Name: ________________________________________ Date: _______________________ AA Trig

ADVANCED ALGEBRA/TRIGONOMETRY MIDTERM EXAM REVIEW! On the pages that follow, is a compilation of all chapter review packets distributed throughout the semester for all chapters tested on the AA Trig Midterm Exam. TOPICS COVERED: I. Advanced Algebra Topics

A. Linear and Quadratic Functions (Chapter 1) 1. Points, slopes and lines 2. Finding equations of lines 3. Complex numbers 4. Solving and graphing quadratics 5. Linear and quadratic models

B Polynomial Functions (Chapter 2) 1. Remainder and factor theorems, synthetic division 2. Graphing polynomial functions 3. Finding maximums and minimums 4. Solving polynomial equations by technology and by factoring

C. Inequalities (Chapter 3) 1. Linear inequalities 2. Polynomial inequalities in one variable 3. Polynomial inequalities in two variables

D. Functions (Chapter 4) 1. Properties of functions 2. Operations on functions 3. Symmetry, graphing 4. Periodic Functions, graphing 5. Inverse Functions

MIDTERM EXAM STUDYING RECOMMENDATIONS: Set a specific schedule for studying & stick to it!

Study a little bit every day - DO NOT CRAM!

Any problems used as review should have answers available to be checked against – you need to know you are doing things correctly!

Allow yourself time to be able to ask your teacher questions about problems/concepts you are still unsure of.

DO NOT STRESS!

Get a good night’s sleep the night before the exam and eat a decent breakfast!

Name: ________________________________________ Date: _______________________ AA Trig

Chapter 1 Part 1: Review of Linear Functions 1. Let A = (-4, 7) and and B = (4, -5). a. Find the length of segment AB. b. Find the coordinates of the midpoint of segment AB. 2. Find the value of a if it is known that the point (-3, 7) lies on the line 2𝑥 + 𝑎𝑦 = 26. 3. Solve the system of equations 3𝑥 − 2𝑦 = 3 and 5𝑥 + 4𝑦 = 16 for the variables. What does your solution mean in reference to the graph of the system? 4. Find the slope and the y-intercept of the line 3𝑥 + 2𝑦 = −1.

Name: ________________________________________ Date: _______________________ AA Trig

5. Tell which of the following equations represent parallel lines. Which of the following represent

perpendicular lines?

a. 3𝑥 − 2𝑦 = 12 b. 𝑦 =3

2𝑥 + 1 c. 4𝑥 + 6𝑦 + 20 = 0

6. Write an equations of the line through the points (6, -2) and (3, 7). 7. Write an equation of the line through the point (2, 5) and parallel to the line 4𝑥 − 3𝑦 = −50. 8. Write an equation of the line with x-intercept 2 and y-intercept -3.

Name: ________________________________________ Date: _______________________ AA Trig

9. Write an equation of the vertical line through the point (5, 1). 10. RECALL: The median of a triangle is a segment that joins one vertex of a triangle with the

midpoint of the opposite side. a. Writing: Describe the steps for finding an equation of the median from vertex A in ∆ABC. b. Write an equation of the median from A if A(2,1), B(4, 8), and C(8,-2) are the vertices of ∆ABC.

11. Twyla buys a 10-ride pass so that she can use public transportation to go to and from work. The pass costs $15 and is worth $0 after the tenth ride. a. Give an equation of the linear function that models the value of the pass as a function of the number of rides completed. b. Use the equation to find the value of the pass after the sixth ride.

Name: ________________________________________ Date: _______________________ AA Trig

Chapter 1 Part 2: QUADRATIC FUNCTIONS (Non-Calculator Section) 1. Explain the method you would use to solve each of the quadratic equations below.

a. 2𝑥2 + 19𝑥 = −10 b. 𝑥 + 9 = 3√𝑥

c. 𝑥2 − 8𝑥 + 1 = 12 − 2𝑥 d. (𝑥 + 2)2 − 10 = −15 2. Solve each equation. Circle your final answer. Do not use approximations for square roots. Be

sure to check for extraneous solutions!

a.) 2x2 + 12x + 20 = 0 b.) 7x2 + 5x – 2 = 0 c.) 4

𝑧=

3𝑧

𝑧−3

d.) (4y+4)2 = -16 e.) 2𝑥2 − 5𝑥 + 8 = 0

Name: ________________________________________ Date: _______________________ AA Trig

3. Simplify a.) (9 + i)(6 – i) b.) (2 + 3i)2

c.) √−50 − √−8 d.) √3+𝑖

√3−𝑖

4. When given 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, how do you find? AOS: X-Intercepts: Vertex: Y-Intercepts: 5. Write the equation of the parabola below in the forms specified.

a.) Vertex form:

b.) Factored form:

c.) Standard form:

Name: ________________________________________ Date: _______________________ AA Trig

6. Graph. Show all work!

a.) 𝑦 = −1

2(𝑥 + 4)(𝑥 − 1)

vertex: ______________ x-intercepts: ___________ A.O.S.: ______________ y-intercept: ______________ b.) 𝑦 = −2(𝑥 + 2)2 + 3 vertex: ______________ x-intercepts: ___________ A.O.S.: ______________ y-intercept: ______________ c.) 𝑦 = 7𝑥2 − 5𝑥 − 2 vertex: ______________ x-intercepts: ___________ A.O.S.: ______________ y-intercept: ______________

1 2 3 4 5 6 7–1–2–3–4–5–6–7 x

1

2

3

4

5

6

7

8

9

10

11

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

y

Name: ________________________________________ Date: _______________________ AA Trig

(Calculator Section) 7. A farmer planting wheat has found that the

yield per acre in bushels depends on the amount of seed sown. By experimenting, he has collected the data given in the following table.

a.) Draw a quick sketch of the graph based on the data in the table. b.) Does a quadratic function make sense for this data? Why? c.) Find a function that models the information in the table. d.) If I wanted to, how could I verify that this function fits the data (explain but do not DO)? e.) If the farmer planted 6 bags of seed, how many bushels per acre would the farmer yield? f.) If the farmer yields 75 bushels per acre, what’s the smallest number of bags of seed needed?

Number of bags of seed 3 4 5

Number of bushels per acre

70 85 80

Name: ________________________________________ Date: _______________________ AA Trig

8. Find the equation of the quadratic function that meets the given criteria. Write the equation in standard form.

a.) Vertex: (1, 7) b.) x-intercepts: (4,0) and (6,0)

Goes through (–4, –11) y-intercept = (0,9) Equation:_____________________ Equation:_____________________

9. Simplify: 𝑖26−𝑖14

𝑖2 10. Explain the relationship of a graph

of a parabola and the solutions to a quadratic equation that models that parabola. 11. Give an example of a quadratic equation with imaginary solutions. Explain your reasoning.

Name: ________________________________________ Date: _______________________ AA Trig

Chapter 2: Polynomial Functions

1. If 𝑃(𝑥) = 4𝑥3 − 5𝑥2 + 1, use synthetic substitution to find:

a. 𝑃(2) b. 𝑃 (1

2)

2. If −2 is a zero of the polynomial 𝑃(𝑥) = 2𝑥2 + 𝑥 + 𝑘, find the value of 𝑘.

3. A polynomial 𝑃(𝑥) is divided by 𝑥 − 2. The quotient is 2𝑥2 + 𝑥 − 2 and the remainder is −1. Find 𝑃(𝑥).

4. Two roots of the equation below are 𝑥 = 2 and 𝑥 = −3. Find the remaining roots.

𝒙𝟒 + 𝒙𝟑 − 𝟓𝒙𝟐 + 𝒙 − 𝟔 = 𝟎

5. Sketch the graph of 𝑓(𝑥) = −𝑥2(𝑥 − 1)(𝑥 + 2).

Name: ________________________________________ Date: _______________________ AA Trig

6. Give the equation fo the cubic graph shown below in standard form.

7. A rectangular enclosure, subdivided into three congruent pens is to be made using a barn as one side. 120 m of fencing will be used for the rest of the enclosure. Find the value of 𝑥 that gives the maximum area for the enclosure.

8. If 𝑃(𝑥) = 2𝑥4 − 2𝑥 − 1, a. Explain how you know that the equation 𝑃(𝑥) = 0 has at least one root between 𝑥 =

1 and 𝑥 = 2.

b. Use an appropriate method to find all of the roots of the equation 𝑃(𝑥) = 0.

x x x

Name: ________________________________________ Date: _______________________ AA Trig

9. Solve 2𝑥3 + 2𝑥2 − 4𝑥 − 4 = 0.

10. Find all roots of 2𝑥4 + 3𝑥3 + 7𝑥2 = 7𝑥 + 5.

Name: ________________________________________ Date: _______________________ AA Trig

Chapter 3: Polynomial Inequalities

Section 3.1: Solve. Graph your solution.

1. 4𝑥−9

6−

11−7𝑥

3 < –4 2.

3−𝑥

4≥

−1−2𝑥

2

3. |2x – 7| > 3 4. |5x + 1| ≤ 11

5. 5. -2|8x + 2|+ 3 < –5 6. Think about which of the following inequalities would result in “All Real Numbers” and which would result in “No Solution.” Without actually solving, decide!

|9x + 6| < –5 |9x + 6| ≥ 0

Name: ________________________________________ Date: _______________________ AA Trig

Solve then graph. Show all work. sdfsdfsdfdsfsdf

7. 4

3(𝑥 −

1

2) +

1

2𝑥 ≥

2

3(2𝑥 −

5

2) ssfsdfsdfsfsdfsdfsdfsfsd

Section 3.3: Graph each inequality.

8. f(x) > x3 – 3x2 – 18x 9. h(x) ≤ x3 – 3x2 – x + 3

Solve using a graphing calculator. Write your solution set using interval notation.

10. 4x3 – 3x2 – 9x – 2 ≤ 0 Solution Set: __________________________

Explain how you found your answer:

Name: ________________________________________ Date: _______________________ AA Trig

Section 3.2: Solve without the use of a calculator. Show all work.

11. (x – 3)(x + 2)(3x – 2)2 > 0 12. 3x3 – x ≤ –2x2

13. x4 – 16 < 0 14. (2𝑥−1)(𝑥−6)

(𝑥−5)3 ≤ 0

15. 4x4 – 4x3 – 3x2 + 4x – 1 ≤ 0

Name: ________________________________________ Date: _______________________ AA Trig

Chapter 4: Functions Use the following review materials as resources:

Quiz Review Sheets

Homework Quizzes/Exit Tickets

Chapter 4 Review Packet