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.