algebra ii name: do all work on looseleaf or … all work on looseleaf or provided graph paper. show...
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Algebra II Name: ____________________________________
Summer Packet Date: _______________________ Period: _______
DO ALL WORK ON LOOSELEAF OR PROVIDED GRAPH PAPER. SHOW ALL WORK to receive credit. Graph paper provided at the end for graphing problems.
I. Solving Linear Equations with One Variable
Linear equation - an equation that represents a straight line
The solution to a linear equation with one variable is the value or values for the variable that make the
equation true.
A linear equation with one variable can have:
- one solution
- no real solutions
(if you end in a false statement like 2=5)
- infinitely many solutions/all real numbers
(if you end in a true statement like 2=2)
Practice Problems:
1. 6.
2. 7.
3.
8.
4.
9.
5. 10.
Goal: Solve the linear equation by finding the value(s)
for the variable that make the equation true.
How: Simplify the equation using the Order of Operations.
Parentheses Exponents Multiplication Division Addition Subtraction
Isolate the variable by getting the variable by itself on one side of the equation.
Example
Solve the following:
a) b) c)
<-- true <-- false
all real numbers ( ) no solution ( )
II. Graphing Linear Equations
Things to Remember: In standard form, slope can be found by
the following equation.
Practice Problems: (Graph on the attached graph paper.)
11. 16.
12. 17.
13. 18.
14. 19.
15. 20.
"rise" & "run"
<-- start at this point on the y-axis
Find your intercepts.
# is a horizontal line Mark -3 on the y-axis and draw the line through it.
# is a vertical line Mark 4 on the x-axis and draw the line through it.
III. Writing Equations of Straight Lines
is the y-intercept (b) -2 is the slope (m)
is the y-intercept (b)
Remember, in standard form the slope can be found by the equation
.
So,
1 1
1 1
Remember, slopes of perpendicular lines are opposite reciprocals. The slope given is . So the slope
of the perpendicular line is
Practice Problems:
Write the equation of each line described. Put your final answer in slope-intercept form.
21. the line that crosses through (0,2) with a slope of
22. the line that crosses through the point (8, 13) with a slope of -9
23. the line through the points (0,7) and (3,5)
24. the line through the points (-2, -3) and (2, -1)
25. the line through (2, 8) that is parallel to
26. the line through (3, -1) that is perpendicular to
IV. Multiplying Binomials - FOIL
A binomial is an expression with two terms. When multiplying binomials, the FOIL method ensures that all
parts are multiplied together.
Practice Problems: Multiply the following.
27. 32. 37.
28. 33. 38.
29. 34. 39.
30. 35. 40.
31. 36.
1 1
First
Outer
Inner
Last
Examples: Multiply the following.
a) b)
V. Simplifying Square Roots
Some radicals have exact values. For example
If the radicand (or number under the radical sign) is not a perfect square, it must be simplified.
Practice Problems: Simplify
41. 42. 43. 44. 45.
46.
47.
48.
49.
50.
Examples: Simplify.
a) b)
Simplifying Square Roots: 1. Make a factor tree. 2. Circle pairs. 3. Pull one number from each pair out of the radical symbol. 4. Multiply them together. 5. Any numbers in the factor tree that do not have pairs must stay inside the radical symbol. Multiply them together.
Square Roots and Fractions: 1. If you can simplify a fraction that is under
a radical, do that first. 2. Split up the fraction so the numerator is a
radical and the denominator is a radical. 3. Simplify each radical. 4. NOTE: You can’t have a radical in the
denominator! Rationalize the denominator. (Multiply the number and denominator by any radicals in the exponent to eliminate them.)
Examples: Simplify.
a)
b)
c)
b)
VI. Functions
A function is a relation in which there is only one output (y) for every input (x).
A function can be expressed in function notation, . [Read “f of x” NOT “f times x”]
This means that any number can be replaced for in the function.
NOTE: Any variable can be used instead of , and any letter can be used to represent the function (not just ).
Practice Problems:
51. Given the function
, find .
52. Given the function , find .
53. Given the function , find the value of
for which .
54. Given the function , find the value of for
which .
55. Given the function , find .
56. Given the function , find .
57. Given the function , find
58. Given the function , find the value of
for which .
59. Given the function , find .
60. Given the function , find .
11. 12.
Example: Given the function ,
a) find .
b) find the value of for which .
13. 14.
15. 16.
17. 18.
19. 20.
Answers:
1.
2.
3. (all real numbers)
4.
5.
6.
7.
8.
9.
10.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.