A Partial List of Topics: Math 112 - Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose of this review sheet to remind you of the different types of material we have covered throughout the semester. You should review your homework and previous exams. You will not be given any formulas and you may not bring any notes to the exam. No calculators will be permitted on this exam. Review & Appendix B: exponent rules, rational and negative expressions with radicals operations with fractions compound fractions Simplification Factoring Sections 1.1-1.4: absolute value as a distance if pq = 0, then either p = 0, q = 0, or both distance formula d = (x 2 x 1 ) 2 (y 2 y 1 ) 2 ( x1 + x 2 midpoint formula M =, y ) 1 + y 2. 2 2 Sections 1.5 & 1.7 Characteristics of graphs x and y intercepts symmetry Reflecting a line segment test an equation for symmetry Equation for a circle: (x h) 2 + (y k) 2 = r 2 completing x 2 + bx to be a square Section 1.6 slope formula m = y 2 y 1 x 2 x 1 = y x point-slope form: y y 1 = m(x x 1 ) slope-intercept form: y = mx + b standard form: Ax + By + C = 0, where A, B, C are integers. two non-vertical lines are parallel if their slopes satisfy m 1 = m 2. two non-vertical lines are perpendicular if their slopes satisfy m 1 = 1 m 2.
Section 2.1 ax 2 + bx + c = 0 has solutions x = b ± b 2 4ac, which come from 2a completing the square. write a quadratic equation with given roots and coefficients. discriminant sum and product of roots factor to solve Section 2.2 taking n th roots of equations (even vs. odd) equations of quadratic type equations with fractions equations with roots extraneous roots (always check your answer) equations with absolute value Section 2.4 Practice these Section 2.5 and 2.6 interval notation compound inequalities (strings of inequalities) inequalities with absolute value rational and polynomial inequalities Section 3.1 & 3.2 Definition of a function Domain/Range, from a function definition or its graph Vertical line test Graph the 6 basic functions Piecewise functions Section 3.3 Average rate of change of f(x) on [a, b] is slope) Simplifying a difference quotient f(b) f(a) b a (related to Definitions: turning point, maximum value, minimum value, increasing, decreasing Velocity Section 3.4 Translations, horizontal and vertical Reflections over the x and y axes. Determine how two graphs are related Getting order correct (Ex. f( x + c) ) Section 3.5 Definitions of (f + g)(x), (f g)(x), (fg)(x), and Definition of (f g)(x). ( ) f (x). g Getting a value for a composition from a graph, table, or algebra Domain and range Express a function as a composition of simpler functions
Section 3.6 Definition Conditions for a function to have an inverse Horizontal line test Finding an inverse function Graphing inverse functions Domain and range of inverse functions Section 4.1 Uses of linear functions (cost, depreciation, position) What does slope represent? What does the constant term represent? Find a linear function from initial data. Which linear functions have inverses? Section 4.2 Standard form for a quadratic function: y = a(x h) 2 + k. Completing the square Graphing: Vertex, Axis of Symmetry, intercepts Maximum or Mininum outputs, and their corresponding inputs. Maximum or minimum outputs of a function related to a quadratic. Section 4.4 Section 4.5 Similar to 4.4, where the corresponding function is always either quadratic or related to a quadratic Use ideas from 4.2 to find the maximum or minimum Section 4.6 & 4.7 Finding intercepts of polynomial/rational functions finding vertical and horizontal asymptotes for rational functions Establishing when a graph is positive, negative Finding where the graph of a rational function crosses its horizontal asymptote, if any Graphing Section 5.1 & 5.2 Simplification: Exponent rules Solve basic exponential equations Characteristics of y = b x for b > 0 and b 1 (Domain, Range, intercepts, Asymptotes) shape for b > 1, as opposed to 0 < b < 1. Graphing (as per 3.4) Inverses, properties of inverses e 2.718 (note: e > 1 ) Geometry: Formulas for rectangles, circles, triangles, boxes. Assigning variables Setting up equations Substitute to get one equation with one variable
Section 5.3 log b (x) is the inverse function of b x. Evaluating basic logarithmic expressions Characteristics of y = log b (x) for b > 1. (Domain, Range, Intercepts, Asymptotes) Estimating logarithms Graphing logarithms (per 3.4) Finding the domain of logarithmic functions Solving equations with logarithms Section 5.4 log b (P Q) =..., log b (P/Q) =..., log b (P n ) =..., b log b (P ) =..., log b (b) =..., log b (1) =... Use properties to shorten or expand expressions Solving equations Finding intercepts Changing base Section 5.5 Checking solutions Use of log and exponent rules Apply logarithms or exponentials to both sides to an equation or inequality Quadratic equation types If b > 1, then b x and log b (x) are both increasing in their domains (which means applying either to both sides of an inequality does not change the direction of the inequality.) If 0 < b < 1, then b x and log b (x) are both decreasing in their domains (which means applying either to both sides of an inequality DOES change the direction of the inequality.) Solve inequalities with variables in exponents or in logarithms Checking domain Section 5.6 Formulas for compound interest and continual compound interest. Effective interest rate Doubling time Section 5.7 Population models Finding growth constants Relative growth rates Doubling time (tripling time) Radioactive decay Half Life
Section 6.1 Solving by substitution Solving by Addition-Subtraction (Linear combinations) Word problems Possible solution sets for two linear equations and two unknowns Graphing linear systems in two variables Section 6.6 Substitution Graphing nonlinear equations in two variables (intersections correspond to real solutions) No solutions, how to tell Converting to a linear system, if possible Exponentials and logarithms Section 7.1 Section 7.3 zero / root of a polynomial Remainder theorem and its applications Factor theorem and its applications Finding all roots of a given polynomial Construct a polynomial satisfying a list of criteria Section 7.4 The Fundamental Theorem of Algebra (only applies to polynomials) Linear Factors theorem Write a polynomial as the product of linear factors polynomials of degree n 1 have n roots, counting multiplicity Make a polynomial satisfying given criteria, if possible The relationship between roots and factors. i 2 = 1, and other computations with i. real and imaginary parts of a complex number equality, operations, and complex conjugates z roots of quadratics being complex Section 7.2 What is a polynomial The division algorithm: p(x) = d(x)q(x) + R(x) (quotient and remainder) Synthetic division, when you can use it