MATH-1150-HBL41

Welcome to the permanent home page for Section HBL41 of MATH-1150 (College Algebra) at Southeast Community College in the second five-week Summer session of 2024. I am Toby Bartels, your instructor.

Course administration

Canvas page (where you must log in).
Help with DjVu (if you have trouble reading the DjVu files on this page).
Official syllabus (DjVu).
Course policies (DjVu).
Class hours: Mondays through Thursdays from 12:30 to 1:55 PM in room U103 (B29 on July 1 Monday).
Final exam: August 1 Thursday from 12:30 to 1:55 PM in room U103.

Contact information

Name: Toby Bartels, PhD.
Canvas messages.
Email: TBartels@Southeast.edu.
Voice mail: 1-402-323-3452.
Text messages: 1-402-805-3021.
Office hours: on Mondays and Thursdays from 11:00 to 12:00 in room U9C and over Zoom meeting 610-024-2876.

Feel free to send a message at any time, even nights and weekends (although I'll be slower to respond then).

Readings

The official textbook for the course is the 11th Edition of Algebra & Trigonometry by Sullivan published by Prentice-Hall (Pearson). You automatically get an online version of this textbook through Canvas, although you can use a print version instead if you like. This comes with access to Pearson MyLab, integrated into Canvas, on which many of the assignments appear. Warning: The dates below are mostly wrong, and even some of the material may be changed.

Graphs and functions

Review:
- Reading:
  - My online introduction;
  - Skim Chapter R (except Section R.6) and Chapter 1 (except Section 1.6) from the textbook, and review anything that you are shaky on;
  - Review Section 2.1 through "Rectangular Coordinates" (pages 150&151) from the textbook;
  - Read Section 2.2 (pages 158–165) from the textbook (this should be review at the start but might be new material by the end);
  - My online notes on symmetry and intercepts.
- Exercises due on July 2 Tuesday (submit these here on Canvas or in class):
  1. Which of the following are equations? (Say Yes or No for each.)
    1. 2x + y;
    2. 2x + y = 0;
    3. z = 2x + y.
  2. You probably don't know how to solve the equation x⁵ + 2x = 1, but show what numerical calculation you make to check whether x = 2 is a solution.
  3. Write the set {x | x < 3} in interval notation and draw a graph of the set.
  4. Suppose that ax² + bx + c = 0 but a ≠ 0; write down a formula for x.
  5. Fill in the blanks with vocabulary words: The two number lines that mark the coordinates in a rectangular coordinate system are the coordinate _____, and the point where they intersect is the _____.
  6. In which number quadrant are both coordinates positive?
  7. Fill in the blanks with a vocabulary word: Given a graph in a coordinate plane, a point on the graph that lies on at least one coordinate axis is a(n) _____ of that graph.
  8. Fill in the blank: If for each point (x, y) on a graph, the point (−x, y) is also on the graph, then the graph is symmetric with respect to the _____.
- Exercises from the textbook due on July 3 Wednesday (submit these through MyLab in the Next item): O.1.1, O.1.2, O.1.3, O.1.4, O.1.5, O.1.6, O.1.7, O.1.8, O.1.10, O.1.12, 1.1.27, 1.1.39, 1.2.23, 1.2.49, 1.3.63, 1.5.71, 1.5.75, 1.7.33, 1.7.47, 2.1.15, 2.1.17, 2.2.13, 2.2.17, 2.2.23, 2.2.29, 2.2.31, 2.2.33, 2.2.35, 2.2.41, 2.2.43, 2.2.45, 2.2.47, 2.2.53, 2.2.55, 2.2.61, 2.2.67, 2.2.71, 2.2.77.
Graphing lines:
- Reading:
  - The rest of Section 2.1 (pages 151–154) from the textbook;
  - My online notes on lines and line segments;
  - Section 2.3 (pages 169–179) from the textbook.
- Exercises due on July 3 Wednesday (submit these here on Canvas or in class):
  1. Fill in the blank with a number: If the legs of a right triangle have lengths 3 and 4, then the length of its hypotenuse is ___.
  2. Fill in the blanks with algebraic expressions: The distance between the points (x₁, y₁) and (x₂, y₂) is _____, and the midpoint between them is (___, ___).
  3. Write an equation for the line in the (x, y)-plane with slope m and y-intercept (0, b).
  4. Fill in the blanks with words or numbers: The slope of a vertical line is _____, and the slope of a horizontal line is _____.
  5. Fill in the blanks with words or numbers: Suppose that a line L has slope 2. The slope of any line parallel to L is ___, and the slope of any line perpendicular to L is ___.
- Exercises from the textbook due on July 8 Monday (submit these through MyLab in the Next item): 2.1.4, 2.1.19, 2.1.21, 2.1.23, 2.1.27, 2.1.33, 2.1.39, 2.1.43, 2.1.47, 2.1.63, 2.1.71, 2.3.2, 2.3.7, 2.3.8, 2.3.13, 2.3.15, 2.3.17, 2.3.19, 2.3.21, 2.3.23, 2.3.25, 2.3.27, 2.3.29, 2.3.31, 2.3.45, 2.3.51, 2.3.53, 2.3.57, 2.3.63, 2.3.67, 2.3.73, 2.3.75, 2.3.79, 2.3.85, 2.3.91, 2.3.93, 2.3.111, 2.3.113.
Systems of equations and inequalities:
- Reading:
  - Section 12.1 (pages 868–878) from the textbook;
  - Section 12.6 (pages 933–938) from the textbook;
  - My online notes and video on systems of equations;
  - Section 12.7 (pages 942–947) from the textbook.
- Exercises due on July 8 Monday (submit these here on Canvas or in class):
  1. Given a system of two equations in the two variables x and y, if the graphs of the two equations intersect at (and only at) the point (3, 5), then what is the solution of the system? (Give explicitly the value of x and the value of y.)
  2. Answer Yes or No: Suppose that you have a system of equations and a point that might be a solution. If the point is a solution to one equation in the system but not a solution to another equation in the system, then is that point a solution to the system of equations?
  3. Consider the system of equations consisting of x + 3y = 4 (equation 1) and 2x + 3y = 5 (equation 2).
    1. If I solve equation (1) for x to get x = 4 − 3y and apply this to equation (2) to get 2(4 − 3y) + 3y = 5 (and continue from there), then what method am I using to solve this system?
    2. If instead I multiply equation (1) by −2 to get −2x − 6y = −8 and combine this with equation (2) to get −3y = −3 (and continue from there), then what method am I using to solve this system?
  4. Fill in the blank: If a system of equations or inequalities has no solutions, then the system is _____.
  5. When graphing an inequality in two variables, if the inequality is weak (written with < or >, instead of ≤ or ≥), then should the boundary be solid or dashed?
- Exercises from the textbook due on July 9 Tuesday (submit these through MyLab in the Next item): 12.1.3, 12.1.4, 12.1.6, 12.1.11, 12.1.13, 12.1.15, 12.1.17, 12.1.19, 12.1.21, 12.1.27, 12.1.31, 12.1.45, 12.1.47, 12.1.65, 12.1.73, more TBA.
Functions:
- Reading:
  - Section 3.1 (pages 203–215) from the textbook;
  - My online notes on functions;
  - Section 3.2 (pages 219–223) from the textbook.
- Exercises due on July 9 Tuesday (submit these here on Canvas or in class):
  1. Fill in the blank with a mathematical expression: If g(x) = 2x + 3 for all x, then g(___) = 2(5) + 3 = 13.
  2. Fill in the blank with an equation, inequality, or other statement: If a function f is thought of as a relation, then it's the relation {x, y | _____}.
  3. Fill in the blanks with vocabulary words: If f(3) = 5, then 3 belongs to the _____ of the function f, and 5 belongs to its _____.
  4. Fill in the blank with an arithmetic operation: If f(x) = 2x for all x, and g(x) = 3x for all x, then (f ___ g)(x) = 5x for all x.
  5. Fill in the blanks with mathematical expressions: If (3, 5) is a point on the graph of a function f, then f(___) = ___.
  6. Fill in the blank with a geometric word: The graph of a relation is the graph of a function if and only if every _____ line goes through the graph at most once.
  7. Which of these is true, and which of these is false?
    1. The graph of a function can have any number of x-intercepts.
    2. The graph of a function can have any number of y-intercepts.
- Exercises from the textbook due on July 10 Wednesday (submit these through MyLab in the Next item): 3.1.1, 3.1.2, 3.1.3, 3.1.10, 3.1.33, 3.1.35, 3.1.43, 3.1.49, 3.1.51, 3.1.53, 3.1.55, 3.1.59, 3.1.63, 3.1.71, 3.1.79, 3.1.81, 3.1.103, 3.2.7, 3.2.9, 3.2.11, 3.2.13, 3.2.15, 3.2.17, 3.2.19, 3.2.21, 3.2.27, 3.2.29, 3.2.31, 3.2.33, 3.2.39, 3.2.45, 3.2.47.

Quiz 1, covering the material in Problem Sets 1–4, is on July 12 Friday.

Properties and types of functions

Properties of functions:
- Reading:
  - Section 3.3 (pages 229–237) from the textbook;
  - My online notes on properties of functions.
- Exercises due on February 12 Monday (submit these here on Canvas or in class): Fill in the blanks with vocabulary words:
  1. Suppose that f is a function and, whenever f(x) exists, then f(−x) also exists and equals f(x). Then f is _____.
  2. If c is a number and f is a function, and if f(c) = 0, then c is a(n) _____ of f.
  3. Suppose that a function f is defined on (at least) a nontrivial interval I and that, whenever a ∈ I and b ∈ I, if a < b, then f(a) < f(b). Then f is (strictly) _____ on I.
  4. Suppose that f(x) = M for at least one value of x, and f(x) ≤ M for every value of x. Then M is the absolute _____ of f.
- Exercises from the textbook due on February 14 Wednesday (submit these through MyLab in the Next item): 3.3.2, 3.3.3, 3.3.5, 3.3.13, 3.3.15, 3.3.17, 3.3.19, 3.3.21, 3.3.23, 3.3.26, 3.3.27, 3.3.29, 3.3.31, 3.3.37, 3.3.39, 3.3.41, 3.3.43, 3.3.45, 3.3.49, 3.3.51.
Word problems with functions:
- Reading:
  - Section 3.6 (pages 267–269) from the textbook;
  - My online notes and video on functions in word problems;
  - Section 4.1 (pages 281–287) from the textbook.
- Exercises due on February 16 Friday (submit these here on Canvas or in class):
  1. Suppose that you have a problem with three quantities, A, B, and C; and suppose that you have two equations, equation (1) involving A and B, and equation (2) involving B and C. If you wish to find A as a function of C, then which equation should you solve first, and which variable should you solve it for? (Although there is a single best answer in my opinion, there is more than one answer that will progress the solution, and I'll accept either of them.)
  2. Suppose that y is linear function of x. If the rate of change of the function is m and the initial value of the function is b, then write an equation relating x and y.
  3. Suppose that f is a linear function. If you know f(a) and f(b) for two distinct real numbers a and b, then give a formula for the slope of the graph of f using a, b, f(a), and/or f(b).
- Exercises from the textbook due on February 19 Monday (submit these through MyLab in the Next item): 3.6.5, 3.6.13, 3.6.15, 3.6.21, 3.6.23, 4.1.2, 4.1.13, 4.1.15, 4.1.17, 4.1.19, 4.1.21, 4.1.23, 4.1.25, 4.1.27, 4.1.37, 4.1.43, 4.1.45, 4.1.47, 4.1.49.
The library of functions:
- Reading:
  - Section 3.4 Objective 1 (pages 242–246) from the textbook.
  - My online notes and video on partially-defined functions;
  - The rest of Section 3.4 (pages 247–249) from the textbook.
- Exercises due on February 21 Wednesday (submit these here on Canvas or in class): Fill in the blanks with vocabulary words:
  1. In the _____ function, the output is always defined and equal to the input.
  2. If you reflect the graph of the cube function across the diagonal line where y = x, then you get the graph of the _____ function.
  3. A _____-defined function is defined by a formula together with a condition restricting its inputs.
  4. A _____-defined function is defined by more than one formula, each with a condition restricting its inputs.
- Exercises from the textbook due on February 23 Friday (submit these through MyLab in the Next item): 3.4.9, 3.4.10, 3.4.11–18, 3.4.19, 3.4.20, 3.4.21, 3.4.22, 3.4.23, 3.4.24, 3.4.25, 3.4.26, 3.4.27, 3.4.29, 3.4.31, 3.4.33, 3.4.35, 3.4.43, 3.4.45, 3.4.51.
Composite functions:
- Reading:
  - Section 6.1 (pages 415–419) from the textbook;
  - My online notes on composite functions.
  - Section 6.2 (pages 423–430) from the textbook;
  - My online notes on inverse functions.
  - Section 3.5 (pages 254–263) from the textbook;
  - My online notes on linear coordinate transformations.
- Exercises due on March 1 Friday (submit these here on Canvas or in class):
  1. Fill in the blanks with a vocabulary word and a mathematical expression: If f and g are functions, then their _____ function, denoted f ∘ g, is defined by (f ∘ g)(x) = _____.
  2. Fill in the blanks with mathematical expressions: A number x is in the domain of f ∘ g if and only if ___ belongs to the domain of g and ___ belongs to the domain of f.
  3. Fill in the blank with a geometric word: A function is one-to-one if and only if every _____ line goes through its graph at most once.
  4. Fill in the blank with a vocabulary word: If f is a one-to-one function, then its _____ function, denoted f⁻¹, exists.
  5. Fill in the blank with an ordered pair: If f is one-to-one and (2, −3) is on the graph of f, then ___ is on the graph of f⁻¹.
  6. Fill in the blanks with vocabulary words: If f is one-to-one, then the domain of f⁻¹ is the _____ of f, and the range of f⁻¹ is the _____ of f.
  7. Assume that the axes are oriented in the usual way (positive x-axis to the right, positive y-axis upwards).
    1. Fill in the blank with a direction: To change the graph of y = f(x) into the graph of y = f(x − 1), shift the graph to the ___ by 1 unit.
    2. To change the graph of y = f(x) into the graph of y = −f(x), do you reflect the graph left and right or up and down?
    3. To change the graph of y = f(x) into the graph of y = f(2x), do you compress or stretch the graph left and right?
- Exercises from the textbook due on March 4 Monday (submit these through MyLab in the Next item): 6.1.2, 6.1.9, 6.1.11, 6.1.15, 6.1.19, 6.1.25, 6.1.27, 6.1.29, 6.1.33, 6.1.55, 6.2.4, 6.2.5, 6.2.7, 6.2.8, 6.2.9, 6.2.12, 6.2.21, 6.2.23, 6.2.25, 6.2.27, 6.2.29, 6.2.31, 6.2.35, 6.2.37, 6.2.41, 6.2.43, 6.2.45, 6.2.55, 6.2.57, 6.2.59, 6.2.61, 6.2.75, 6.2.77, 6.2.79, 6.2.87, 3.5.5, 3.5.6, 3.5.7–10, 3.5.11–14, 3.5.15–18, 3.5.19, 3.5.21, 3.5.23, 3.5.25, 3.5.29, 3.5.30, 3.5.33, 3.5.35, 3.5.37, 3.5.41, 3.5.43, 3.5.45, 3.5.47, 3.5.53, 3.5.61, 3.5.63, 3.5.73, 3.5.89.

Quiz 2, covering the material in Problem Sets 5–8, is on July 19 Friday.

Exponential and logarithmic functions

Exponential functions:
- Reading:
  - Section 6.3 (pages 435–446) from the textbook;
  - My online notes on exponential functions.
- Exercises due on March 22 Friday (submit these here on Canvas or in class): Let f(x) be Cb^x for all x.
  1. What is f(x + 1)/f(x)?
  2. What are f(−1), f(0), and f(1)?
  (Write your answers using b and/or C, and simplify them as much as possible.)
- Exercises from the textbook due on March 25 Monday (submit these through MyLab in the Next item): 6.3.1, 6.3.15, 6.3.16, 6.3.21, 6.3.23, 6.3.25, 6.3.27, 6.3.29, 6.3.31, 6.3.33, 6.3.35, 6.3.37–44, 6.3.45, 6.3.47, 6.3.51, 6.3.53, 6.3.57, 6.3.59, 6.3.61, 6.3.65, 6.3.67, 6.3.71, 6.3.73, 6.3.76, 6.3.77, 6.3.79, 6.3.83, 6.3.85, 6.3.91, 6.3.93.
Logarithmic functions:
- Reading:
  - Section 6.4 through Objective 4 (pages 452–457) from the textbook;
  - Section 6.4 Summary (page 460) from the textbook;
  - My online notes on logarithmic functions.
- Exercises due on March 25 Monday (submit these here on Canvas or in class): Suppose that b > 0 and b ≠ 1.
  1. Rewrite log_b M = r as an equation involving exponentiation.
  2. What are log_b b, log_b 1, and log_b (1/b)?
- Exercises from the textbook due on March 27 Wednesday (submit these through MyLab in the Next item): 6.4.11, 6.4.13, 6.4.15, 6.4.17, 6.4.19, 6.4.21, 6.4.23, 6.4.25, 6.4.27, 6.4.29, 6.4.31, 6.4.33, 6.4.35, 6.4.37, 6.4.39, 6.4.43, 6.4.51, 6.4.53, 6.4.55, 6.4.57, 6.4.65–72, 6.4.73, 6.4.79, 6.4.83, 6.4.85.
More about logarithms:
- Reading:
  - Section 6.4 Objective 5 (pages 457–460) from the textbook;
  - Section 6.5 Objective 4 (pages 469–471) from the textbook.
- Exercises due on April 3 Wednesday (submit these here on Canvas or in class): Given b > 0, b ≠ 1, and u > 0, write log_b u in these two ways:
  1. Using only common logarithms (logarithms base 10);
  2. Using only natural logarithms (logarithms base e).
- Exercises from the textbook due on April 5 Friday (submit these through MyLab in the Next item): 6.4.89, 6.4.91, 6.4.93, 6.4.95, 6.4.97, 6.4.99, 6.4.101, 6.4.103, 6.4.105, 6.4.107, 6.4.109, 6.4.111, 6.4.119, 6.4.129, 6.4.131, 6.5.7, 6.5.11, 6.5.71, 6.5.73, 6.5.75, 6.5.78.
Properties of logarithms:
- Reading:
  - Section 6.5 through Objective 3 (pages 465–469) from the textbook;
  - Section 6.5 Summary (page 471) from the textbook;
  - My online notes on laws of logarithms.
- Exercises due on April 5 Friday (submit these here on Canvas or in class): Fill in the blanks to break down these expressions using properties of logarithms. (Assume that b, u, and v are all positive and that b ≠ 1.)
  1. log_b (uv) = ___.
  2. log_b (u/v) = ___.
  3. log_b (u^x) = ___.
- Exercises from the textbook due on April 8 Monday (submit these through MyLab in the Next item): 6.5.13, 6.5.15, 6.5.17, 6.5.19, 6.5.21, 6.5.23, 6.5.25, 6.5.27, 6.5.37, 6.5.39, 6.5.41, 6.5.43, 6.5.45, 6.5.47, 6.5.49, 6.5.51, 6.5.53, 6.5.55, 6.5.57, 6.5.61, 6.5.63, 6.5.65, 6.5.67, 6.5.69, 6.5.87, 6.5.91, 6.5.97.
Logarithmic equations:
- Reading: Section 6.6 (pages 474–478) from the textbook.
- Exercises due on April 8 Monday (submit these here on Canvas or in class): In solving which of the following equations would it be useful to have a step in which you take logarithms of both sides of the equation? (Say Yes or No for each one.)
  1. log₂ (x + 3) = 5.
  2. (x + 3)² = 5.
  3. 2^x+3 = 5.
- Exercises from the textbook due on April 10 Wednesday (submit these through MyLab in the Next item): 6.6.1, 6.6.2, 6.6.5, 6.6.7, 6.6.9, 6.6.15, 6.6.19, 6.6.21, 6.6.23, 6.6.25, 6.6.27, 6.6.29, 6.6.31, 6.6.39, 6.6.43, 6.6.45, 6.6.49, 6.6.57, 6.6.61.
Compound interest:
- Reading:
  - Section 6.7 (pages 481–487) from the textbook;
  - My online notes on compound interest.
- Exercises due on April 10 Wednesday (submit these here on Canvas or in class):
  1. The original amount of money that earns interest is the _____.
  2. If you borrow P dollars at 100r% annual interest compounded n times per year, then how much will you owe after t years (if you make no payments)?
- Exercises from the textbook due on April 12 Friday (submit these through MyLab in the Next item): 6.7.1, 6.7.2, 6.7.7, 6.7.11, 6.7.13, 6.7.15, 6.7.21, 6.7.31, 6.7.33, 6.7.41, 6.7.43.
Applications of logarithms:
- Reading:
  - Section 6.8 (pages 478–485) from the textbook;
  - My online notes on applications of logarithms.
- Exercises due on April 12 Friday (submit these here on Canvas or in class):
  1. Suppose that a quantity A undergoes exponential growth with a relative growth rate of k and an initial value of A₀ at time t = 0. Write down a formula for the value of A as a function of the time t.
  2. Suppose that a quantity A undergoes exponential decay with a halflife of h and an initial value of A₀ at time t = 0. Write down a formula for the value of A as a function of the time t.
- Exercises from the textbook due on April 15 Monday (submit these through MyLab in the Next item): 6.8.1, 6.8.3, 6.8.5, 6.8.7, 6.8.9, 6.8.11, 6.8.13, 6.8.15, 6.8.17, 6.8.19, 6.8.21, 6.8.23.

Quiz 3, covering the material in Problem Sets 9–11, is on July 26 Friday.

Polynomial and rational functions

Quadratic functions:
- Reading:
  - Section 4.3 (pages 299–308) from the textbook;
  - My online notes on quadratic functions.
- Exercises due on March 8 Friday (submit these here on Canvas or in class):
  1. Fill in the blank with a vocabulary word: The shape of the graph of a nonlinear quadratic function is a(n) _____.
  2. Fill in the blanks with algebraic expressions: Given a ≠ 0 and f(x) = ax² + bx + c for all x, the vertex of the graph of f is (___, ___).
  3. Given a ≠ 0, b² − 4ac > 0, and f(x) = ax² + bx + c for all x, how many x-intercepts does the graph of y = f(x) have?
- Exercises from the textbook due on March 18 Monday (submit these through MyLab in the Next item): 4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.15–22, 4.3.31, 4.3.33, 4.3.43, 4.3.49, 4.3.53, 4.3.57, 4.3.61, 4.3.63, 4.3.67, 4.3.70.
Applications of quadratic functions:
- Reading: Section 4.4 through Objective 1 (pages 312–316) from the textbook.
- Exercises due on March 18 Monday (submit these here on Canvas or in class): Suppose that x and y are variables, and y = ax² + bx + c for some constants a, b, and c.
  1. Fill in the blank with an algebraic equation or inequality: y has a maximum value if _____.
  2. Fill in the blank with an algebraic expression: In this case, y has its maximum when x = ___.
- Exercises from the textbook due on March 20 Wednesday (submit these through MyLab in the Next item): 4.3.93, 4.3.95, 4.4.7, 4.4.9, 4.4.11, 4.4.13, 4.4.15.
Applications to economics:
- Reading: My online notes on economic applications.
- Exercises due on March 20 Wednesday (submit these here on Canvas or in class):
  1. If you make and sell x items per year at a price of p dollars per item, then what is your revenue (in dollars per year)?
  2. If a business's revenue is R dollars per year and its costs are C dollars per year, then what is its profit (in dollars per year)?
- Exercises from the textbook due on March 22 Friday (submit these through MyLab in the Next item): 4.3.87, 4.3.89, 4.4.3, 4.4.5.
Power functions:
- Reading: Section 5.1 through Objective 2 (pages 331–336) from the textbook.
- Exercises due on April 15 Monday (submit these here on Canvas or in class): For any real number n, the power function with exponent n is the function f such that f(x) = xⁿ for all possible x. Give the coordinates of:
  1. A point on the graph of every power function;
  2. Another point (different from the answer to the previous part) on the graph of every power function with a positive exponent;
  3. Another point on the graph of every power function with an even exponent; and
  4. Another point on the graph of every power function with an odd exponent.
- Exercises from the textbook due on April 17 Wednesday (submit these through MyLab in the Next item): 5.1.2, 5.1.15, 5.1.17, 5.1.19, 5.1.21, 5.1.27, 5.1.29, 5.1.33.
Graphing polynomials:
- Reading:
  - The rest of Section 5.1 (pages 336–342) from the textbook;
  - My online notes on graphing polynomials (but the last paragraph is optional);
  - Section 5.2 Objective 1 (pages 346–348) from the textbook.
- Exercises due on April 17 Wednesday (submit these here on Canvas or in class):
  1. If a root (aka zero) of a polynomial function has odd multiplicity, then does the graph cross (go through) or only touch (bounce off) the horizontal axis at the intercept given by that root? Which does the graph do if the root has even multiplicity?
  2. If the leading coefficient of a polynomial function is positive then does the graph's end behaviour go up on the far right, or down? Which does the graph do if the leading coefficient is negative?
- Exercises from the textbook due on April 19 Friday (submit these through MyLab in the Next item): 5.1.1, 5.1.11, 5.1.41, 5.1.43, 5.1.47, 5.1.49, 5.1.59, 5.1.61, 5.1.69, 5.1.71, 5.1.73, 5.1.75, 5.2.1, 5.2.2, 5.2.5, 5.2.11.
Advanced factoring:
- Reading:
  - Section R.6 (pages 57–60) from the textbook;
  - Section 5.6 through Objective 1 (pages 387–390) from the textbook;
  - Section 5.6 Objectives 3–5 (pages 391–395) from the textbook.
- Exercises due on April 19 Friday (submit these here on Canvas or in class):
  1. Suppose that f is a polynomial function and c is a number. If you divide f(x) by x − c, then what will the remainder be?
  2. Suppose that f is a polynomial function with rational coefficients and c is an integer. If x − c is a factor of f(x), then what is f(c)?
- Exercises from the textbook due on April 22 Monday (submit these through MyLab in the Next item): 5.6.2, 5.6.3, 5.6.4, 5.6.11, 5.6.15, 5.6.19, 5.6.33, 5.6.35, 5.6.37, 5.6.45, 5.6.51, 5.6.53, 5.6.57, 5.6.59, 5.6.65, 5.6.67, 5.6.93, 5.6.99, 5.6.101.
Imaginary roots:
- Reading: Section 5.7 (pages 401–406) from the textbook.
- Exercises due on April 22 Monday (submit these here on Canvas or in class): Suppose that f is a polynomial function with real coefficients, a and b are real numbers with b ≠ 0, and the complex number a + bi is a root (aka zero) of f.
  1. What other complex number must be a root of f?
  2. What non-constant polynomial in x (with real coefficients) must be a factor of f(x)?
- Exercises from the textbook due on April 24 Wednesday (submit these through MyLab in the Next item): 5.7.1, 5.7.2, 5.7.9, 5.7.11, 5.7.13, 5.7.15, 5.7.17, 5.7.19, 5.7.21, 5.7.23, 5.7.25, 5.7.29, 5.7.35, 5.7.39.
Rational functions and asymptotes:
- Reading:
  - Section 5.3 (pages 354–361) from the textbook;
  - My online notes on rational functions.
- Exercises due on April 24 Wednesday (submit these here on Canvas or in class):
  1. If a graph gets arbitrarily close to a line (without necessarily reaching it) in some direction, then the line is a(n) _____ of the graph.
  2. Suppose that when you divide R(x) = P(x)/Q(x), you get a linear quotient q(x) and a linear remainder r(x). Write an equation in x and y for the non-vertical linear asymptote of the graph of R. (Warning: Don't mix up lowercase and uppercase letters in your answer!)
- Exercises from the textbook due on April 26 Friday (submit these through MyLab in the Next item): 5.3.2, 5.3.3, 5.3.4, 5.3.15, 5.3.17, 5.3.19, 5.3.23, 5.3.27, 5.3.29, 5.3.31, 5.3.35, 5.3.45, 5.3.47, 5.3.49, 5.3.51.
Graphs of rational functions:
- Reading: Section 5.4 (pages 365–375) from the textbook.
- Exercises due on April 26 Friday (submit these here on Canvas or in class):
  1. If the reduced form of a rational function is defined somewhere where the original (unreduced) form is not, then the graph of the original function has a(n) _____ there.
  2. Suppose that when you divide R(x) = P(x)/Q(x), you get a linear quotient q(x) and a linear remainder r(x). Write an equation in x that you might solve to find where the graph of R meets this asymptote. (Warning: Don't mix up lowercase and uppercase letters in your answer!)
- Exercises from the textbook due on April 29 Monday (submit these through MyLab in the Next item): 5.4.1, 5.4.5, 5.4.7, 5.4.9, 5.4.11, 5.4.17, 5.4.19, 5.4.21, 5.4.23, 5.4.31, 5.4.33, 5.4.35, 5.4.51, 5.4.53.
Inequalities:
- Reading:
  - Section 5.5 (pages 380–384) from the textbook;
  - My online notes on solving inequalities.
- Exercise due on April 29 Monday (submit this here on Canvas or in class): Suppose that you have a rational inequality in one variable that you wish to solve. You investigate the inequality and discover the following facts about it:
  - the left-hand side is always defined;
  - the right-hand side is undefined when x is 2 but is otherwise defined;
  - the left-hand side and right-hand side are equal when x is −3/2 and only then;
  - the original inequality is true when x is −3/2 or 3 but false when x is −2 or 0.
  What are the solutions to the inequality?
- Exercises from the textbook due on May 1 Wednesday (submit these through MyLab in the Next item): 5.5.1, 5.5.5, 5.5.7, 5.5.9, 5.5.13, 5.5.15, 5.5.19, 5.5.21, 5.5.23, 5.5.27, 5.5.29, 5.5.35, 5.5.39, 5.5.41, 5.5.43, 5.5.47.

Quiz 4, covering the material in Problem Sets 12–15, is due on August 2 Friday.

Quizzes

Graphs and functions:
- Date: July 12 Friday.
- Corresponding problem sets: 1–4.
- Help allowed: Your notes, calculator.
- NOT allowed: Textbook, my notes, other people, websites, etc.
Properties and types of functions:
- Date: July 19 Friday.
- Corresponding problem sets: 5–8.
- Help allowed: Your notes, calculator.
- NOT allowed: Textbook, my notes, other people, websites, etc.
Exponential and logarithmic functions:
- Date: July 26 Friday.
- Corresponding problem sets: 9–11.
- Help allowed: Your notes, calculator.
- NOT allowed: Textbook, my notes, other people, websites, etc.
Polynomial and rational functions:
- Date: August 2 Friday.
- Corresponding problem sets: 12–15.
- Help allowed: Your notes, calculator.
- NOT allowed: Textbook, my notes, other people, websites, etc.

Final exam

There is a comprehensive final exam on August 1 Thursday, in our normal classroom at the normal time. To speed up grading at the end of the term, the exam is multiple choice, with no partial credit.

For the exam, you may use one sheet of notes that you wrote yourself. However, you may not use your book or anything else not written by you. You certainly should not talk to other people! Calculators are allowed (although you shouldn't really need one). but not communication devices (like cell phones).

The exam consists of questions similar in style and content to those in the practice exam (DjVu).

This web page and the files linked from it (except for the official syllabus) were written by Toby Bartels, last edited on 2024 June 28. Toby reserves no legal rights to them.

The permanent URI of this web page is https://tobybartels.name/MATH-1150/2024SU2/.