Helen Troupis Ralph Raskas Micaila Edlin Chapter 1: List of Topics 1.1­ Functions Topics: * Describe subsets of real numbers. * Identify and evaluate functions and state their domains. Terms: * Set­Builder Notation: An expression that describes a set of numbers by using the properties of numbers in the set to define the set. EX. {x|x ≥ 8,x ε ω } *Interval Notation: An expression the uses inequalities to describe subsets of real numbers. * Function: A relation that assigns to each element in the domain one element in the range. * Function Notation: An equation of y in terms of x can be rewritten *Independent Variable: The variable in a function that represents any value in the domain. *Dependent Variable: The variable in a function the represents any value in the range. *Implied Domain: In a function with an unspecified domain, the set of all real numbers for which the expression used to define the function is real. *Piecewise­defined function: A function that is defined using two or more expressions for different intervals of the domain. Key Concepts: * Q=rationals * I=irrationals * Z=integers * W=wholes * N=naturals 1.2 ­ Analyzing Graphs of Equations and Relations Topics: * Use graphs of functions to estimate function values and find domains, ranges, y­intercepts, and zeros of functions. * Explore symmetries of graphs, and identify even and odd functions. Terms: * Zeros: The x­intercepts of the graph of a function. * Roots: For a function f(x), a solution of the equation f(x)=0. * Line symmetry: Describes graphs that can be folded along a line so the two halves match. *Point symmetry: Describes graphs that can be rotated 180 ° with respect to a point and appear unchanged. * Even functions: A function that is symmetric with respect to the y­axis. To determine if something is an even function algebraically, for every x in the domain of f, f(­x)= f(x). * Odd functions: A function that is symmetric with respect to the origin. To determine if something is an odd function algebraically, for every x in the domain of f, f(­x)= ­f(x). 1.3 ­ Continuity, End Behavior, and Limits * Continuous Function: a function that can be graphed without any holes, breaks, or gaps * Limit: the unique value that a function approaches as x­values of the function approach c from the left and right sides * Discontinuous Function: a function that is not continuous, cannot be graphed without picking up pencil * Infinite Discontinuity: a characteristic of a function in which the absolute value of the function increases or decreases indefinitely as x­values approach c from the left and right * Jump Discontinuity: a characteristic of a function in which the function has two distinct limit values as x­values approach c from left and right * Removable Discontinuity: a characteristic of a function in which the function is continuous everywhere except for a hole at x=c * Nonremovable Discontinuity: describes infinite and jump discontinuities because they cannot be eliminated by redefining the function at that point * End Behavior: describes what happens to the value f(x) as x increases or decreases without bound ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­ * Limits * Types of discontinuity * Continuity test * Identifying a point of discontinuity * Approximating zeros * Describe end behavior 1.4 ­ Extrema and Average Rates of Change * Increasing Function: a function where a positive change in x results in a positive change of f(x) * Decreasing Function: a function where a positive change in x results in a negative change of f(x) * Constant Function: a function where a positive change in x results in a zero change of f(x) * Critical Points: points in a function at which a line drawn tangent to the curve at that point is vertical or horizontal * Extrema: critical points at which a function has a maximum or minimum value * Point of Inflection: point where graph changes shape, but not its increasing or decreasing behavior, change of the bending direction * Average Rate of Change: linear slope of the line between two points on a graph Avg. Rate of Change = (y2 ­ y1) / (x2 ­ x1) * Secant Line: the line going through two points on a curve * Relative max/min: greatest or least value when a certain domain is defined * Absolute max/min: greatest or least value when considering the entire function ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­ * Increasing, decreasing, and constant functions * Estimating and identifying extrema of a function * Find average rates of change * Find average speed * Use extrema for optimization * Use graphing calculator to approximate extrema: 1. graph function and adjust window, so that all of the graph’s behavior is visible 2. analyze the graph to determine what type of extrema the graph might have 3. on the “calc” menu on the calculator, select min. or max. to find the approximate extrema of the function 1.5 ­Parent Functions and Transformations ● Topics: ○ Identify, graph, and describe parent functions ○ Identify and graph transformations of parent functions ● Terms: ○ Parent function: the simplest of the functions in a family of functions. A function that is transformed to create other members in a family of functions. ○ Constant function: a function of the form f(x) = c, where c is any real number ○ Zero function: the function sometimes known as the zero function is the constant function with constant c = 0. In other words, f(x) = 0. ○ Identity function: the function f(x) =x, which passes through all points with coordinates (a,a). 2 ○ Quadratic function: a function of the form f(x) = aX + bx + c, where a =/ 0, with a parent function of f(x) = x2 . ○ Cubic function: A function of the form f(x) = ax3 + bx2 + cx + d, where a =/ 0, with a parent function of f(x) = x3 . ○ Square root function: A function that contains a square root of the independent variable, with a parent function f(x) = √x . ○ Reciprocal function: 1) A function of the form f(x) = 1/a(x), where a(x) is a linear function and a(x) =/ 0, with parent function f(x) = 1/x. 2) Trigonometric functions that are reciprocals of each other. ○ Absolute value function: A function that contains an absolute value of the independent variable, with parent function f(x) = |x|. ○ Step function: A piecewise­defined function in which the graph is a series of line segments that resemble a set of stairs. ○ Greatest integer function: Has the parent function f(x) = [x] , which is defined as the greatest integer less than or equal to zero. ○ Transformation: A change in the position or shape of the graph of a parent function. ○ Translation: A rigid transformation that has the effect of shifting the graph of a function. ○ Reflection: A transformation in which a mirror image of the graph of a function is produced with respect to a specific line. ○ Dilation: A transformation in which the graph of a function is compressed or expanded vertically or horizontally. 1.6 ­ Function Operations and Composition of Functions ● Topics: ○ Perform operations with functions ○ Find compositions of functions ● Formulas: ○ Sum (f + g) (x) = f(x) + g(x) ○ Difference (f − g) (x) = f(x) + g(x) ○ Product (f • g) (x) = f(x) • g(x) ○ Quotient (f ÷ g) (x) = f(x) ÷ g(x), g(x) =/ 0 ○ [f ° g] (x) = f [g(x)] ● Terms: ○ Composition: The combining of functions by using the result of one function to evaluate a second function. The composition of function f with function g is defined by [f ° g] (x) = f [g(x)] . 1.7 ­ Inverse Relations and Functions ● Topics: ○ Use the horizontal line test to determine inverse functions ○ Find inverse functions algebraically and graphically ● Formulas ● Terms: ○ Inverse relation: The two relations are inverse relations if and only one relation contains the element (b,a) whenever the other relation contains the element (a,b). −1 ○ Inverse function: Two functions f are inverse functions if and only f [f−1(x)] = x for every x in the domain of f−1(x), and f−1 [f(x)] = x for every x in the domain of f(x) . ○ One­to­one: 1) A function in which no x­value is matched with more than one y­value and no y­value is matched with more than one x­value. 2) A function whose inverse is a function..