Readings and homework

Here are the assigned readings and exercises (Reading 1, Reading 2, Reading 3, Reading 4, Reading 5, Reading 6, Reading 7, Reading 8, Reading 9, Reading 10, Reading 11, Reading 12, Reading 13, Reading 14, Reading 15, Reading 16, Reading 17); but anything whose assigned date is in the future is subject to change!

1. Introduction and review:
   • Date assigned: October 3 Wednesday.
   • Date due: October 8 Monday.
   • Reading: My online introduction.
   • Exercises due:
     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.
2. Graphing:
   • Date assigned: October 8 Monday.
   • Date due: October 10 Wednesday.
   • Reading:
     • Section 2.1 (pages 74–83) from the textbook, but objective 3 (Graphing equations with a graphing utility) is optional;
     • Section 2.2, objectives 3–5 (pages 92–99, starting with Finding a linear equation);
     • My online notes on lines and line segments.
   • Exercises due:
     1. In which number quadrant are both coordinates positive?
     2. Fill in the blank: 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.
     3. Fill in the blank: The slope of a vertical line is ___, and the slope of a horizontal line is ___.
3. Systems of equations:
   • Date assigned: October 10 Wednesday.
   • Date due: October 15 Monday.
   • Reading:
     • Section 11.1 (pages 876–888);
     • Section 11.2 (pages 892–898);
     • Section 11.3, objective 1 (pages 903–906, Solving a system of nonlinear equations using substitution).
   • Exercises due:
     1. Given a system of two equations in the two variables x and y, if the graphs of the two equations intersect at the point (3, 5), then what is the solution of the system? (That is, what are the values of the two variables?)
     2. Fill in the blank: If a system of equations has no solutions, then the system is ___.
     3. Consider the system of equations consisting of x + 3y = 4 (equation 1) and 2x + 3y = 5 (equation 2). 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?
4. Functions:
   • Date assigned: October 15 Monday.
   • Date due: October 17 Wednesday.
   • Reading:
     • Section 3.1 (pages 160–175);
     • My online notes on functions.
   • Exercises due:
     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 blanks with geometric words: The graph of a relation is the graph of a function if and only if every ___ line goes through the graph at most once; a function is one-to-one if and only if every ___ line goes through its graph at most once.
     3. In the ___ function, the output is always defined and equal to the input.
5. Piecewise-defined functions:
   • Date assigned: October 17 Wednesday.
   • Date due: October 22 Monday.
   • Reading:
     • Most of Section 3.2 (pages 180–189);
     • My online notes on partial functions;
     • The rest of Section 3.2 (Graphing Piecewise-Defined Functions, pages 189–192).
   • Exercises due: Fill in the blanks with vocabulary words:
     1. The set of all arguments (inputs) of a function is the function's ___, and the set of all values (outputs) of the function is its ___.
     2. A function defined by multiple formulas, each with its own interval of applicability, is said to be ___-defined.
6. Properties of functions:
   • Date assigned: October 22 Monday.
   • Date due: October 24 Wednesday.
   • Reading:
     • My online notes on properties of functions.
     • Section 3.5, objective 3 (page 233 and the top of page 234, Determining Even and Odd Functions).
     • Section 4.1, objective 3 (the bottom of page 283 through the middle of page 285, Interpreting Slope as a Rate of Change).
     • Section 3.3 (pages 196–205).
   • Exercises due: 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.
7. Composite and inverse functions:
   • Date assigned: October 24 Wednesday.
   • Date due: October 29 Monday.
   • Reading:
     • Section 3.4 (pages 209–217).
     • Section 3.7 (pages 254–263).
     • My online notes on composite and inverse functions.
   • Exercises due:
     1. Fill in the blanks with a vocabulary word and a mathematical formula: If f and g are functions, then their ___ function, denoted f ∘ g, is defined by (f ∘ g)(x) = ___.
     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.
8. Linear functions:
   • Date assigned: October 29 Monday.
   • Date due: October 31 Wednesday.
   • Reading:
     • Section 4.1, objective 1 (pages 280–282, Representing Linear Functions).
     • Section 4.2 (pages 309–316).
   • Exercises due:
     1. Fill in the blank with a vocabulary word: A(n) ___ function is a function that is defined everywhere and has the same average rate of change between any two inputs.
     2. Identify which of the following functions are linear: f(x) = 3x − 2; g(x) = 3/x + 2; h(x) = 3/2.
     3. Fill in the blank with a mathematical formula: If f is a linear function and you know its values at x₁ and at x₂, then the rate of change of f is ___.
     4. Fill in the blank with an algebraic expression: If a linear function f has rate of change m and initial value b, then f(x) = ___ for all x.
9. Coordinate transformations:
   • Date assigned: October 31 Wednesday.
   • Date due: November 5 Monday.
   • Reading:
     • The rest of Section 3.5 (pages 222–232, the rest of page 234 through page 242);
     • My online notes on linear coordinate transformations;
     • Section 3.6, objective 2 (pages 248–250, Graphing an Absolute Value Function).
   • Exercises due: 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?
10. Quadratic functions:
    • Date assigned: November 5 Monday.
    • Date due: November 7 Wednesday.
    • Reading:
      • Section 5.1 (pages 344–356);
      • My online notes on quadratic functions.
    • Exercises due:
      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?
11. Exponential and logarithmic functions:
    • Date assigned: November 7 Wednesday.
    • Date due: November 12 Monday.
    • Reading:
      • Section 6.1 through objective 3 (pages 463–471);
      • Section 6.2 (pages 479–487);
      • Section 6.3 (pages 491–496);
      • Section 6.4 (pages 499–512);
      • My online notes on exponential and logarithmic functions.
    • Exercises due:
      1. If f(x) = Cb^x for all x, then what is f(x + 1)/f(x)?
      2. Rewrite log_b(M) = r as an equation involving exponentiation.
12. Properties of logarithms:
    • Date assigned: November 12 Monday.
    • Date due: November 14 Wednesday.
    • Reading:
      • Section 6.5 (pages 516–524);
      • My online notes on laws of logarithms;
      • Section 6.6 (pages 526–534).
    • Exercises due:
      • Fill in the blanks to break down the expressions using properties of logarithms:
        1. log_b 1 = ___;
        2. log_b b = ___;
        3. log_b (uv) = ___;
        4. log_b (u^x) = ___.
      • 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.
13. Applications of logarithms:
    • Date assigned: November 14 Wednesday.
    • Date due: November 19 Monday.
    • Reading:
      • The rest of Section 6.1 (page 472–475);
      • Section 6.7 (pages 537–548);
      • My online notes on applications of logarithms.
    • Exercise due: 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.
14. Polynomial functions:
    • Date assigned: November 19 Monday.
    • Date due: November 26 Monday.
    • Reading:
      • Section 5.2 (pages 360–371);
      • Section 5.3 (pages 375–389);
      • My online notes on graphing polynomials (but the last paragraph is optional).
    • Exercises due:
      1. Give the coordinates of a point on the graph of every power function, a point on the graph of every power function with a positive exponent, a point on the graph of every power function with an even exponent, and a 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?
15. Advanced factoring:
    • Date assigned: November 26 Monday.
    • Date due: November 28 Wednesday.
    • Reading:
      • Section 5.4 (pages 393–399);
      • Section 5.5 (pages 402–411).
    • Exercises due:
      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. If a + bi is a root (or zero) of f, then what else must be a root of f?
16. Rational functions:
    • Date assigned: November 28 Wednesday.
    • Date due: December 3 Monday.
    • Reading:
      • Section 5.6 (pages 414–430);
      • My online notes on rational functions.
    • Exercises due:
      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.
17. Inequalities:
    • Date assigned: December 12 Wednesday.
    • Date due: December 17 Monday.
    • Reading:
      • My online notes on solving inequalities;
      • From the OpenStax textbook for Intermediate Algebra:
        • Section 7.6 (pages 721–729, PDF, DjVu);
        • Section 9.8 (pages 968–976, PDF, DjVu).
    • Exercise: 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:
      1. the left-hand side is always defined;
      2. the right-hand side is undefined when x is 1 but is otherwise defined;
      3. the left-hand side and right-hand side are equal when x is 3 and only then;
      4. the original inequality is true when x is 2 or 3 but false when x is 0, 1, or 4.
      What are the solutions to the inequality?

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