# MATH-1150-LN07

Welcome to the permanent home page for Section LN07 of MATH-1150 (College Algebra) at Southeast Community College in the Spring term of 2021. I am Toby Bartels, the instructor.

## Contact information

The official textbook for the course is the 11th Edition of Algebra & Trigonometry by Sullivan published by Prentice-Hall (Pearson). You will automatically get an online version of this textbook through Canvas, although you can also order a print version if you like. This comes with access to Pearson MyLabs, directly integrated into Canvas), on which many of the assignments appear.

### Graphs and functions

1. General review:
• My online introduction;
• Skim Chapter R (except Section R.6) and Chapter 1 (except Section 1.6) and review anything that you are shaky on.
• Exercises due on January 13 Wednesday:
1. Which of the following are equations?
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 = 1 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.
• Exercises from the textbook due on January 20 Wednesday: 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. Graphing points:
• Reading: Section 2.1 (pages 150–154) from the textbook.
• Exercises due on January 20 Wednesday:
1. 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 _____.
2. 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 ___.
3. Fill in the blanks with algebraic expressions: The distance between the points (x1, y1) and (x2, y2) is _____, and the midpoint between them is (___, ___).
• Exercises from the textbook due on January 25 Monday: 2.1.4, 2.1.15, 2.1.17, 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.
3. Graphing equations:
• Exercises due on January 25 Monday: Fill in the blanks with vocabulary words:
1. 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.
2. 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 January 27 Wednesday: 2.2.1, 2.2.2, 2.2.7, 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.
4. Lines:
• Section 2.3 (pages 169–179);
• My online notes on lines.
• Exercises due on January 27 Wednesday: Fill in the blanks with words or numbers:
1. The slope of a vertical line is _____, and the slope of a horizontal line is _____.
2. 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 February 1 Monday: 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.
5. Systems of equations:
• Exercises due on February 1 Monday: 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?
• Exercises from the textbook due on February 3 Wednesday: 12.1.3, 12.1.4, 12.1.6, 12.1.11, 12.1.19, 12.1.21, 12.1.27, 12.1.31, 12.1.45, 12.1.47, 12.1.65, 12.1.73.
6. Functions:
• Section 3.1 (pages 203–215);
• My online notes on functions.
• Exercises due on February 3 Wednesday:
1. Fill in the blanks with vocabulary words: If f(3) = 5, then 3 belongs to the _____ of the function, and 5 belongs to its _____.
2. Fill in the blank with a mathematical expression: If g(x) = 2x + 3 for all x, then g(___) = 2(5) + 3 = 13.
• Exercises from the textbook due on February 8 Monday: 3.1.1, 3.1.2, 3.1.3, 3.1.10, 3.1.31, 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.
7. Graphs of functions:
• Reading: Most of Section 3.2 (pages 219–223), but you may skip parts D and E of Example 4.
• Exercises due on February 8 Monday:
1. Fill in the blanks with mathematical expressions: If (3, 5) is a point on the graph of a function f, then f(___) = ___.
2. 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.
3. True or false: The graph of a function can have any number of x-intercepts.
4. True or false: The graph of a function can have any number of y-intercepts.
• Exercises from the textbook due on February 10 Wednesday: 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 1–7, is available on February 10 Wednesday.

### Properties and types of functions

1. Properties of functions:
• Exercises due on February 10 Wednesday: 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.
• Exercises from the textbook due on February 17 Wednesday: 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.
2. Word problems with functions:
• Most of Section 3.6 (pages 267–269), but you may skip the parts involving graphing calculators;
• My online notes and video on functions in word problems;
• Section 4.1 (pages 281–287).
• Exercises due on February 17 Wednesday:
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.
• Exercises from the textbook due on February 22 Monday: 3.6.5, 3.6.13, 3.6.15, 3.6.17, 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.
3. Examples of functions:
• Section 3.4 Objective 1 (pages 242–246);
• My online notes and video on partially-defined functions;
• The rest of Section 3.4 (pages 247–249).
• Exercises due on February 22 Monday: Fill in the blanks with vocabulary words:
1. In the _____ function, the output is always defined and equal to the input.
2. A _____-defined function is defined by a formula together with a condition restricting its inputs.
3. A _____-defined function is defined by more than one formula, each with a condition restricting its inputs.
• Exercises from the textbook due on February 24 Wednesday: 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.
4. Composite functions:
• Exercises due on February 24 Wednesday:
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.
• Exercises from the textbook due on March 1 Monday: 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.
5. Inverse functions:
• Most of Section 6.1: from page 403 to the top of page 407;
• Section 6.2 (pages 423–430);
• My online notes on inverse functions.
• Exercises due on March 1 Monday:
1. 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.
2. Fill in the blank with a vocabulary word: If f is a one-to-one function, then its _____ function, denoted f−1, exists.
3. 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.
• Exercises from the textbook due on March 3 Wednesday: 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.
6. Coordinate transformations:
• Exercises due on March 3 Wednesday: 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 8 Monday: 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.
• Exercises due on March 8 Monday:
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 10 Wednesday: 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.
• Exercises due on March 10 Wednesday:
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)?
3. If the width of a rectangle is w metres and its length is l metres, then what is its area (in square metres)?
• Exercises from the textbook due on March 24 Wednesday: 4.3.87, 4,3,89, 4,3.93, 4.3.95, 4.4.3, 4.4.5, 4.4.7, 4.4.9, 4.4.11, 4.4.13, 4.4.15.
Quiz 2, covering the material in Problem Sets 8–15, is available on March 10 Wednesday.

### Logarithms and polynomials

1. Exponential functions:
• Exercises due on March 24 Wednesday: Let f(x) be Cbx 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 C, and simplify them as much as possible.)
• Exercises from the textbook due on March 29 Monday: 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.
2. Logarithmic functions:
• Exercises due on March 29 Monday: Suppose that b > 0 and b ≠ 1.
1. Rewrite logb(M) = r as an equation involving exponentiation.
2. What are logb(b), logb(1), and logb(1/b)?
• Exercises from the textbook due on March 31 Wednesday: 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, 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.
3. Properties of logarithms:
• Section 6.5 (pages 465–471);
• My online notes on laws of logarithms;
• Section 6.6 (pages 474–478).
• Exercises due on March 31 Wednesday:
• 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. logb (uv) = ___;
2. logb (u/v) = ___;
3. logb (ux) = ___.
• 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. 2x + 3 = 5.
• Exercises from the textbook due on April 5 Monday: 6.5.7, 6.5.11, 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, 6.5.71, 6.5.73, 6.5.75, 6.5.78, 6.5.87, 6.5.91, 6.5.97, 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.
4. Applications of logarithms:
• Exercises due on April 5 Monday:
1. The original amount of money that earns interest is the _____.
2. 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.
• Exercises from the textbook due on April 7 Wednesday: 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, 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.
5. Polynomial functions:
• Section 5.1 (pages 331–342);
• My online notes on graphing polynomials (but the last paragraph is optional);
• Section 5.2 Objective 1 (pages 346–348).
• Exercises due on April 7 Wednesday:
1. Give the coordinates of a point on the graph of every power function, another point (different from the previous point) on the graph of every power function with a positive exponent, another point on the graph of every power function with an even exponent, and another point on the graph of every power function with an odd exponent.
2. If a root (zero) of a polynomial function has odd multiplicity, 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?
• Exercises from the textbook due on April 12 Monday: 5.1.1, 5.1.2, 5.1.11, 5.1.15, 5.1.17, 5.1.19, 5.1.21, 5.1.27, 5.1.29, 5.1.33, 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.
• Section R.6 (pages 57–60);
• Section 5.6 through Objective 1 (pages 387–390);
• Section 5.6 Objectives 3–5 (pages 391–395).
• Exercises due on April 12 Monday:
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)?
3. Suppose that f is a polynomial function with real coefficients and a and b are real numbers with b ≠ 0. If the imaginary complex number a + bi is a root (or zero) of f, then what other number must be a root of f?
• Exercises from the textbook due on April 14 Wednesday: 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, 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.
7. Rational functions:
• Section 5.3 (pages 354–361);
• Section 5.4 (pages 365–375);
• My online notes on rational functions.
• Exercises due on April 14 Wednesday:
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. 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.
• Exercises from the textbook due on April 19 Monday: 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, 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.
8. Inequalities:
• Exercise due on April 19 Monday: 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, 0, or 2.
What are the solutions to the inequality?
• Exercises from the textbook due on April 21 Wednesday: 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 3, covering the material in Problem Sets 16–23, is available on April 21 Wednesday.

## Quizzes

1. Graphs and functions:
• Review date: February 10 Wednesday (in class).
• Date due on MyLab: February 15 Monday (before class).
• Corresponding problem sets: 1–7.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
• Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except #2 and #9.
2. Properties and types of functions:
• Review date: March 10 Wednesday (in class).
• Date due on MyLab: March 22 Monday (before class).
• Corresponding problem sets: 8–15.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
• Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except #1, #5, and #9.
3. Logarithms and polynomials:
• Review date: April 21 Wednesday (in class).
• Date due on MyLab: April 26 Monday (before class).
• Corresponding problem sets: 16–23.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
• Work to show: TBA.

## Final exam

There will be a comprehensive final exam on May 3 Monday, in our normal classroom at the normal time but lasting until 7:10 PM. (You can also arrange to take it at a different time from April 30 to May 6.) To speed up grading at the end of the term, the exam will be multiple choice, with no partial credit.

For the exam, you may use one sheet of notes that you wrote yourself; please take a scan or a picture of this (both sides) and submit it on Canvas. 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 final exam on MyLab.

This web page and the files linked from it were written by Toby Bartels, last edited on 2021 April 16. Toby reserves no legal rights to them.

The permanent URI of this web page is `http://tobybartels.name/MATH-1150/2021SP/`.