Dear Honors Pre-Calculus Students June 2016

The packet attached to this letter contains a series of problems that will overview the Algebra II

skills you should have mastered in order to have a good start in Pre-Calculus. Completing this packet over

the summer will help you practice and maintain these skills. I expect that you will work through these as

best you can, and that your work will be neat and accurate. Moreover, you should work on this periodically

throughout the summer, as it will take several hours to complete. Be sure to use the selected answers at the

end to check your work. This assignment will be checked on the first day of classes; after spending class

time reviewing it, you will be tested on the material. That grade will count as your first test grade of quarter

one.

Be sure to present your work in an organized and systematic way. In this course, the way you arrive

at an answer is often as important as the answer itself. Please be mindful of this when you are working

through this packet and provide sufficient work to justify your answers.

For your convenience, problems in this packet are preceded by the section number in parentheses in

your Algebra II notes, indicating when the concept was covered. If you retained your notes from Algebra II,

you can refer to those sections for a refresher as needed.

Also, you will need a Texas Instruments TI-83, TI-83 Plus, TI-84, TI-84 Plus or TI-84 Silver Edition

graphing calculator for this packet as well as for the course.

Lastly, I recommend that you purchase the appropriate materials for this course:

A 3-ring binder with 3 dividers: Homework, Class Notes, Assessments.

A pack of filler paper for the Homework section of your binder

A graph paper notebook

I look forward to seeing you in the fall. Expect an exciting and challenging learning experience!

Have a great summer

Mrs. Warren

warrena@mersd.org

Prerequisite Algebra II skills for Honors Pre-Calculus Directions:

1. All work must be done NEATLY, on separate paper. You are expected to number each problem, show work

clearly, and circle your final answers.

2. All graphs must be done on graph paper. (print some from www.staff.mersd.org/brown if necessary)

3. Questions with the icon indicate that a calculator may not be used.

4. A number in parentheses that immediately follows the problem number indicates the section of your Algebra II

notes to which you should refer, if needed.

I. Point-Slope Equation of a Line

1. (2.2) a. Provide an equation, in point-slope form, of the line that contains the points 6,5 and 8,7 .

(2.4) b. Use the linear regression capability of the graphing calculator to find the slope-intercept equation of the

line in (a).

(2.4) c. Verify your answer to (b) by solving that equation for y and comparing it to the linear regression

equation you got in part (a).

2. (2.2) Write a point-slope equation of the line whose x-intercept is 3 and y-intercept is -4.

3. (2.2) a. Write a point-slope equation of the line perpendicular to the line 543 yx that shares its y-intercept.

b. Write a point-slope equation of the line that is parallel to the line in (a) and contains the point (6, -7)

c. Sketch three lines: the given line in (a), your answer to (a), and your answer to (b).

4. (2.2) a. Write an equation of a line that has an x-intercept but no y-intercept and only runs through quadrants

II and III. Then, sketch the line.

b. What is the slope of your line?

c. Write an equation of a line that is perpendicular to your line in (a). Then, state its slope and sketch it.

5. (2.4) A school district purchases a high-volume copy machine for $25,000. After 10 years, the equipment will

have to be replaced (due to wear and tear). Its value at that time is expected to be $2000.

a. Write a linear equation giving the value v of the equipment as a function of the number of years t since the

machine was purchased.

b. Write a sentence in which you describe the slope of your linear model in the context of this problem.

II. Operations with Functions, Composition of Functions

6. (2.1, 5.5, 5.6, 7.6) Given the function xxxf 2)( 2 ,

a. evaluate:

i. )3(f ii.

4

1f iii. ))10(( ff iv. 0ff

b. evaluate, and fully simplify:

i. )23(f ii. )24( f iii.

h

fhf 4)4( , where 0h (why?)

c. For what values of x is

i. 0)( xf ? ii. 48)( xf ?

7. (5.5, 7.6) If 12)( 2 xxxf and 57)( xxg , find the values of x for which

a. f(x) = 0 b. g(x) = 0 c. f(x) = g(x) d. 0xgf e. 0xgg

8. (2.1, 6.2, 7.6) Please refer to the graph of y = g(x) shown.

a. Is g(x) a function? Briefly explain.

b. State the value of each:

i. g(1) ii. g(-3) iii. g(g(0))

c. For what values of x is g(x) = 0?

d. Write a possible equation of g(x)

i. in factored form

ii. in standard form

III. Graphs of Functions, Relations, and Inverses

9. For each relation:

i. Sketch its graph. ii. (2.1) State its domain and range. iii. (2.1) Tell whether or not it is a function.

a. (2.2) y b. (2.5) 14 xy c. (5.2) 132 xy

d. (5.2) 22 xy e. (6.1) 132 xxxy f. (6.1) 33 xy

g. (6.1) 23 xxy h. (7.8) 5 xy i. (7.8) 5 xy

j. (7.8) 3 5 xy k. (8.1) x

y

3

13 l. (8.6) xy log

10. (7.7) i. Find the equation of the inverse 1f (x) of each function.

ii. Sketch f(x), 1f (x), and the line of identity y = x for each.

iii. State the domain and range of f(x) and 1f (x).

a. xxf2

14)( b. xxf 2)( c. 32)( 3 xxf

IV. Quadratic Functions

11. (5.7) i. Rewrite each equation in vertex form 2hxaky by the method of completing the square.

ii. State the coordinates of the vertex and the equation of the axis of symmetry of each parabola.

a. 2410)( 2 xxxf b. 96)( 2 xxxf c. 782)( 2 xxxf

12. (5.2) Use the maximum/minimum feature of the grapher to find the vertex of each parabola in #11. Use these

answers to confirm that your answers in #11 are correct.

13. (5.2) The path of a diver jumping off of a diving board is given by the equation 129

24

9

4)( 2 xxxf , where y

is the diver’s height (in feet) above the surface of the water and x is the horizontal distance (in feet) of the

diver from the end of the diving board.

.

V. Polynomial Functions

14. (5.8, 6.2) Find all real zeroes of each function using quadratic formula or square roots. Answers must be

expressed in simplest radical form.

a. 4123)( 2 xxxf b. 2036)( 2 xxf c. 94)( 2 xxxf

15. (6.2) Write a polynomial function, in standard form, that has the given zeroes:

a. 4,3,2x b. 0,5,5x c. 4x , multiplicity 3

16. (6.3) a. Use polynomial long division to find the quotient: 5451174 23 xxxx

b. Use your answer from part (a) to factor 51174 23 xxx completely.

c. Use your answer from part (b) to state all real zeroes of the function 51174)( 23 xxxxf .

17. (5.4, 6.4) Algebraically determine all real zeroes of each function by using factoring techniques such

as “splitting the middle”, trial & error, Greatest Common Factor, and/or grouping:

a. 8103)( 2 xxxf b. 8116)( 2 xxf c. 10116)( 2 xxxf

d. xxxxf 25204)( 23 e. 755032)( 23 xxxxf f. 3613)( 24 xxxf

18. (6.3) a. Use synthetic division to find the quotient: 31834 23 xxxx

b. Use your answer to part (a) to find all real zeroes of the function 1834)( 23 xxxxf .

a. How high above the water is the diving board?

b. What is the maximum height of the diver?

c. Find the diver’s horizontal distance from the end of

the diving board when the diver hits the water.

VI. Operations with Radicals, Rationalizing the Denominator

19. (5,5, 7.1, 7.2) Simplify each of the following radical expressions. Express all answers in simplest radical form,

with rationalized denominators (i.e. radicals may not be left in the denominator of any fraction.)

a. 3

1 b.

27

3 c.

2

2 d.

6

2 e.

2

6 f.

8

8 g.

26

6

20. (7.3) Write each expression in simplest radical form by rationalizing the denominator with the conjugate.

a. 32

1

b.

63

9

c.

222

4

d.

26

26

21. (5.5, 5.8) Use square roots or quadratic formula to find all complex zeroes. Express answers in simplest

radical form, with rationalized denominators.

a. 012 2 x b. 0542 xx c. 0322 xx

VII. Operations with Imaginary Numbers, Rationalizing the Denominator

(For these problems, i represents the imaginary number 1i )

22. Given the function xxxf 2)( 2 , evaluate and fully simplify each. Express answers in a + bi form.

i. )(if ii. )3( if iii. )3( if iv. )31( if

23. (5.6) Show that each number satisfies the equation 0910 24 xx by substituting them and simplifying: i. ix ii. ix iii. ix 3 iv. ix 3

24. (5.6) Find all real zeroes of the equation 0910 24 xx by factoring. Do your answers look familiar?

25. (5.6) Simplify each fraction by rationalizing the denominator. Express answers in a + bi form.

a. i

1 b.

i2

1 c.

2

1

i d.

i2

1

e. i1

1 f.

i2

1 g.

i1

2 h.

i22

12

SELECTED SOLUTIONS

1. a. or

b.

c. (Check)

3. a.

b.

c.

5. a.

b. Each year, the value of the copy machine drops by $2300.

6. b. i. ii. iii. 6+h,

7. a. x = -1

c. x = 2 or 3

e. x =

8. a. yes, passes VLT

b. i. 0 ii. 2 iii. -1

c. x = -4, 1, 4

d. i. ii.

9. a. i. ii. domain: ; range:

c. i. ii. domain: ; range:

e. i. ii. domain: ; range:

h. i. ii. domain: ; range:

l. i. ii. domain: ; range:

10. c. i. iii. Domain: ; range: y

11. b. i. ii. Vertex (-3, 0); axis of symmetry:

13. a. 12 feet b. 16 feet c. 9 feet

14. a.

15. b.

16. a. b. c.

17. c. e. f.

19. b. c. g.

20. b. d.

21. b.

22. i. -1 – 2i iii. 2 + 4i

25. b. d. g.