Download Algebra 2 Honors – Unit 2, Packet 1: Linear Equations

Name:_______________________________________________________________ Period: _____________
Algebra 2 Honors – Unit 2, Packet 1: Linear Equations
As with the previous packet, you will be working in your groups to solve as many problems as possible each
day in class. You will earn a group grade for this effort. You ALL must show proper notation and complete
work to earn full credit.
You will again be responsible for creating and solving your own problems similar to the problems in this
packet. For every ten problems in the packet, you will invent one of your own problems that is similar to one
from each group of ten questions in the packet.
1. Draw a line that has a positive slope.
2. Draw a line that has a negative slope.
3. What is the slope of the line below?
4. What is the slope of any horizontal line?
Slope:_________________
5. What is the slope of the line below?
Slope:___________________
6. What is the slope of any vertical line?
7. State in your own words what is meant by “slope.” ______________________________
_______________________________________________________________________.
8. Write two equations or formulas for expressing the slope of a line. _________________
_______________________________________________________________________.
Use your formula(s) in number 8 to find the slope of each line below that passes through each pair of
points listed. Remember that slope expressions should always be fully reduced.
9. (2, 6) and (-8, 18)
10. (5,-4) and (-13, 2)
11. (5, 3) and (5, 1)
12. Discuss in your groups what a “y-intercept” is. Write a brief explanation of “y-intercept” in your
own words: ____________________________________________________________.
13. Draw a line with a y-intercept of 3.
Label that point on the graph.
14. Draw a line with a y-intercept of -2.
Label that point on the graph.
15. In general, a y-intercept is a point on a graph that has the ___ coordinate = 0.
16. What is an x-intercept? ______________________________________________________
_____________________________________________________________________________.
17. Draw a line with an x-intercept of 2.
Label that point on the graph.
18. Draw a line with an x-intercept of -3.
Label that point on the graph.
19. One common format for writing an equation of a line is called “Slope-Intercept Form”. What is the
general formula for a “Slope-Intercept Form” of a line? ___________________.
For problems 20 & 21, First graph the line passing through the given point with the given slope. Then,
write an equation for the line in Slope-Intercept Form.
20. Slope = -1 and passes through (-4, 2)
21. Slope = 3/2 and passes through (-2,-4)
Equation: ______________________
Equation: ______________________
For problems 22 – 24, rewrite the following equations in the Slope-Intercept Form if they are not already
in that form. Then, state the slope and y-intercept given in each equation.
22. y = x – 6
23. 3x + y = 10
24. 16 – 4y = 28x
Slope:_______
Slope:_______
Slope:_______
y-intercept: ________
y-intercept: ________
y-intercept: ________
25. Draw a pair of parallel lines.
26. Draw a pair of perpendicular lines.
27. What is true about the slopes of parallel lines? ___________________________________.
28. One way of talking about the slopes of perpendicular lines is to say that the product of the slopes of
the two lines is -1. What is another way to express that same relationship? ___________________
________________________________________________________________________________.
29. Fill in the following grid:
Original Line’s Equation or Slope
Slope of a Parallel Line
Slope of a Perpendicular Line
Slope = 2/5
y = 6x + 1
y=3–x
y=8
x = -2
Are the lines described below parallel, perpendicular, or neither?
30. The lines passing through (-4, 3) and (1,-3) and the line passing through (1, 2) and (-1, 3).
31. The line passing through (3, 9) and (-2, -1) and the graph of y = -2.
32. The line with x-intercept -2 and y-intercept 5 and the line with x-intercept 2 and y-intercept -5.
33. The line passing through (8, -4) and (4, 6) and the graph of 2x – 5y = 5.
34. Write an equation of a line parallel to the line of y = 13x + 1: ____________________.
35. Write an equation of a line that is perpendicular to y = 13x + 1: __________________.
36. The Point-Slope Form of a line going through the point (-2, 5) having a slope of 3 is written as
y  5  3 x  2 . Write the Point-Slope Form of a line going through (6, -3) and having a
slope of -4: ____________________________________________.
37. Write an equation of a line in Point-Slope Form that passes through (3, -7) and (-1, -5):
________________________________ (there are two correct answers here, either one is ok )
For problems 38-43, write an equation in Slope-Intercept Form for the line that satisfies each set of
conditions.
38. Slope 1/3, passes through the origin
39. Slope 4, passes through (2, -1)
40. x-intercept 5, y-intercept -6
41. Passes through (2, 4) and (-4, 7)
42. Passes through (-3, -7), parallel to the
2
graph of y  x  8 .
3
43. Passes through (6, -5), perpendicular to
1
the line whose equation is 3 x  y  7 .
2
(Hint: solve for y to find the slope)
44. Write the equation of a line in Slope
Intercept Form given the table below.
x
y
2
-1
3
-4
4
-7
5
-10
Equation: _________________________
45. Write the equation of a line in Slope
Intercept Form given the graph of the
line below.
Equation: __________________________
46. The equation 3x + 2y = 12 is in Standard Form. We can easily find the x and y intercepts from this
format.
a. What is the x-intercept of this line? ___________
b. What is the y-intercept of this line? ___________
c. This equation can be transformed into another equation where we can easily see the slope,
what is the name of this second format? ___________________________________
d. How can you transform any Standard Form equation into this other format?
____________________________________________________________________
e. What would be the result of the equation above written in this format? (Show work below)
__________________________
47. One way to remember something is to explain it to another person. Suppose that you are studying
this lesson with a friend who thinks that she should let x = 0 to find the x-intercept and let y = 0 to
find the y-intercept. How would you explain to her how to remember the correct way to find the
intercepts of a line?
48. For Meg’s long-distance calling plan, the monthly cost C in dollars is given by the linear equation
C = 6 + 0.05t, where t is the number of minutes talked. (Show all your work below)
a. What is the total cost of talking 8 hours? _______________
b. Of talking 20 hours? ____________
c. How much long distance talking time (in hours) did she use up last month if her bill was $
156.00? __________
49. Paul the plumber is coming in to fix a leak in your water pipe. He charges $80 to show up and
$60 for each hour. (Show all your work below)
a. Write an equation in Slope Intercept Form that models this situation: _________________
b. How much would he charge you if he had 3.5 hours of work? ______________
c. If he charged you $260, how long was he there? _______________
For problems 50-57, graph the following equations on each coordinate plane.
1
50. y   x  7 .
4
51. y  x  3
52. y  2 
1
 x  5
3
53. y  4    x 1
54. y  3
55. x  7
56. 3x  4 y  24 (hint: use the intercept
method )
57. 2 x  y  6 (hint: use the intercept
method )
Algebra 2 Honors – Unit 2, Packet 2: Functions
1. Explain two ways to determine whether a relation is a function. Use specific examples. Then
write a relation that is NOT a function.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Fill in the blanks. For problem numbers 2-8, choose the word from the answer choices below that fits
each definition below.
Relation
Domain
Range
Function
Vertical Line Test
Dependent Variable
Independent Variable
2. A mathematical relationship between two values where the second value depends on the first one
is _________________________
3. The set of values of the independent variable where the function can be defined
____________________________
4. The set of values of the dependent variable where the function can be defined
____________________________
5. A variable that depends on one or more other variables
____________________________
6. A set of ordered pairs
____________________________
7. A test used to determine if a relation is a function
____________________________
8. A variable in an equation that may have its value freely chosen without considering values of any
other variables
____________________________
Use the vertical line test to determine whether the graph represents a function or not:
9.
10.
Graph each relation or equation and find the domain and range. Then determine whether the relation
or equation is a function.
11. {(3,-4),(1,0),(2,-2),(3,2)}
12. y=x-4
13. What makes an equation a Linear Equation?
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
For questions 13-16, determine if each equation is linear. If the equation is linear, re-write it in the
standard form of a linear equation (ie. Ax + By = C). If the equation is not linear, explain why it isn’t a
linear equation.
14. 6y – x = 7
15. 9x = 18/y
16. y= 4𝑥 2
17. y – 3 =
x
For questions 18-20,find the x-intercept and y-intercept of the graph of each equation. Then graph the
equation.
18. 2x + 7y = 14
19. 5x – 5y +7.5 = 0
20. y = -2x – 4
For questions 21-27, graph each function. Identify the domain and range for each function.
21. f(x) = 8x + 12
22. f(x) = -5/3 + 10
-Include explanation of absolute value and piece wise functions for students to work on.
23. f(x) = | x |
24. f(x) = 2|x|
25. f(x) = - | 2x +1| - 4
26. f(x) = {x if x<0,
2 if x≥0}
27. f(x) = {x+2 if x ≤ -2,
3x if x > -2}
For questions 28-31, graph each inequality. Choose a test point that is not on the boundary to determine
which region to shade.
28. x – y > -2
29. y + 1 ≤ 2x
30. y > |x| - 1
31. y < -3|x+1| - 2
32. Describe some ways in which graphing one inequality in one variable on a number line is
similar to graphing an inequality in two variables in a coordinate plane. How can what you
know about graphing inequalities on a number line help you to graph inequalities in a
coordinate plane?
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Include:





function notation
graphing absolute value
graphing piece wise
graphing inequalities
ACT style questions
33. Name and describe the three methods to solve a system of linear equations.
Classify systems of equations: Graphs of systems of equations may be intersecting lines, parallel lines
or the same lines. A system of equation is consistent if it has one solution (meaning if it only intersects at
one point) and inconsistent if it has no solution (meaning if it doesn’t intersect anywhere). Consistent
system is independent if it only has one solution (intersects in one place) or dependent if it has infinite
number of solutions (if the lines are the same).
Graph each system of equations and describe it as consistent and independent, consistent and
dependent, or inconsistent. If the system is consistent and independent, find the solution.
34. x + 1/2y = 5
3y – 2x = 6
35. 9x – 6y = 24
6x – 4y = 16
36. 3x + 4y = 12
6x + 8y = -16
37. Give an example of a system of equations that is consistent AND independent.
38. To solve systems of equations, when it is most efficient to use graphing? To use substitution?
To use elimination? Explain and give examples.
Use the method of your choice to solve the following systems of equations:
39. y = -3x +6
y = 2x – 4
63. c +2d = -2
-2c – 5d = 3
64. x – 4y = 4
2x + 12y = 13
65. 3x – 2y = 12
2x – 2/3y = 14
66. 1/2x +3y = 11
8x – 5y = 17
67. 5g – 4k = 10
-3g – 5k = 7
68. How can a system of equations be used to predict sales? Give an example.
69. Megan exercises every morning for 40 minutes. She does a combination of step aerobics, which
burns about 11 Calories per minute, and stretching, which burns about 4 calories per minute.
Her goal is to burn 355 calories during her routine.
a. Write a system of equation that represents Megan’s morning workout.
b How long should she participate in each activity in order to burn 335 Calories?
70. To graph a system of inequalities, you must graph two or more boundary lines. When you graph
each of these lines, how can the inequality symbols help you remember whether to use a dashed
or solid line?
Solve each system of inequalities by graphing. Shade the region that satisfies both inequalities.
71. x – y ≤ 2
x + 2y ≥ 1
72. y < 3
x +2y < 12
73. Without actually drawing the graph, describe the boundary lines for the system of inequalities
below.
|x| < 3
|y| ≤ 5
74. Write a system of inequalities that has no solution.
75. A painter has exactly 32 inuts of yellow dye and 54 units of green dye. He plans to mix as many
gallons as possible to color A and color B. Each gallon of color A requires 4 units of yellow dye
and 1 unit of green dye. Each gallon of color B requires 1 unit of yellow dye and 6 units of
green dye. Find the maximum number of gallons he can mix.
a. Define the variables.
b. Write a system of inequalities.
c. Graph the system of inequalities.
d. Find the maximum number of gallons of color A and color B combined that he can make.