per hour of service. Write an equation in slope-intercept form for the cost, C, after h hours of service. What will be the total cost for 8 hours of work? 10 hours of work?
7. y = x, y = 2x, and y = 3x. How are the graphs alike? How are they different? Compare the slopes.
2.3.1: Writing equations given the slope and a point
I Can _____________________________________________________________________.
2-75. DOWN ON THE FARM
Colleen wants to learn as much about her baby chicks as possible. In particular, she wants to know how much a baby chick weighs when it is hatched.
Given: Chick grew steadily by about 5.2 grams each day, and she assumes that it has been doing so since it hatched.
Nine days after it hatched, the chick weighed 98.4 grams.
Create an XY table Create A Graph
a. How much did the chick weigh when it hatched?
b. When will the chick weigh 140 grams?
2-78. FINDING AN EQUATION WITHOUT A TABLE OR GRAPH
a. Since Colleen is assuming that the chick grows linearly, the equation will be in the form
y = mx + b. Without graphing, what do m and b represent?
Do you know either of these values? If so, what are their units?
b. You already know the chicken’s rate of growth. Place this into the equation of the line. What information is still unknown?
c. Using the point (9, 98.4); find the y-intercept.
d. Does the y-intercept you found algebraically match the one you found using the graph?
Does it match the one you found using the table?
How accurate do you think your algebraic answer is?
What are the units for the y-intercept?
e. Use your equation to determine when Colleen’s chicken will weigh 140 grams.
2-79. Find equations for lines with the following properties:
a. A slope of −3, passing through the point (15, −50).
b. A slope of 0.5 with an x-intercept of (28, 0)
PRACTICE
Write an equation of line with the given slope that passes through the given
point.
1. Slope = 5, (3,13) 2. Slope = -5/3, (3,-1) 3. Slope = -4, (-
2,9)
4. Slope = 3/2, (6,8) 5. Slope = 3 (-7,-23) 6. Slope = 2,
(5/2, -2)
Write an equation of line parallel to the given line that passes through the
given point.
2.3.2: Finding the equation of a line through two points
I Can _____________________________________________________________________.
2-87. In this problem, you will find the equation of the line that goes through the points in the table below. Use the questions below to help you organize your work.
IN (x) 29 18 −8 14 −27OUT (y) 97 64 −14 52 −71
a. What is the slope of the line?b. Does it matter which points you used to find the slope of your line?
Find the slope with two other points to verify your answer.
c. Find the equation of the line.
d. Verify that your equation is correct?
2-88. LINE FACTORY LOGO
a. What are the equations of the four line segments that make up Design A?
b. What are the equations of the four line segments that make up Design B?
c. What are the domain and range of each of the line segments in Design A?
d. What are the domain and range of each of the line segments in Design B?
Practice
Complete the following on notebook paper. Show all work. Circle your answer.
HOMEWORK Day 12.1.1 & 2.1.2: Seeing Growth in Linear Representations / Slope
2-6. A tile pattern has 5 tiles in Figure 0 and adds 7 tiles in each new figure. Write the equation of the line that represents the growth of this pattern.
2-7. Evaluate each expression if r = −3, s = 4, and t = −7.
A. B. C. 2s3 + r – t D.
2-8. Examine the relation h(x) defined at right. Then estimate the values below.
A. h(1)
B. h(3)
C. x when h(x) = 0
D. h(−1)
E. h(−4)
2-9. Which of the relations below are functions? Justify your answer.
A.
B. For each graph above, state the domain and range.
2-10. Examine the graphs in problem 2-9 again. Which, if any, have lines of symmetry? Draw the lines of symmetry on each graph if they are present.
2-19. What shape will the graph of y = x2 + 2 be? How can you tell? Justify your prediction by making a table and graphing y = x2 + 2 on graph paper.
2-20. Evaluate each expression for x = −2 and y = −5.
A. 1 − 2x + 3y
B.
C.
D.
2-22. Figure 2 of a tile pattern is shown at right. If the pattern grows linearly and if Figure 5 has 15 tiles, then find a rule for the pattern.
2-23. Find the output for the relation with the given input. If there is no possible output for the given input, explain why not.
2-24. Find the slope of the line shown on the graph below.
HOMEWORK Day 22.1.3 & 2.1.4: Steepness / y=mx+b
2-31. Does the table below appear to represent a function? If so, write an equation using function notation that represents the table. If not, explain why it cannot represent a function.
2-32. When Yoshi graphed the lines y = 2x + 3 and y = 2x − 2, she got the graph shown at right.
A. One of the lines at right matches the equation y = 2x + 3, and the other matches y = 2x − 2. Which line matches which equation?
B. Yoshi wants to add the line y = 2x + 1 to her graph. Predict where it would lie and sketch a graph to show its position.
C. Where would the line y = −2x + 1 lie? Again, justify your prediction and add the graph of this line to your graph from part (b).
2-33. Graph a line with y-intercept (0, −4) and x-intercept (3, 0). Find the equation of the line.
2-35. What number is not part of the domain of ? How can you tell?
2-41. If y = x − 4:
A. What is the slope of the line?B. What is the y-intercept of the line?C. Graph the line. Use the graph provide for problem 2-33. Label this line 2-41
2-42. Without graphing, find the slope of each line described below.
A. A line that goes through the points (4, 1) and (2, 5).
B. A line that goes through the origin and the point (10, 5).
C. A vertical line (one that travels “up and down”) that goes through the point (6, −5).
D. A line that goes through the points (1, 6) and (10, 6).
2-43. Ms. Cai’s class is studying a tile pattern. The rule for the tile pattern is y = 10x − 18. Kalil thinks that Figure 12 of this pattern will have 108 tiles. Is he correct? Justify your answer.
2-44. State the slope and y-intercept of each line.
A. y = x − 4
B. y = x + 3
C. y = −5
2-45. Evaluate the expressions below for the given values.
A. −x2 + 3x for x = −3
B. 5 − (x − 2)2 for x = −1
C.
D.
HOMEWORK Day 3 2.2.2: Rate of Change (The Big Race)
2-59. Find the rule for the following tile pattern.
2-60. Copy and complete each of the Diamond Problems below.
2-61. THE BIG RACE − HEAT 3
Barbara thought she could win with a 3 meter head start & pedaled 3 meters every 2 seconds.
Mark began at the starting line and finished the 20 meter race in 5 seconds.
Carlos rode his tricycle so that his distance (y) from the starting line in meters could be
represented by the equation y = + 1, where x represents time in seconds.
A. What is the dependent variable? What is the independent variable?
B. Graph lines for each contestant. Who won the 20 meter race and will advance to the final
race?
C. Find equations that describe Barbara’s and Mark’s motion.
D. How fast did Carlos pedal? Write your answer as a unit rate.
E. When did Carlos pass Barbara? Confirm your answer algebraically.
2-62. Create a table and a graph for the line y = 5x − 10. Find the x-intercept and y-intercept in the table and on the graph.
2-63. Find the slope of the line containing the points in the table below
IN (x) 2 4 6 8 10OUT (y) 4 10 16 22 28
2-64. Use what you know about y = mx + b to graph each of the following equations quickly on the same set of axes.
A. y = 3x + 5B. y = −2x + 10C. y = 1.5x
2-65.
A. Find the equation of the line graphed at right.B. What are its x- and y-intercepts?
2-66. Use the idea of cube root from problem 1-35 to evaluate the following expressions.
A. B. C. D.
2-67. Each part (a) through (d) below represents a different tile pattern. For each one, determine how the pattern is growing and the number of tiles in Figure 0.
A. C. y = 3x − 14
Growth: _____ Fig #0 __________ Growth: _______ Fig #0 __________
B. D.
Growth: ______ Fig #0 __________ Growth: ______ Fig #0 __________
x −3 −2 −1 0 1 2 3y 18 13 8 3 −2 −7 −1
2
HOMEWORK DAY 4 2.2.3: Equations of Lines in Situations
2-70. Sometimes the quickest and easiest two points to use to graph a line that is not in slope-intercept form are the x- and y-intercepts. Find the x- and y-intercepts for the two lines below and then use them to graph each line. Write the coordinates of the x- and y-intercepts on your graph
A. x − 2y = 4B. 3x + 6y = 24
2-71. Find the slope of the line passing through each pair of points below.
A. (1, 2) and (4, −1)B. (7, 3) and (5, 4)C. (−6, 8) and (−8, 5)D. (55, 67) and (50, 68)E. Azizah got 1 for the slope of the line through points (1, 2) and (4, −1). Explain to her
the mistake she made and how to find the slope correctly.
2-72. Evaluate the following expressions.
A.
B.
C.
D.
2-73. Complete the table below. Then write the corresponding equation.
IN (x) 2 4 6 7 10OUT (y) −7 −17 −37
2-74. MATCH-A-GRAPH
Match the following graphs with their equations. Pay special attention to the scaling of each set of axes. Explain how you found each match.
a. y = + 4 b. y = + 4 c. y = 2x + 4 d. y = + 4C.
1. 2.
3. 4.
HOMEWORK Day 5 2.3.1: Writing equations given the slope and a point
2-82. The point (21, 32) is on a line with slope 1.5.
A. Find the equation of the line.B. Find the coordinates of another point on the line.
2-83. Copy and complete each of the Diamond Problems below. The pattern used in the Diamond Problems is shown at right.
2-84. The graph of the equation 2x − 3y = 7 is a line.
A. Find the x- and y-intercepts and graph the line using these two points. B. If a point on this line has an x-coordinate of 10, what is its y-coordinate?
2-85. Without graphing, identify the slope and y-intercept of each equation below.
A. y = 3x + 5 m = _______ b = ________
B. y = m = _______ b = ________C. y = 3 m = _______ b = ________D. y = 7 + 4x m = _______ b = ________
2-86. Graph the line .
HOMEWORK Day 6 2.3.2: Finding the equation of a line through two points
2-90. Explain what the slope of each line below represents. Then find the slope and give its units.
A. B.
2-91. Find the equation of the line that goes through the points (–15, 70) and (5, 10)
2-92. This problem is the checkpoint for evaluating expressions and the Order of Operations. It will be referred to as Checkpoint 2.
Evaluate each expression if x = −2, y = −3, and z = 5.
A. 2x + 3y + z B. x – y C.
D. 3x2 − 2x + 1 E. 3y(x + x2 − y) F.
2-93. Greta is opening a savings account. She starts with $100 and plans to add $50 each week. Write an equation she can use to calculate the amount of money she will have after any number of weeks. How much money will she have after 1 year.
2-94. Paula found a partially completed table that her friend Donna was using to determine how fast water evaporated from a bucket during the summer. Every other day she measured the height of the water remaining in the bucket in centimeters.
A. Complete the table. B. For this table, what is the rate of change, including the units? C. Write an equation to represent the height of the water after any number of days.
