Math 3 Unit 2: Solving Equations and Inequalities

Unit Title Standards

2.1 Analyzing Piecewise Functions

F.IF.9

2.2 Solve and Graph Absolute Value Equations F.IF.7B F.BF.3

2.3 Solve and Graph Absolute Value Inequalities A.CED.3

2.4 Factoring and Solving Quadratic Equations A.SSE.2

2.5 Solve and Graph Quadratic Equations

F.IF.7A

2.6 Factoring Sum and Difference of Cubes F.IF.8, A.REI.4b

2.7 Solutions of Functions

F.IF.9

2.8 Graphing Systems of Inequalities

A.CED.3

Unit 2 Review

Additional Clovis Unified Resources http://mathhelp.cusd.com/courses/math-3 Clovis Unified is dedicated to helping you be successful in Math 3. On the website above you will find videos from Clovis Unified teachers on lessons, homework, and reviews. Digital copies of the worksheets, as well as hyperlinks to the videos listed on the back are also available at this site.

Math 3 Unit 2: Online Resources 2.1 Analyzing

Piecewise Functions

• Patrick JMT: Find the Formula for a Piecewise Function from a Graph http://bit.ly/21pwfa

• Patrick JMT: Finding the Domain and Range of a Piecewise Function http://bit.ly/21pwfb

• Cool Math: Finding Relative Maximums and Minimums http://bit.ly/21pwfc

2.2 Solve and Graph Absolute Value Equations

• Virtual Nerd: Graph an Absolute Value Function http://bit.ly/22avea

• Purple Math: Solving Absolute Value Equations http://bit.ly/22aveb

• Patrick JMT: Simple Problems Solving Absolute Value Equations http://bit.ly/22avec

• eHowEducation: Solving Absolute Value Equations http://bit.ly/22avee

2.3 Solve and Graph Absolute Value Inequalities

• Khan Academy: Solving Absolute Value Inequalities http://bit.ly/23avia

• Mathispower4u: Solve and Graph Absolute Value Inequalities http://bit.ly/23avib

2.4 Factoring and Solving Quadratic Equations

• Khan Academy: Solving Quadratics by Factoring http://bit.ly/24fqea

• Purple Math: Solving Quadratic Equations by Factoring http://bit.ly/24fqeb

• Mroldridge: Factoring any Quadratic Equation http://bit.ly/24fqec

2.5 Solve and Graph Quadratic Equations

• Khan Academy: Graphing Quadratic Equations http://bit.ly/25sqea

• Purple Math: Solving Quadratic Equations by Graphing http://bit.ly/25sqeb

• Math Planet: Use Graphing to Solve Quadratic Equations http://bit.ly/25sqec

• Purple Math: Solving Quadratic Equations by Taking Square Roots http://bit.ly/25sqed

• Solving Quadratic Equations by Taking Square Roots http://bit.ly/25sqee

2.6 Factoring Sum and Difference of Cubes

• Purple Math: Factoring Sums & Differences of Cubes & Perfect Squares http://bit.ly/26sdca

• Patrick JMT: Factoring Sums and Differences of Cubes http://bit.ly/26sdcb

• Khan Academy: Factoring Sum of Cubes http://bit.ly/26sdcc

• Melissa Gresham: Solving Polynomial Equations Using the Sum and Difference of Cubes http://bit.ly/26sdcd

2.7 Solutions of Functions

• M-Squared Tutorials: Finding solutions from a graph for when f(x) =0 http://bit.ly/27sofa

• Purple Math: Finding solutions from a graph where f(x) = g(x) http://bit.ly/27sofb

• Bethany M: Finding values of a function from a graph ( ex: f(2.5), f(0) ) http://bit.ly/27sofc

• Virtual Nerd: Finding zeros of a function from a table of values http://bit.ly/27sofd

2.8 Graphing Systems of Inequalities

• Khan Academy: Graphing Linear Systems of Inequalities – No Solution http://bit.ly/28gsia

• Patrick JMT: Graphing Linear Systems of Inequalities http://bit.ly/28gsib http://bit.ly/28gsic

Math 3 Unit 2 Worksheet 1

Math 3 Unit 2 Worksheet 1 Name: Analyzing Piecewise-defined Functions Date: Per: Answer the following questions about the piecewise-defined functions below. 1. (a) State the open interval(s) on which 𝑓𝑓 is increasing.

(b) State the open interval(s) on which 𝑓𝑓 is decreasing.

(c) State the domain and range of 𝑓𝑓.

(d) State the coordinates of any relative minimums of 𝑓𝑓.

(e) State the coordinates of any relative maximums of 𝑓𝑓.

(f) Write a two pieced piecewise-defined function, 𝑓𝑓, that accurately represents the graph of f shown above.

2. (a) State the open interval(s) on which 𝑓𝑓 is increasing.

(b) State the open interval(s) on which 𝑓𝑓 is decreasing.

(c) State the domain and range of 𝑓𝑓.

(d) State the coordinates of any relative minimums of 𝑓𝑓.

(e) State the coordinates of any relative maximums of 𝑓𝑓.

(f) Write a two pieced piecewise-defined function, 𝑓𝑓, that accurately represents the graph of 𝑓𝑓 shown above.

Math 3 Unit 2 Worksheet 1

3. (a) State the open interval(s) on which 𝑓𝑓 is increasing.

(b) State the open interval(s) on which 𝑓𝑓 is decreasing. (c) State the domain and range of 𝑓𝑓. (d) State the coordinates of any relative minimums of 𝑓𝑓.

(e) State the coordinates of any relative maximums of 𝑓𝑓. (f) Write a three pieced piecewise-defined function, 𝑓𝑓, that accurately represents the graph of f shown above. 4.

(a) State the open interval(s) on which 𝑓𝑓 is increasing. (b) State the open interval(s) on which 𝑓𝑓 is decreasing. (c) State the open interval(s) on which 𝑓𝑓 is constant. (d) State the domain and range of 𝑓𝑓. (e) State the coordinates of any relative minimums of 𝑓𝑓.

(f) State the coordinates of any relative maximums of 𝑓𝑓. (g) Write a three pieced piecewise-defined function, 𝑓𝑓 that accurately represents the graph of f shown above.

(2, 2) (−2, 2) (6, 2)

(−4,−2) (4,−2)

(5, 3) (3, 3)

(1, 2)

(2, 0)

Math 3 Unit 2 Worksheet 2

Math 3 Unit 2 Worksheet 2 Name: Solving and Graphing Absolute Value Equations Date: Per: [1-4] Accurately graph 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥) on the same set of axes. 1. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥 − 2|and 𝑔𝑔(𝑥𝑥) = 3

a) 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 = __________ b) Solve algebraically for 𝑥𝑥, |𝑥𝑥 − 2| = 3

2. 𝑓𝑓(𝑥𝑥) = 2|𝑥𝑥 + 3|and 𝑔𝑔(𝑥𝑥) = 2 a) 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 = __________ b) Solve algebraically for 𝑥𝑥, 𝑓𝑓(𝑥𝑥) = 2 3. 𝑓𝑓(𝑥𝑥) = 3|𝑥𝑥 − 4|and 𝑔𝑔(𝑥𝑥) = −2 a) 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 = __________ b) Solve algebraically for 𝑥𝑥,𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) 4. 𝑓𝑓(𝑥𝑥) = −1

2|𝑥𝑥 + 2| − 1and 𝑔𝑔(𝑥𝑥) = −1

a) 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 = __________ b) Solve algebraically for 𝑥𝑥, 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥)

Math 3 Unit 2 Worksheet 2

[5-14] Solve the following absolute value equations for 𝑥𝑥 and graph the solution(s) on a number line. If there is no solution write ‘none’ and explain why. 5. 7 = |8𝑥𝑥 − 1| 6. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 9 Solve: 𝑓𝑓(𝑔𝑔(𝑥𝑥)) = −11 7. 1

7|6 − 3𝑥𝑥| = 3 8. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| and 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 − 5

Solve: 3(𝑓𝑓 ∘ 𝑔𝑔)(𝑥𝑥) = 0 9. −2 �1

2𝑥𝑥 + 4� = −12 10. 𝑓𝑓(𝑥𝑥) = 5 + 2𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = |𝑥𝑥|

Solve: 7 + 𝑔𝑔(𝑓𝑓(𝑥𝑥)) = 16 11. 8 − |4𝑥𝑥 + 1| = 11 12. 𝑓𝑓(𝑥𝑥) = 7𝑥𝑥 − 10 and 𝑔𝑔(𝑥𝑥) = |𝑥𝑥| Solve: 2(𝑔𝑔 ∘ 𝑓𝑓)(𝑥𝑥) + 1 = 9 13. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| and 𝑔𝑔(𝑥𝑥) = 3𝑥𝑥 − 6 14. 5 + 2|4𝑥𝑥 + 7| = 1 Solve: 7 − 3(𝑓𝑓 ∘ 𝑔𝑔)(𝑥𝑥) = −14

Math 3 Unit 2 Worksheet 3

Math 3 Unit 2 Worksheet 3 Name: Solving and Graphing Absolute Value Inequalities Date: Per: [1-4] Accurately graph 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥) on the same set of axes. 1. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| and 𝑔𝑔(𝑥𝑥) = 5

a) Highlight the portion of 𝑔𝑔(𝑥𝑥) where 𝑓𝑓(𝑥𝑥) ≤ 𝑔𝑔(𝑥𝑥) and

state the interval along the x-axis.

b) Solve algebraically for x, |𝑥𝑥| ≤ 5

2. 𝑓𝑓(𝑥𝑥) = 2 and 𝑔𝑔(𝑥𝑥) = −|𝑥𝑥|

a) Highlight the portion of 𝑓𝑓(𝑥𝑥) where 𝑔𝑔(𝑥𝑥) ≥ 𝑓𝑓(𝑥𝑥) and

state in interval notation the interval along the x-axis.

b) Solve algebraically for 𝑥𝑥, 𝑔𝑔(𝑥𝑥) ≥ 𝑓𝑓(𝑥𝑥)

3. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| + 1 and 𝑔𝑔(𝑥𝑥) = −3

a) Highlight the portion of 𝑔𝑔(𝑥𝑥) where 𝑓𝑓(𝑥𝑥) ≥ 𝑔𝑔(𝑥𝑥) and

state the interval along the x-axis

b) Solve algebraically for 𝑥𝑥 in inequality notation, 𝑓𝑓(𝑥𝑥) ≥ 𝑔𝑔(𝑥𝑥)

4. 𝑔𝑔(𝑥𝑥) = 5 and 𝑓𝑓(𝑥𝑥) = |𝑥𝑥 − 2|

a) Highlight the portion of 𝑔𝑔(𝑥𝑥) where 𝑔𝑔(𝑥𝑥) ≤ 𝑓𝑓(𝑥𝑥) and

state the interval along the x-axis

b) Solve algebraically for 𝑥𝑥 in inequality notation, 𝑔𝑔(𝑥𝑥) ≤ 𝑓𝑓(𝑥𝑥)

Math 3 Unit 2 Worksheet 3

[5-14] Solve the following absolute value inequalities for 𝑥𝑥 and graph the solution on a number line. Write the solution in interval notation. If there is no solution, explain why. 5. |3𝑥𝑥 − 15| ≥ 30 6. −1

2|𝑥𝑥 + 4| < 10

7. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| and 𝑔𝑔(𝑥𝑥) = 4𝑥𝑥 + 1 8. |𝑥𝑥| < −6 Solve: 𝑓𝑓(𝑔𝑔(𝑥𝑥)) − 14 < −5

9. 𝑓𝑓(𝑥𝑥) = 7 − 2𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = |𝑥𝑥| 10. �17𝑥𝑥 + 2� − 5 > 3

Solve: 2𝑔𝑔(𝑓𝑓(𝑥𝑥)) − 1 ≤ 37

11. �5−𝑥𝑥6� < 2 12. −3

7|3𝑥𝑥 + 4| < −21

13. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 1 14. �23𝑥𝑥� + 4 > 2

Solve: 𝑓𝑓(𝑔𝑔(𝑥𝑥)) − 3 ≤ −5

Math 3 Unit 2 Worksheet 3

[15-16] Selected Response. Select ALL answers that apply

15. What is the solution of |6𝑥𝑥 − 9| ≤ 33 ?

a. −4 ≤ 𝑥𝑥 ≤ 7 b. −7 ≤ 𝑥𝑥 ≤ 4 c. 𝑥𝑥 ≤ −4 𝑜𝑜𝑜𝑜 𝑥𝑥 ≥ 7 d. 𝑥𝑥 ≤ 7 𝑜𝑜𝑜𝑜 𝑥𝑥 ≥ −4 16. Which inequalities are equivalent to −2|𝑥𝑥 − 3| < 8 ?

a. |𝑥𝑥 − 3| < −4 b. |𝑥𝑥 − 3| > −4 c. |𝑥𝑥 − 3| < 10 d. |−2𝑥𝑥 + 6| < 8 [17-24] Use the definitions of absolute value equations and inequalities to determine if the statement is True or False. 17. True or False: |3𝑥𝑥 + 7| = 13 is equivalent to 3𝑥𝑥 + 7 = 13 𝑜𝑜𝑜𝑜 3𝑥𝑥 + 7 = −13

18. True or False: |9𝑥𝑥 + 1| < 19 is equivalent to 9𝑥𝑥 + 1 < 19 𝑜𝑜𝑜𝑜 9𝑥𝑥 + 1 > −19

19. True or False: |2𝑥𝑥 − 4| < 12 is equivalent to −12 < 2𝑥𝑥 − 4 < 12

20. True or False: |6𝑥𝑥 − 4| = −10 is equivalent to 6𝑥𝑥 − 4 = 10 𝑜𝑜𝑜𝑜 6𝑥𝑥 − 4 = −10

21. True or False: |2𝑥𝑥 + 5| > −1 has no solution

22. True or False: 𝑥𝑥 < 2 or 𝑥𝑥 > 7 is equivalent to 7 < 𝑥𝑥 < 2

23. True or False: 𝑥𝑥 ≤ 5 or 𝑥𝑥 > −1 is equivalent to −1 < 𝑥𝑥 ≤ 5

24. True or False: 𝑥𝑥 ≥ −3 or 𝑥𝑥 < −5 is equivalent to −3 ≤ 𝑥𝑥 < −5

25. An archery store carries bows that are from 40 to 52 inches long. They recommend that bows be 23 times a

person’s arm span, 𝑠𝑠. a) Write a compound inequality, in terms of 𝑠𝑠, that represents the problem. b) Solve the compound inequality from part a) c) Graph the solution from part b) on a number line. d) If a person’s height is equivalent to their arm span, what is the height of the tallest person that the archery store carries bows for?

Math 3 Unit 2 Worksheet 3

26. A sporting goods store carries softball bats that are from 25 to 35 inches long. They recommend that softball bats be 5

8 of a person’s height, ℎ.

a) Write a compound inequality, in terms of ℎ, that represents the problem. b) Solve the compound inequality from part a) c) Graph the solution from part b) on a number line. d) What is the height of the shortest person the sporting goods store carries a softball bat for? 27. A space-themed miniature golf course named Puttnik carries putters from 24 to 40 inches long. They recommend that putters be 4

9 times a person’s height, ℎ.

a) Write a compound inequality, in terms of ℎ, that represents the problem. b) Solve the compound inequality from part a) c) Graph the solution from part b) on a number line. d) What is the height of the tallest person that the golf course carries a putter for?

Math 3 Unit 2 Worksheet 4

Math 3 Unit 2 Worksheet 4 Name: Factoring and Solving Quadratic Equations Date: Per: [1 – 16] Completely Factor the Following 1. 𝑥𝑥2 − 8𝑥𝑥 + 12 2. 𝑥𝑥2 + 𝑥𝑥 − 6 3. 𝑥𝑥2 − 3𝑥𝑥 − 10 4. 𝑥𝑥2 − 16 5. 𝑥𝑥2 − 12𝑥𝑥 6. 𝑥𝑥2 + 4𝑥𝑥 − 5 7. 2𝑥𝑥2 + 5𝑥𝑥 + 2 8. 3𝑥𝑥2 + 4𝑥𝑥 + 1 9. 4𝑥𝑥2 + 13𝑥𝑥 + 3 10. 2𝑥𝑥2 + 7𝑥𝑥 + 5 11. 8𝑥𝑥2 + 30𝑥𝑥 12. 25𝑥𝑥2 − 49 13. 𝑥𝑥(𝑥𝑥 − 5) + 3(𝑥𝑥 − 5) 14. 7(𝑥𝑥 + 1) − 𝑥𝑥(𝑥𝑥 + 1) 15. 3𝑥𝑥2 − 8𝑥𝑥 + 4 16. 3𝑥𝑥2 + 5𝑥𝑥 + 2 [17 – 25] Solve By Factoring 17. 𝑥𝑥2 − 7𝑥𝑥 + 10 = 0 18. 𝑥𝑥2 − 4𝑥𝑥 − 12 = 0 19. 2𝑥𝑥2 + 9𝑥𝑥 − 5 = 0 20. 𝑥𝑥(3𝑥𝑥 − 1) = 4 21. (𝑥𝑥 + 2)(2𝑥𝑥 − 1) = 3 22. (2𝑥𝑥 + 3)(3𝑥𝑥 + 1) = 3 23. 4𝑥𝑥2 − 8𝑥𝑥 = 0 24. 5𝑥𝑥2 = 10𝑥𝑥

Math 3 Unit 2 Worksheet 4

25. 8𝑥𝑥2 + 2𝑥𝑥 = 3 26. 9𝑥𝑥2 − 16 = 0 27. 𝑥𝑥(𝑥𝑥 − 1) + 9(𝑥𝑥 − 1) = 0 28. 36𝑥𝑥2 = 121 [29 – 34] Solve By Using Square Roots 29. (𝑥𝑥 − 5)2 = 9 30. 2(𝑥𝑥 + 1)2 = 24 31. (𝑥𝑥 + 2)2 − 7 = 17 32. 1 + 3(𝑥𝑥 + 4)2 = 13 33. 31 − 2(𝑥𝑥 − 5)2 = 7 34. 1 + 2(2𝑥𝑥 − 3)2 = 17 [35 – 38] Completely Simplify Each Expression

35. 6±12√3

4 36.

9±�(−9)2−4(1)(4)2(1)

37. −5±�52−4(5)(−1)

2(5) 38.

6±�(−6)2−4(1)(−3)2(1)

Math 3 Unit 2 Worksheet 5

Math 3 Unit 2 Worksheet 5 Name: Solving and Graphing Quadratic Equations Date: Per:

[1-4] Accurately graph 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥) on the same set of axes.

1. 𝑓𝑓(𝑥𝑥) = −(𝑥𝑥 − 1)2 + 4 and 𝑔𝑔(𝑥𝑥) = 0 a) 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 = __________ b) Solve algebraically for 𝑥𝑥: −(𝑥𝑥 − 1)2 + 4 = 0

2. 𝑓𝑓(𝑥𝑥) = 1

2(𝑥𝑥 − 2)2 − 2 and 𝑔𝑔(𝑥𝑥) = 0

a) 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 = __________ b) Solve algebraically for 𝑥𝑥: 12

(𝑥𝑥 − 2)2 − 2 = 0 3. 𝑓𝑓(𝑥𝑥) = 2(𝑥𝑥 + 3)2 + 2 and 𝑔𝑔(𝑥𝑥) = 0 a) 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 = __________ b) Solve algebraically for 𝑥𝑥: 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) 4. 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 1)2 − 2 and 𝑔𝑔(𝑥𝑥) = 0

a) 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 ≈ __________ b) Solve algebraically for 𝑥𝑥 to the nearest tenth, 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥)

Math 3 Unit 2 Worksheet 5

[5-16] Solve the following quadratic equations for x.

5. 2(𝑥𝑥 + 3)2 + 9 = 5 6. 3𝑥𝑥2 − 11𝑥𝑥 − 4 = 0 7. (𝑥𝑥 + 1)(𝑥𝑥 + 5) = 3 8. 𝑥𝑥2 + 4𝑥𝑥 − 6 = 0 9. 𝑥𝑥2 + 2𝑥𝑥 + 5 = 0 10. (𝑥𝑥 + 4)2 − 5 = 6 11. 2𝑥𝑥2 − 6𝑥𝑥 + 5 = −2 12. 𝑥𝑥(𝑥𝑥 − 3) = 7 13. 2𝑥𝑥 − 8 = 𝑥𝑥2 + 𝑥𝑥 14. −3𝑥𝑥2 + 𝑥𝑥 + 2 = 0 15. 20 + 3(𝑥𝑥 + 7)2 = −34 16. 16 − 3(𝑥𝑥 − 5)2 = 88

Math 3 Unit 2 Worksheet 6

Math 3 Unit 2 Worksheet 6 Name: Factoring Sum and Difference of Cubes Date: Per:

Factor completely: 1. 𝑥𝑥3 + 8 2. 𝑥𝑥3 − 𝑦𝑦3 3. 125 − 8𝑦𝑦3 4. 64𝑎𝑎3 − 125𝑟𝑟3 5. 27𝑚𝑚3 + 216𝑛𝑛3 6. 81𝑦𝑦4 + 3𝑦𝑦 For each equation, write in factored form then determine the real and imaginary solutions.

7. 𝑥𝑥3 − 125 = 0 Factored Form: Real solutions: Imaginary Solutions:

8. 0 = 1000 + 𝑥𝑥3 Factored Form: Real solutions: Imaginary Solutions:

9. 8𝑥𝑥3 + 1 = 0 Factored Form: Real solutions: Imaginary Solutions:

Math 3 Unit 2 Worksheet 6

10. 64 − 27𝑥𝑥3 = 0 Factored Form: Real solutions: Imaginary Solutions:

11. 0 = 250𝑥𝑥4 − 54𝑥𝑥 Factored Form: Real solutions: Imaginary Solutions:

12. 24𝑥𝑥 + 81𝑥𝑥4 = 0 Factored Form: Real solutions: Imaginary Solutions: 13. Is it possible to solve an equation using sum or difference of cubes factoring and have all solutions be

imaginary? Please explain your thinking.

Math 3 Unit 2 Worksheet 7

Math 3 Unit 2 Worksheet 7 Name: Solutions of Functions Date: Per: 1. The graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) is shown in the graph below.

2. The graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) is shown in the graph below.

3. The graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) is shown in the graph below.

𝑔𝑔(𝑥𝑥) 𝑓𝑓(𝑥𝑥)

a) How many solutions are there for 𝑔𝑔(𝑥𝑥) = 0?

b) How many solutions are there for 𝑔𝑔(𝑥𝑥) = 𝑓𝑓(𝑥𝑥)?

c) 𝑓𝑓(2.5) = _______________.

d) How many times does 𝑔𝑔(𝑥𝑥) = 3.5?

e) On what approximate interval is 𝑔𝑔(𝑥𝑥) < −1?

f) How many roots does 𝑔𝑔(𝑥𝑥) have? g) Is 𝑔𝑔(𝑥𝑥) = 0 between 𝑥𝑥 = 4 and 𝑥𝑥 = 5?

A

C

B D

A

B

D

C

E

𝑓𝑓(𝑥𝑥)

𝑔𝑔(𝑥𝑥)

E

a) List all of the labeled points that are solutions for 𝑓𝑓(𝑥𝑥) = 0. b) List all of the labeled points that are solutions for 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). c) List all of the labeled points that are solutions for 𝑥𝑥 = 0. d) 𝑓𝑓(0) =

a) List all of the labeled points that are solutions for 𝑓𝑓(𝑥𝑥) = 0. b) List all of the labeled points that are solutions for 𝑔𝑔(𝑥𝑥) = 𝑓𝑓(𝑥𝑥). c) List all of the labeled points that are solutions for 𝑥𝑥 = 0 on the graph of 𝑓𝑓(𝑥𝑥). d) List all of the labeled points that are solutions for 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). e) List all of the labeled points that are solutions for 𝑦𝑦 = 𝑔𝑔(𝑥𝑥).

Math 3 Unit 2 Worksheet 7

4. State whether the following statements are correct (A) or incorrect (B)

a) 𝑓𝑓(4) > 0 b) 𝑓𝑓(−2) < 0 c) 𝑓𝑓(0) = −2 d) 𝑓𝑓(2) = 0 e) 𝑓𝑓(5.123) = 2

f) y = 0 at 𝑓𝑓(2) g) 𝑓𝑓(𝑥𝑥) = 0 between 𝑥𝑥 = 6 and 𝑥𝑥 = 8 h) 𝑓𝑓(𝑥𝑥) has a relative maximum at (4, 2) i) 𝑓𝑓(𝑥𝑥) equals zero five times j) . 𝑓𝑓(𝑥𝑥) has a relative minimum at (0,−2) 5. The table below shows several points on two continuous functions, 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥). a) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) must exist. If it is not necessary that a solution exists, explain why.

b) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑓𝑓(𝑥𝑥) = 0 must exist. If it is not necessary that a solution exists, explain why.

c) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑔𝑔(𝑥𝑥) = 4 must exist. If it is not necessary that a solution exists, explain why.

0 1 2 3

0 1 2 3

0 1 2 3

𝑥𝑥 0 1 2 3 𝑓𝑓(𝑥𝑥) –5 –2 1 –5 𝑔𝑔(𝑥𝑥) –7 –3 2 –1

Math 3 Unit 2 Worksheet 7

6. The table below shows several points on two continuous functions, 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥).

x 0 1 2 3 𝑓𝑓(𝑥𝑥) 5 3 1 –1 𝑔𝑔(𝑥𝑥) –1 –2 4 6

a) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) must exist. If it is not necessary that a solution exists, explain why.

b) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑓𝑓(𝑥𝑥) = 0 must exist. If it is not necessary that a solution exists, explain why.

c) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑔𝑔(𝑥𝑥) = 3 must exist. If it is not necessary that a solution exists, explain why.

7. The table below shows several points on three continuous functions, 𝑓𝑓(𝑥𝑥), 𝑔𝑔(𝑥𝑥) and ℎ(𝑥𝑥). a) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) must exist. If it is not necessary that a solution exists, explain why.

b) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑓𝑓(𝑥𝑥) = ℎ(𝑥𝑥) must exist. If it is not necessary that a solution exists, explain why.

c) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑔𝑔(𝑥𝑥) = ℎ(𝑥𝑥) must exist. If it is not necessary that a solution exists, explain why.

d) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑔𝑔(𝑥𝑥) = 2 and ℎ(𝑥𝑥) = 2 must exist. If it is not necessary that a solution exists, explain why.

e) On the number line below, shade the interval(s) between the integers where the solution(s) to ℎ(𝑥𝑥) = 0 must exist. If it is not necessary that a solution exists, explain why.

0 1 2 3

0 1 2 3

0 1 2 3 4

𝑥𝑥 0 1 2 3 4 𝑓𝑓(𝑥𝑥) 3 1 3 5 7 𝑔𝑔(𝑥𝑥) 0 –3 1 7 –2 ℎ(𝑥𝑥) –4 –1 0 4 –3

0 1 2 3

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

Math 3 Unit 2 Worksheet 7

[8-15]: Solve. Show all work. 8. 𝑥𝑥2 + 2𝑥𝑥 = 5𝑥𝑥 + 10 9. 2𝑥𝑥2 + 4𝑥𝑥 = 5𝑥𝑥 + 28 10. 4𝑥𝑥(𝑥𝑥 + 3) = 12𝑥𝑥 + 25 11. 100 + 2(𝑥𝑥 − 3)2 = 12 12. 14 − 3(𝑥𝑥 + 2)2 = −106 13. 3|𝑥𝑥 + 4| − 17 = 13 14. 5 − 2|𝑥𝑥 − 7| = −21 15. 𝑥𝑥3 − 125 = 0 Hint: Quadratic formula is not needed for any of the above problems except #16. All answers may be found below; however, they are in no particular order. {−14, 6} �3 ± 2𝑖𝑖√11 � {−2, 5} �±

52� �5,

−5 ± 5𝑖𝑖√32 � {−6, 20} �−

72 , 4� �−2 ± 2√10 �

Math 3 Unit 2 Worksheet 8

Math 3 Unit 2 Worksheet 8 Name: Graphing Systems of Inequalities Date: Per: Graph the system of inequalities on the graph provided. If the solution exists, then name 2 points in the solution set.

1. �𝑦𝑦 ≤ 4 − 𝑥𝑥

𝑦𝑦 > 2𝑥𝑥 − 3 2. � 𝑦𝑦 < −𝑥𝑥2 + 1𝑦𝑦 ≤ 2𝑥𝑥 − 3

3. � 𝑦𝑦 ≥ −|𝑥𝑥 + 1| 𝑦𝑦 < 2|𝑥𝑥| − 2 4. �

𝑦𝑦 ≥ 2𝑥𝑥 − 1 𝑦𝑦 < −1

2𝑥𝑥 − 1

5. � 𝑦𝑦 ≥|𝑥𝑥| − 2

𝑦𝑦 ≤ 2 − |𝑥𝑥| 6. �𝑦𝑦 < −2

𝑦𝑦 ≥ (𝑥𝑥 + 1)2 + 2

Math 3 Unit 2 Worksheet 8

7. � 𝑦𝑦 < 4 𝑦𝑦 ≤ 2𝑥𝑥𝑦𝑦 > −𝑥𝑥

8. �𝑦𝑦 ≤ 3 − 𝑥𝑥

𝑦𝑦 > 3 − |𝑥𝑥|

9. � 𝑦𝑦 ≤ −2 𝑦𝑦 > (𝑥𝑥 − 2)2 − 1 10. �

𝑥𝑥 > −1 𝑦𝑦 ≥ 2𝑥𝑥

𝑦𝑦 ≤ 1 − 𝑥𝑥

11. � 𝑦𝑦 < 0 𝑦𝑦 ≤ 2 − |𝑥𝑥|𝑥𝑥 ≥ 0

12. � 𝑦𝑦 ≤ |𝑥𝑥| − 4 𝑦𝑦 ≥ 1

2𝑥𝑥2 − 4

Math 3 Unit 2 Worksheet 8

[13-21]: Solve the following for 𝑥𝑥. 13. 2|𝑥𝑥 + 3| − 17 < −5 14. 100 − 3|𝑥𝑥 + 11| ≤ 28 15. 100 + 3|𝑥𝑥 + 5| < 25 16. 2|𝑥𝑥 − 4| − 21 ≥ −63 17. 15 − |2𝑥𝑥 + 5| > −9 18. 20 + 3|𝑥𝑥 + 9| > 50 19. 2𝑥𝑥(𝑥𝑥 + 3) = 5(8 − 𝑥𝑥) 20. 50 − (𝑥𝑥 + 7)2 = 68 21. 1000 + 𝑥𝑥3 = 0 Hint: Quadratic formula is only needed for question 21. All answers may be found below; however, they are in no particular order.

(−∞,∞) { ∅ } �−7 ± 3𝑖𝑖√2 � �−292 ,

192 � �−8,

52� (−∞,−35] ∪ [13,∞) {−10, 5 ± 5𝑖𝑖√3 } (−9, 3) (−∞,−19) ∪ (1,∞)

Math 3 Unit 2 Worksheet 8

Math 3 Unit 2 Review Worksheet

Math 3 Unit 2 Name: Review Date: Per: 1.

(a) State the open interval(s) on which 𝑓𝑓 is increasing. (b) State the open interval(s) on which 𝑓𝑓 is decreasing. (c) State the domain and range of 𝑓𝑓. (d) State the coordinates of any relative minimums of 𝑓𝑓.

(e) State the coordinates of any relative maximums of 𝑓𝑓. (f) Write a two pieced piecewise-defined function, 𝑓𝑓, that accurately represents the graph of 𝑓𝑓 shown above.

2.

(a) State the open interval(s) on which 𝑓𝑓 is increasing. (b) State the open interval(s) on which 𝑓𝑓 is decreasing. (c) State the domain and range of 𝑓𝑓. (d) State the coordinates of any relative minimums of 𝑓𝑓. (e) State the coordinates of any relative maximums of 𝑓𝑓.

(f) Write a three pieced piecewise-defined function, 𝑓𝑓, that accurately represents the graph of 𝑓𝑓 shown above.

Math 3 Unit 2 Review Worksheet

3. Graph 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥) on the same set of axes. Use the graph and verify algebraically, where 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥).

𝑓𝑓(𝑥𝑥) = 3|𝑥𝑥 + 1| and 𝑔𝑔(𝑥𝑥)

[4-7] Solve the following absolute value equations for x and graph the solution(s) on a number line. If there is no solution write ‘none’ and explain why. 4. 𝑓𝑓(𝑥𝑥) = |𝑥𝑥| and 𝑔𝑔(𝑥𝑥) = 3𝑥𝑥 + 2 5. − �𝑥𝑥

2− 5� = 4

Solve: 𝑓𝑓(𝑔𝑔(𝑥𝑥)) + 1 = 12

6. 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 + 2 and 𝑔𝑔(𝑥𝑥) = |𝑥𝑥| 7. 3 �𝑥𝑥4− 10� = 0

Solve: 25

(𝑔𝑔 ∘ 𝑓𝑓)(𝑥𝑥) = 20

[8-11] Solve the following absolute value inequalities for x and graph the solution(s) on a number line. If there is no solution write ‘none’ and explain why. 8. |3𝑥𝑥 − 10| ≤ 2 9. −2

3|𝑥𝑥 + 4| < 6

Math 3 Unit 2 Review Worksheet

10. 3 �2𝑥𝑥3− 1� + 4 < −2 11. 2 + 2|𝑥𝑥 − 5| ≥ 0

12. The graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) is shown in the graph below.

a) List all of the labeled points that are solutions for 𝑔𝑔(𝑥𝑥) = 0.

b) List all of the labeled points that are solutions for 𝑔𝑔(𝑥𝑥) = 𝑓𝑓(𝑥𝑥).

c) List all of the labeled points that are solutions for 𝑥𝑥 = 0 on the graph of 𝑓𝑓(𝑥𝑥).

d) List all of the labeled points that are solution(s) for 𝑔𝑔(𝑥𝑥) < 𝑓𝑓(𝑥𝑥).

e) List all of the labeled points that are solution(s) for 𝑔𝑔(𝑥𝑥) > 𝑓𝑓(𝑥𝑥).

13. The table below shows several points on two continuous functions, 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥)

𝑥𝑥 0 1 2 3 𝑓𝑓(𝑥𝑥) 0 2 4 5 𝑔𝑔(𝑥𝑥) –1 3 2 –2

a) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) must exist. If no solutions must exist, explain why.

b) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑔𝑔(𝑥𝑥) = 0 must exist. If no solutions must exist, explain why.

c) On the number line below, shade the interval(s) between the integers where the solution(s) to 𝑓𝑓(𝑥𝑥) = 3 must exist. If no solutions must exist, explain why.

0 1 2 3

0 1 2 3

0 1 2 3

𝑓𝑓(𝑥𝑥)

A B

C

E

D

𝑔𝑔(𝑥𝑥)

F

Math 3 Unit 2 Review Worksheet

[14-17] Graph the system of inequalities on the graph provided.

14. � 𝑦𝑦 ≤ 4 𝑦𝑦 > 𝑥𝑥 15. � 𝑦𝑦 > 𝑥𝑥2 + 1

𝑦𝑦 < 1 − 𝑥𝑥2

16. � 𝑦𝑦 ≤13

|𝑥𝑥 + 2|𝑦𝑦 ≥ −|𝑥𝑥|

17. � 𝑦𝑦 > 2𝑥𝑥 − 2𝑦𝑦 < −𝑥𝑥 + 1

Solve for x.

18. 3(𝑥𝑥 − 2)2 + 15 = 90 19. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 3 20. 2(𝑥𝑥 − 5)2 − 6 = 120 Solve: −2𝑓𝑓�𝑔𝑔(𝑥𝑥)� + 30 = −212

21. 14𝑥𝑥 = 24 − 3𝑥𝑥2 22. 26𝑥𝑥 + 36 = 6𝑥𝑥 − 𝑥𝑥2 23. 𝑓𝑓(𝑥𝑥) = 8𝑥𝑥 − 4 and 𝑔𝑔(𝑥𝑥) = 7𝑥𝑥2 Solve: (𝑓𝑓 − 𝑔𝑔)(𝑥𝑥) = 0

Math 3 Unit 2 Review Worksheet

24. Use the graph of 𝑓𝑓(𝑥𝑥) below to answer the following questions.

True or False:

a) 𝑓𝑓(5) > 1

b) 𝑓𝑓(8) < −1

c) 𝑓𝑓(𝑥𝑥) = 1 between 𝑥𝑥 = 5 and 𝑥𝑥 = 6

d) 𝑓𝑓(𝑥𝑥) = 1 between 𝑥𝑥 = 0 and 𝑥𝑥 = 2

e) 𝑓𝑓(0) = 0

f) 𝑓𝑓(−7) = 4

g) 𝑓𝑓(3.743) = 1

h) 𝑓𝑓(9.324) = −2

i) 𝑓𝑓(𝑥𝑥) has a relative maximum at (−2,−4)

j) 𝑓𝑓(𝑥𝑥) has a relative maximum at (3, 2)

k) 𝑓𝑓(𝑥𝑥) has a relative minimum at (9,−2)

l) 𝑓𝑓(𝑥𝑥) has a relative minimum at (−2,−4)

−7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−5

−4

−3

−2

−1

1

2

3

4

x

y

Math 3 Unit 2 Review Worksheet