Section 3.1

• Be able to state the definitions of function, domain, and range.
• Be able to give examples of functions, including functions that relate physical quantities, from several perspectives (words, formulas, tables, graphs).
• Be able to find the domain from a formula and domain and range from a graph.
• Understand how to use and interpret function notation.
• Know how to compute difference quotients.

• Understand how and when a graph represents a function.
• Know graphs of basic functions.
• Know how to read information about a function from its graph.
• Know how to compute average rates of change from a graph, formula, or table using difference quotients.

• Understand the effect of translation on the graph and formula for a function.
• Understand the effect of reflection on the graph and formula for a function.
• Given a graph of y=f(x), be able to sketch graphs of y=f(x+c), f(x-c), f(x)+c, f(x)-c, f(-x), -f(x).
• Given a graph of y=f(x) and translations and/or reflections of it, be able to give the formulas for those graphs.
• Know how to combine operations and when order matters.

• Know how to form f+g, f-g, fg, f/g, and find domains of these functions if given f and g.
• Know how to compose functions using formulas, graphs, tables.
• Know how to find the domain of a composite function.
• Know how to write a given function as a composition of other functions.

• Know the definition of inverse function.
• Be able to find the inverse of f when f is given by formula, graph, table, or words.
• Know the definition of a one-to-one function.
• Be able to test whether a function has an inverse.

• Know the definition of a linear function.
• Be able to construct linear functions if you know two points or one point and the slope.
• Use linear functions (that you construct or are given to you as a regression line) to predict values in applications.

• Know the definitions of a quadratic function and features on the graph of a quadratic function.
• Be able to find features of the graph of a quadratic function (vertex, axis of symmetry, maximum and minimum values, intercepts) from the formula and/or graph of the function.
• Be able to compute the square to rewrite ax^2+bx+c as a(x-h)^2+k.

• Be able to define a function from a description of related quantities in words.

• Be able to find the maximum or minimum value of a key quantity (or another value that gives that maximum or minimum) when the key quantity is given by a quadratic expression (by completing the square).

• Be able to recognize polynomials by their formulas and graphs.
• Be able to identify the degree and leading coefficient of a polynomial.
• Know how the degree and leading coefficient of a polynomial control the overall behavior and the maximum number of turning points on its graph.
• Understand how the factored form of a polynomial controls the behavior of its graph near an x-intercept.
• Given a polynomial in factored form or that is easily factored, be able to find its intercepts, the behavior near those intercepts, and be able to graph the polynomial (using excluded regions).

• Know the definition of a rational function and an asymptote.
• Know how to graph basic rational functions and their translations and/or reflections.
• Know how to find the domain, range, intercepts, and asymptotes of a rational function.
• Know how to use properties of a rational function to sketch its graph.

• Know the definition of an exponential function and be able to distinguish an exponential function from a power function.
• Know basic exponent properties and be able to use them.
• Know properties of graphs of exponential functions given by formula, including domain, range, intercepts, asymptotes, and whether they are increasing or decreasing.
• Be able to graph exponential functions and their transformations using the properties.
• Be able to solve basic exponential equations algebraically and from a graph.

• Know properties of graphs of exponential functions, including base e.
• Be able to graph f(x)=e^x and its translations and reflections.
• Be able to estimate values involving e.
• Know how e and e^x arise in applications.

• Understand the definition of logarithmic functions.
• Know how to convert equations from exponential form to logarithmic form and vice versa.
• Know basic graphs of logarithmic functions and their properties.
• Be able to evaluate simple logarithmic expressions.

• Know properties of logarithms and why they are true.
• Be able to use properties of logarithms to manipulate logarithmic expressions.

• Be able to solve equations and inequalitites involving exponential and logarithmic expressions.
• Remember to ALWAYS CHECK solutions.

• Know formulas for compound interest and why they are true.
• Be able to use formulas for compound interest to compute account balances, time, or rates.
• Understand the difference between nominal and effective rates and be able to find one from the other.
• Know why the doubling time formula is true and how to use it.

• Be able to use the formula for exponential growth to find the relative growth rate (or growth constant), predict populations, and calculate times.
• Be able to use the formula for exponential decay to find the decay rate, predict amounts, and calculate half-life or other times.

• Know the meanings of and reasons for using imaginary and complex numbers.
• Know how to perform arithmetic with complex numbers and their conjugates.
• Know how conjugate pairs of roots relate to coefficients of quadratic polynomials.

• Be able to divide one polynomial by another using long division.
• Know the division algorithm and be able to write a division in the form p(x)=d(x)q(x)+R(x) where R(x)=0 or the degree of R(x) is less than the degree of d(x).
• Be able to use synthetic division when dividing by a linear polynomial.

• Know what the multiplicity of a root is and why it matters.
• Be able to use the Remainder Theorem to evaluate a polynomial.
• Understand the resulting Factor Theorem and how to use it to find remaining roots of a polynomial when one or more is given.
• Know how to use division of polynomials to investigate slant asymptotes of rational functions.

• Know what the Fundamental Theorem of Algebra and its consequences says and when they apply.
• Know the implications of the Fundamental Theorem of Algebra for factoring and finding roots of polynomials.
• Be able to express a polynomial in factored form.
• Be able to construct a polynomial from its roots and possibly other conditions.

• Know the Conjugate Roots Theorem.
• Be able to find remaining roots of a polynomial when one or more real or complex roots is given.