MHCA Math Summer Packet 2015

For students entering PreCalculus CP

Directions:

You are to complete all the problems assigned in this packet by Friday,

September 4th. If you don’t turn in your summer packet when it is due, you will be

required to work on your summer packet in the library during your Club period

time, every day until your summer packet is complete. You will also lose 10 points

from your overall summer packet grade.

The summer packet will count as your 1st Marking Period Project. Failure to

complete this on time or to take it seriously will drastically reduce your 1st Marking

Period Grade. Students in Honors class who don’t turn in their packet on time

WILL NOT be recommended for Honors the following year.

You may use as much scrap paper as you want for this packet, but you will

ONLY turn in the answer sheet for a grade. Any answers not written on the answer

sheet will be assumed to have not been answered and you will lose the point for

that problem. When you get your graded answer sheet back you can look back at

your work to see where you made the mistake. If you can’t find it, you are

encouraged to see your teacher before school, during club or after school for help.

All the material in this packet is what you learned in your math class this

past year; so none of it should be new. You will be expected to know all the

material in this packet for your PreCalculus CP class this year. Your teacher

WILL NOT reteach the material found in this packet; they will only review it with

you at appropriate points in the class.

If you do encounter questions where you are not sure how to solve, you

should also watch videos from Khan Academy to help supplement your own notes

from this year. In addition, a simple Google or YouTube search of the topic should

yield numerous videos to help you out.

If you have any questions regarding the completion of the MHCA Math

Summer Packet please contact me at fmcmahon@maryhelp.org. Have a great

summer and we look forward to seeing you at the start of the school year!

Fred McMahon

Mathematics Department Chair

Answer Sheet

Week 1

1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ 9. _____ 10. _____ 11. _____ 12. _____ 13. _____ 14. _____

Week 2

1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ 9. _____ 10. _____ 11. _____ 12. _____ 13. _____ 14. _____

Week 3

1. _____ 2. _____ 3. _____ 4. _____

5. _____ 6. _____ 7. _____ 8. _____ 9. _____ 10. _____ 11. _____ 12. _____ 13. _____ 14. _____

Week 4

1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ 9. _____ 10. _____ 11. _____ 12. _____ 13. _____ 14. _____

Week 5

1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ 9. _____ 10. _____

11. _____ 12. _____ 13. _____ 14. _____

Week 6

1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ 9. _____ 10. _____ 11. _____ 12. _____ 13. _____ 14. _____

Week 7

1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ 9. _____ 10. _____ 11. _____ 12. _____ 13. _____ 14. _____ 15. _____ 16. _____

Week # 1

Chapter 1 – Equations and Inequalities

Exercises

Evaluate each expression.

1. 20 ÷ (5 – 3) + 52(3)

A 235 B 85 C 20

77 D 255

2. 18 – {5 – [34 – (17 – 11)]}

A B C D

3. Evaluate (𝑎 + 𝑦)2 + 2y if a = 5 and y = –3.

A 58 B –2 C 70 D 10

4. Evaluate |–2b| if b = 8.

A –16 B 6 C 10 D 16

5. Name the sets of numbers to which 3

5 belongs.

A rationals C rationals, reals

B naturals, reals D integers, rationals, reals

6. Simplify 2 (x + 3) + 5 (2x – 1) .

A 12x + 1 B 12x + 11 C 12x + 2 D 9x + 1

7. Name the property illustrated by the equation.

5x • (4y + 3x) = 5x • (3x + 4y)

A Associative B Commutative C Distributive D Inverse

For Questions 6–8, solve each equation.

8. 1

2y = 8

A 16 B 4 C 1

4 D 10

9. 4 (2x – 9) = 3x + 4

A –32 B – 32

5 C

40

3 D 8

10. 4|x + 3| = 20

A {2} B {–8} C {2, –8} D ∅

For Questions 11-12, solve each inequality.

11. 2x – 1 ≤ 5 or 7 – x < 1

A 3 ≤ x < 6 B x < 6 C x ≤ 3 or x > 6 D ∅

12. |2x – 5| ≤ 9

A –4 ≤ x ≤ 14 B –2 ≤ x ≤ 7 C x ≤ –2 or x ≥ 7 D all real numbers

13. Identify the graph of the solution set of 9 > 3 + 2x.

A C

B D

14. A parking garage charges $2 for the first hour and $1 for each additional hour. Fran has $7.50 to

spend for parking. What is the greatest number of hours Fran can park?

A 3 B 5 C 6 D 7

Week # 2

Chapter 2- Linear Relations and Functions

Exercises

1. Find the domain of the relation {(0, 0), (1, 1), (2, 0)}. Then determine whether the relation is a

function.

A {0, 1, 0}; function C {0, 1, 2}; function

B {0, 1, 0}; not a function D {0, 1, 2}; not a function

2. Find f(–1) if f(x) = –3x – 5.

A –9 B –8 C –2 D 2

3. Which equation is linear?

A xy = 60 C y = 𝑥2 – 3x + 1 B 3x – 2y = 5 D 𝑦2 + 1 = x

4. Write y – 4x = 7 in standard form.

A 4x – y = –7 B 4x + y = 7 C y = 4x + 7 D 4x = y – 7

5. Find the x-intercept of the graph of –5x + 10y = 20.

A –2 B 2 C 4 D –4

6. Find the slope of the line that passes through (0, 2) and (8, 8).

A 8 B 4

3 C

3

4 D

5

4

7. If a line rises to the right, its slope is ___?____.

A zero B positive C negative D undefined

8. What is the slope of a line that is perpendicular to the graph of y = 2x + 5?

A – 1

2 B

1

2 C 2 D –2

9. Line a through (2, 3) is parallel to line b with equation y = –1. Which point below also lies on line

a?

A (2, 9) B (–5, 3) C (0, 1) D (1, 4)

10. Write an equation in slope-intercept form for the line that has a slope of – 4

5 and passes through

(0, 7).

A y = 7x B y = 7x – 4

5 C y =

4

5x + 7 D y = –

4

5x + 7

11. Write an equation in slope-intercept form for the line that passes through

(0, 1) and is perpendicular to the line whose equation is y = 2x.

A y = –2x + 1 B y = 2x + 1 C y = 1

2x + 1 D y = –

1

2x + 1

12. Identify the range of y = ⎪x⎥ .

A all real numbers B {x | x ≥ 0} C { y | y ≥ 0} D {y | y ≤ 0}

13. The graph of the linear inequality y ≥ 2x – 1 is the region __?__ the graph of the line y = 2x – 1.

A on or above B on or below C above D below

14. Which inequality is graphed at the right?

A y ≥ |x| – 3 B y ≤ |x| – 3 C y > |x| – 3 D y < |x| – 3

Week # 3

Chapter 3- systems of equations and Inequalities

Exercises 1. A system of linear equations may not have

A exactly one solution. C infinitely many solutions. B no solution. D exactly two solutions.

2. Choose the correct description of the system of equations.

4x + 2y = –6

2x – y = 8

F consistent and independent H consistent and dependent

G inconsistent J inconsistent and dependent

3. Which system of equations is graphed?

A y – 1

3 x = 0 C y – 3x = 0

x – y = –2 x – y = 2

B y – 3x = 0 D y – 1

3 x = 0

x – y = –2 x – y = 2

4. Which system of inequalities is graphed?

A y > – 1 B y > – 1

y ≥ – x + 1 y ≤ – x + 1

C y ≥ –1 D y > – 1

y ≥ – x + 1 y < – x + 1

5. Find the minimum value of f(x, y) = 3x + y for the feasible region.

A 6 B 4 C 2 D 0

6. Find the maximum value of f(x, y) = 3x + y for the feasible region.

A 2 B 4 C 6 D 12

7. What is the value of y in the 2x + y + z = 1

solution of the system of equations? 2x – y – 3z = –3

x – 2y – 4z = –2

A –10 B –8 C 2 D 5

For Questions 8-15, use the matrices to find the following.

P = [4 12 0

] Q = [1 60 2

] R = [0

1

2

1 −2] S = [

6 −4 93 −1 −5

]

8. the first row of 4S

A [–2 8 –5] B [12 –4 –20] C [24 –16 36] D not possible

9. the first row of 2P + 2R

A [8 3] B [4 3] C [6 –4] D not possible

10. the first row of SP

A [12 –4 –20] B [–23 21] C [53 –27] D not possible

11. the inverse of matrix R

A P B Q C T D not possible

12. the determinant of Q

A 8 B 4 C 2 D 4

13. Find the value of |5 13 2

|.

A 13 B 7 C 17 D 3

14. Cramer’s Rule is used to solve the system of equations 2m + 3n = 11 and 3m – 5n = 6. Which

determinant represents the numerator for m?

A |11 26 3

| B |2 33 −5

| C |2 113 6

| D |11 36 −5

|

Week # 4

Chapter 4 – Quadratic Functions and Relations

Exercises

1. Find the y-intercept for f(x) = −(𝑥 + 1)2.

A 1 B –1 C x D 0

2. What is the equation of the axis of symmetry of y = –3(𝑥 + 6)2 + 12?

A x = 2 B x = –6 C x = 6 D x = –18

3. The graph of f(x) = –2𝑥2 + x opens _____ and has a _____ value.

A down; maximum C up; maximum B down; minimum D up; minimum

4. The related graph of a quadratic equation is shown at the right. Use the graph to determine the

solutions of the equation.

A –2, 3 B –3, 2 C 0, –6 D 0, 2

5. The quadratic function f(x) = 𝑥2 has ________ .

A no zeros C exactly two zeros B exactly one zero D more than two zeros

6. Solve 𝑥2 – 3x – 10 = 0 by factoring.

A {–5, 2} B (–2, –5) C {–2, 5} D {–10, 1}

7. Which quadratic equation has roots –2 and 3?

A 𝑥2 + x + 6 = 0 C 𝑥2 – 6x + 1 = 0 B 𝑥2 – x – 6 = 0 D 𝑥2 + x – 6 = 0

8. Simplify (5 + 2i)(1 + 3i).

A 5 + 6i B –1 C –1 + 17i D 11 + 17i

9. To solve 𝑥2 + 8x + 16 = 25 by using the Square Root Property, you would first rewrite the

equation as .

A (𝑥 + 4)2 = 25 B 𝑥2 + 8x – 9 = 0 C (𝑥 + 4)2 = 5 D 𝑥2 + 8x = 9

10. Find the value of c that makes 𝑥2 + 10x + c a perfect square.

A 100 B 25 C 10 D 50

11. The quadratic equation 𝑥2 + 6x = 1 is to be solved by completing the square. Which equation

would be the first step in that solution?

A 𝑥2 + 6x – 1 = 0 B 𝑥2 + 6x + 36 = 1 + 36 C x(x + 6) = 1 D 𝑥2 + 6x + 9 = 1 + 9

12. Find the exact solutions to 𝑥2 – 3x + 1 = 0 by using the Quadratic Formula.

A −3 ± √5

2 B

3 ± √13

2 C

−3 ± √13

2 D

3 ± √5

2

13. What is the vertex of y = 2(𝑥 − 3)2 + 6?

A (–3, –6) B (3, –6) C (–3, 6) D (3, 6)

14. Which quadratic inequality is graphed at the right?

A y ≥ (𝑥 + 1)2 + 4 B y ≤ –(𝑥 + 1)2 + 4 C y ≤ –(𝑥 – 1)2 + 4 D y ≤ –(𝑥 − 1)2 – 4

Week # 5

Chapter 5 - Polynomials and Polynomial Functions

Exercises 1. Simplify (3𝑥0)2(2𝑥4).

A 𝑥4 B 12𝑥4 C 18𝑥6 D 18𝑥4

2. Simplify 3𝑦2𝑧

15𝑦5 . Assume that no variable equals 0.

A 𝑧

5𝑦3 B 𝑦3𝑧

5 C 5𝑦3z D

𝑦7𝑧

5

3. Shen is simplifying the expression (3𝑥4 + 4𝑥2)( 𝑥3 – 2𝑥2 – 1). Which of the following shows the

correct product?

A 3𝑥12 – 6𝑥8 + 4𝑥6 – 11𝑥4 – 4𝑥2 C 3𝑥7 + 6𝑥6 – 4𝑥5 + 11𝑥4 + 4𝑥2

B 3𝑥7 – 6𝑥6 + 4𝑥5 – 11𝑥4 – 4𝑥2 D 3𝑥12 – 6𝑥8 – 11𝑥4 + 4𝑥6 – 4𝑥2

4. Simplify 3x(2𝑥2 – y).

A 5𝑥3 + 3xy B 12x – y C 6𝑥2 – 3y D 6𝑥3 – 3xy

5. Simplify (𝑥2 – 2x – 35) ÷ (x + 5).

A 𝑥2 – x – 30 B x – 7 C x + 5 D 𝑥3 + 3𝑥2 – 45x – 175

6. Which represents the correct synthetic division of (𝑥2 – 4x + 7) ÷ (x – 2)?

A C

B D

7. Factor 𝑚2 + 9m + 14 completely.

A m(m + 23) B (m + 14)(m + 1) C (m + 7)(m + 2) D m(m + 9) + 14

8. Simplify 𝑡2 + 𝑡 − 6

𝑡 − 2. Assume that the denominator is not equal to 0.

A t – 5 B t – 2 C t – 3 D t + 3

9. State the number of real zeros for the function whose graph is shown at the right.

A 0 B 2 C 1 D 3

10. Write the expression 𝑥4 + 5𝑥2 – 8 in quadratic form, if possible.

A (𝑥2)2 + 5(𝑥2) – 8 B (𝑥4)2 + 5(𝑥4) – 8 C (𝑥2)2 – 5(𝑥2) – 8 D not possible

11. Use synthetic substitution to find f(3) for f(x) = 𝑥2 – 9x + 5.

A –23 B –16 C –13 D 41

12. One factor of 𝑥3 + 4𝑥2 – 11x – 30 is x + 2. Find the remaining factors.

A x – 5, x + 3 B x – 3, x + 5 C x – 6, x + 5 D x – 5, x + 6

13. Which describes the number and type of roots of the equation 4x + 7 = 0?

A 1 imaginary root B 1 real root and 1 imaginary root C 2 real roots D 1 real root

14. Find all the rational zeros of p(x) = 𝑥3 – 12x – 16.

A –2, 4 B 2, –4 C 4 D –2

Week # 6

Chapter 6 – Inverse and Radical Functions and Relations

Exercises For Questions 1 and 2, use f(x) = x + 5 and g(x) = 2x.

1. Find (f + g)(x).

A 3x + 5 B x + 5 C 2x + 10 D 2𝑥2 + 5

2. Find (f ⋅ g)(x).

A 2𝑥2 + 5 B 3𝑥2 + 10x C 2𝑥2 + 10x D 2x + 10

3. If f(x) = 3x + 7 and g(x) = 2x – 5, find g[f(–3)].

A –26 B –9 C –1 D 10

4. If f(x) = 𝑥2 and g(x) = 3x – 1 find [ g ◦ f](x).

A 𝑥2 + 3x – 1 B 9𝑥2 – 1 C 9𝑥2 – 6x + 1 D 3𝑥2 – 1

5. Find the inverse of g(x) = –3x.

A 𝑔−1(x) = x + 1 B 𝑔−1 (x) = x – 1 C 𝑔−1 (x) = –3x – 3 D 𝑔−1 (x) = – 1

3x

6. Determine which pair of functions are inverse functions.

A f(x) = x – 4 B f(x) = x – 4

g(x) = x + 4 g(x) = 4x – 1

C f(x) = x – 4 D f(x) = 4x – 1

g(x) = 𝑥 – 4

4 g(x) = 4x + 1

7. State the domain and range of the function graphed.

A D = {x │ x > 2}, R = {y │ y > 0} B D = {x │ x < 2}, R = {y │ y > 0}

C D = {x │ x ≥ 2}, R = {y │ y < 0} D D = {x │ x ≥ 2}, R = {y │ y ≥ 0}

8. Which inequality is graphed?

A y ≤ √4𝑥 + 8 B y > √4𝑥 + 8 C y < √4𝑥 + 8 D y ≥ √4𝑥 + 8

9. Use a calculator to approximate √224 to three decimal places.

A 15.0 B 14.97 C 14.966 D 14.967

10. Simplify (2 + √5)(3 – √5).

A 1 + √5 B 1 – √5 C –1 + √5 D –1 – √5

11. Write the expression 51

7 in radical form.

A √517

B 35 C √57

D √75

12. Simplify the expression 𝑚2

5 • 𝑚1

5.

A 𝑚5

3 B 𝑚3

5 C 𝑚2

25 D 𝑚2

5

13. If x is a positive number, then √𝑥5

÷ 𝑥1

5 = ?

A 𝑥5 B 1

5x C 1 D

1

5

14. If 28 • y = 25, then y = ?

A –2−3 B –23 C 21

3 D 2−3

Week # 7

Chapter 7 – Exponential and Logarithmic Functions and Relations

Exercises

1. Find the domain and range of the function whose graph is shown.

A D = {x | x > 0}; R = {y | y > 0} B D = {all real numbers}; R = {y | y > 0}

C D = {x | x > 0}; R = {all real numbers} D D = {all real numbers}; R = {y | y < 0}

2. Which function represents exponential growth?

A y = 9(1

3)

𝑥 B y = 4𝑥4

C y = 12(1

5)

𝑥 D y = 10(3)𝑥

3. The graph of which exponential function passes through the points (0, 4) and (1, 24)?

A y = 4(6)𝑥 B y = 3(8)𝑥 C y = 2(2)𝑥

D y = 10(3)𝑥

4. Solve 8𝑥 + 2 = 322𝑥 + 4.

A –2 B –1 C 0 D 1

5. Solve 23𝑚−4 > 4.

A {x | m < 0} B {x | m > 0} C {x | m > 2} D {𝑥 | 𝑚 > 5

3}

6. Write the equation 43 = 64 in logarithmic form.

A log4 3 = 64 B log3 4 = 64 C log64 4 = 3 D log4 64 = 3

7. Write the equation log12 144 = 2 in exponential form.

A 1442 = 12 B 122

= 144 C 212 = 144 D 14412

= 2

8. Evaluate log2 8.

A 3 B 4 C 16 D 64

9. Solve log3 𝑛 = 2.

A 6 B 5 C 8 D 9

10. Solve log2 2m > log2(𝑚 + 5).

A {𝑥 | 𝑚 > 5

3} B {x | m < 5} C {x | m > 5} D {x | m > –5}

11. Solve log6 10 + log6 𝑥 = log6 40.

A 180 B 4 C 5 D 30

12. Solve 3𝑥 ≥ 21. Round to the nearest ten-thousandth.

A {x | x ≥ 0.8451} B {x | x ≥ 2.7712} C {x | x ≥ 0.3608} D {x | x ≥ 7.0000}

13. Express log9 22 in terms of common logarithms.

A log 22

9 B log 198 C

log 22

log 9 D

log 9

log 22

14. Evaluate 𝑒ln 4.

A 𝑒4 B 4e C ln 4 D 4

15. Martin bought a painting for $5000. It is expected to appreciate at a continuous rate of 4%. How

much will the painting be worth in 6 years? Round to the nearest cent.

A $6200.00 B $5360.38 C $37,647.68 D $6356.25

16. Solve 4𝑥 = 20. Round to the nearest ten-thousandth.

A 0.4628 B 1.5214 C 0.6990 D 2.161