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Pre-calculus · AP · High school · Math
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Unit 1: Polynomial and Rational Functions
- 1.1 * — Change in Tandem A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value. The set of input values is called the domain of the function, and the set of output values is called the range of the function. The variable representing input values is called the independent variable, and the variable representing output values is called the dependent variable. 1.1.A.2 The input and output values of a function vary in tandem according to the function rule, which can be expressed graphically, numerically, analytically, or verbally. 1.1.A.3 A function is increasing over an interval of its domain if, as the input values increase, the output values always increase. That is, for all a and b in the interval, if a < b, then f a( ) < f b( ). 1.1.A.4 A function is decreasing over an interval of its domain if, as the input values increase, the output values always decrease. That is, for all a and b in the interval, if a < b, then f a( ) > f b( ).
- 1.2 * — Rates of Change
- 1.3 * — Rates of Change in Linear and Quadratic Functions For a linear function, the average rate of change over any length input-value interval is constant. 1.3.A.2 For a quadratic function, the average rates of change over consecutive equal-length input value intervals can be given by a linear function. 1.3.A.3
- 1.4 * — Polynomial Functions and Rates of Change Identify key characteristics of polynomial functions related to rates of change.
- 1.5 * — Polynomial Functions and Complex Zeros The degree of a polynomial function can be found by examining the successive differences of the output values over equal-interval input values. The degree of the polynomial function is equal to the least value n for which the successive nth differences are constant. 1.5.B 1.5.B.1 Determine if a polynomial function is even or odd.
- 1.6 * — Polynomial Functions and End Behavior Describe end behaviors of rational functions. A rational function is analytically represented as a quotient of two polynomial functions and gives a measure of the relative size of the polynomial function in the numerator compared to the polynomial function in the denominator for each value in the rational function’s domain. 1.7.A.2 The end behavior of a rational function will be affected most by the polynomial with the greater degree, as its values will dominate the values of the rational function for input values of large magnitude. For input values of large magnitude, a polynomial is dominated by its leading term. Therefore, the end behavior of a rational function can be understood by examining the corresponding quotient of the leading terms. 1.7.A.3 If the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude, then the quotient of the leading terms is a nonconstant polynomial, and the original rational function has the end behavior of that polynomial. If that polynomial is linear, then the graph of the rational function has a slant asymptote parallel to the graph of the line.
- 1.7 * — Rational Functions and End Behavior Describe end behaviors of rational functions. If neither polynomial in a rational function dominates the other for input values of large magnitude, then the quotient of the leading terms is a constant, and that constant indicates the location of a horizontal asymptote of the graph of the original rational function. 1.7.A.5 If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, then the quotient of the leading terms is a rational function with a constant in the numerator and nonconstant polynomial in the denominator, and the graph of the original rational function has a horizontal asymptote at y = 0. 1.7.A.6 When the graph of a rational function r has a horizontal asymptote at y = b, where b is a constant, the output values of the rational function get arbitrarily close to b and stay arbitrarily close to b as input values increase or decrease without bound. The corresponding mathematical notation is lim r x( ) = b or x lim r x( ) = b. x
- 1.8 * — Rational Functions and Zeros Determine the zeros of rational functions.
- 1.9 * — Rational Functions and Vertical Asymptotes Determine holes in graphs of rational functions. If the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator, then the graph of the rational function has a hole at the corresponding input value. 1.10.A.2 If the graph of a rational function r has a hole at x = c, then the location of the hole can be determined by examining the output values corresponding to input values sufficiently close to c. If input values sufficiently close to c correspond to output values arbitrarily close to L, then the hole is located at the point with coordinates (c L, ). The corresponding mathematical notation is lim r x( ) = L x c . It should be noted that lim r x( ) = lim r x( ) = lim r x( ) = L. − + x c x c x c
- 1.10 * — Rational Functions and Holes INSTRUCTIONAL
- 1.11 * — Equivalent Representations of Polynomial and Rational Expressions
- 1.12 * — Transformations of Functions Linear functions model data sets or aspects of contextual scenarios that demonstrate roughly constant rates of change. 1.13.A.2 Quadratic functions model data sets or aspects of contextual scenarios that demonstrate roughly linear rates of change, or data sets that are roughly symmetric with a unique maximum or minimum value. 1.13.A.3 Geometric contexts involving area or two dimensions can often be modeled by quadratic functions. Geometric contexts involving volume or three dimensions can often be modeled by cubic functions. 1.13.A.4 Polynomial functions model data sets or contextual scenarios with multiple real zeros or multiple maxima or minima. 1.13.A.5 A polynomial function of degree n models data sets or contextual scenarios that demonstrate roughly constant nonzero nth differences. 1.13.A.6 A polynomial function of degree n or less can be used to model a graph of n + 1 points with distinct input values.
- 1.13 * — Function Model Selection and Assumption Articulation A piecewise-defined function consists of a set of functions defined over nonoverlapping domain intervals and is useful for modeling a data set or contextual scenario that demonstrates different characteristics over different intervals. 1.13.B 1.13.B.1 Describe assumptions and restrictions related to building a function model.
- 1.14 * — Function Model Construction and Application A model can be constructed based on restrictions identified in a mathematical or contextual scenario. 1.14.A.2 A model of a data set or a contextual scenario can be constructed using transformations of the parent function. 1.14.A.3 A model of a data set can be constructed using technology and regressions, including linear, quadratic, cubic, and quartic regressions. 1.14.A.4 A piecewise-defined function model can be constructed through a combination of modeling techniques. 1.14.B 1.14.B.1 Construct a rational function model based on a context.
Unit 2: Exponential and Logarithmic Functions
- 2.1 * — Change in Arithmetic and Geometric Sequences A sequence is a function from the whole numbers to the real numbers. Consequently, the graph of a sequence consists of discrete points instead of a curve. 2.1.A.2 Successive terms in an arithmetic sequence have a common difference, or constant rate of change. 2.1.A.3 The general term of an arithmetic sequence with a common difference d is denoted by an and is given by an = a d 0 + n, where a0 is the initial value, or by an = a + d(n k − ) k , where ak is the kth term of the sequence. 2.1.B 2.1.B.1 Express geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers.
- 2.2 * — Change in Linear and Exponential Functions Describe similarities and differences between linear and exponential functions.
- 2.3 * — Exponential Functions Identify key characteristics of exponential functions. For an exponential function in general form, as the input values increase or decrease without bound, the output values will increase or decrease without bound or will get arbitrarily close to zero. That is, for an exponential function in general form, x lim ab , x lim ab x = − , or lim ab x = 0. x x
- 2.4 * — Exponential Function Manipulation Exponential functions model growth patterns where successive output values over equal length input-value intervals are proportional. When the input values are whole numbers, exponential functions model situations of repeated multiplication of a constant to an initial value. 2.5.A.2 A constant may need to be added to the dependent variable values of a data set to reveal a proportional growth pattern. 2.5.A.3 An exponential function model can be constructed from an appropriate ratio and initial value or from two input-output pairs. The initial value and the base can be found by solving a system of equations resulting from the two input-output pairs. 2.5.A.4 Exponential function models can be constructed by applying tr o ansformations t f x( ) = ab x based on characteristics of a contextual scenario or data set. 2.5.A.5 Exponential function models can be constructed for a data set with technology using exponential regressions. 2.5.A.6 The natural base e, which is approximately 2.718, is often used as the base in exponential functions that model contextual scenarios.
- 2.5 * — Exponential Function Context and Data Modeling For an exponential model in general form f x) = ab x ( , the base of the exponent, b, can be understood as a growth factor in successive unit changes in the input values and is related to a percent change in context. 2.5.B.2 Equivalent forms of an exponential function can reveal different properties of the function. For example, if d represents number of days, then the base of f d( ) = 2 d
- 2.6 * — Competing Function Model Validation Construct linear, quadratic, and exponential models based on a data set.
- 2.7 * — Composition of Functions Construct a representation of the composition of two or more functions.
- 2.8 * — Inverse Functions Determine the input-output pairs of the inverse of a function.
- 2.9 * — Logarithmic Expressions The general form of a logarithmic function is f x( ) = alogb x, with baseb, where b > 0, b 1, and a 0. 2.10.A.2 The way in which input and output values vary together have an inverse relationship in exponential and logarithmic functions. Output values of general-form exponential functions change proportionately as input values increase in equal-length intervals. However, input values of general-form logarithmic functions change proportionately as output values increase in equal-length intervals. Alternately, exponential growth is characterized by output values changing multiplicatively as input values change additively, whereas logarithmic growth is characterized by output values changing additively as input values change multiplicatively. 2.10.A.3 x f x( ) = logb x and g (x) = , where b > 0 and
- 2.10 * — Inverses of Exponential Functions Identify key characteristics of logarithmic functions.
- 2.11 * — Logarithmic Functions Rewrite logarithmic expressions in equivalent forms.
- 2.12 * — Logarithmic Function Manipulation
- 2.13 * — Exponential and Logarithmic Equations and Inequalities Solve exponential and
- 2.14 * — Logarithmic Function Context and Data Modeling
- 2.15 * — Semi-log Plots In a semi-log plot, one of the axes is logarithmically scaled. When the y-axis of a semi-log plot is logarithmically scaled, data or functions that demonstrate exponential characteristics will appear linear. 2.15.A.2 An advantage of semi-log plots is that a constant never needs to be added to the dependent variable values to reveal that an exponential model is appropriate. 2.15.B 2.15.B.1 Construct the linearization of exponential data.
Unit 3: Trigonometric and Polar Functions
- 3.1 * — Periodic Phenomena Construct graphs of periodic relationships based on verbal representations.
- 3.2 * — Sine, Cosine, and Tangent Determine the sine, cosine, and tangent of an angle using the unit circle.
- 3.3 * — Sine and Cosine Function Values Construct representations of the sine and cosine functions using the unit circle.
- 3.4 * — Sine and Cosine Function Graphs cos = sin + . 2 2 . Identify key characteristics
- 3.5 * — Sinusoidal Functions − 2 The graph of the additive transformation g ( ) = sin + d of the sine function f ( ) = sin is a vertical translation of the graph of f , including its midline, by d units. The same transformation of the cosine function yields the same result. 3.6.A.3
- 3.6 * — Sinusoidal Function Transformations 1 b 1 2 b The graph of the multiplicative transformation g ( ) = sin(b ) of the sine function f ( ) = sin is a horizontal dilation of the graph of f and differs in period by a factor of . The same transformation of the cosine function yields the same result. 3.6.A.6 The graph of y = f ( ) = asin(b( + c)) + d has an amplitude of a units, a period of units, a midline vertical shift of d units from y = 0, and a phase shift of −c units. The same transformations of the cosine function yield the same results.
- 3.7 * — Sinusoidal Function Context and Data Modeling
- 3.8 * — The Tangent Function = + k 2 Construct representations of the tangent function using
- 3.9 * — Inverse Trigonometric Functions Solve equations and inequalities involving
- 3.10 * — Trigonometric Equations and Inequalities coscot = sin
- 3.11 * — The Secant, Cosecant, and Cotangent Functions
- 3.12 * — Equivalent Representations of Trigonometric Functions
- 3.13 * — Trigonometry and Polar Coordinates Construct graphs of polar functions. The graph of the function r = f ( ) in polar coordinates consists of input-output pairs of values where the input values are angle measures and the output values are radii. 3.14.A.2 The domain of the polar function r = f ( ), given graphically, can be restricted to a desired portion of the function by selecting endpoints corresponding to the desired angle and radius. 3.14.A.3 When graphing polar functions in the form of r = f ( ), changes in input values correspond to changes in angle measure from the positive x-axis, and changes in output values correspond to changes in distance from the origin.
- 3.14 * — Polar Function Graphs Describe characteristics of the graph of a polar function. If a polar function, r = f ( ), is positive and increasing or negative and decreasing, then the distance between f ( ) and the origin is increasing. 3.15.A.2 If a polar function, r = f ( ), is positive and decreasing or negative and increasing, then the distance between f ( ) and the origin is decreasing. 3.15.A.3 For a polar function, r = f ( ), if the function changes from increasing to decreasing or decreasing to increasing on an interval, then the function has a relative extremum on the interval corresponding to a point relatively closest to or farthest from the origin. 3.15.A.4 The average rate of change of r with respect to over an interval of is the ratio of the change in the radius values to the change in over an interval of . Graphically, the average rate of change indicates the rate at which the radius is changing per radian. 3.15.A.5 The average rate of change of r with respect to over an interval of can be used to estimate values of the function within the interval.
- 3.15 * — Rates of Change in Polar Functions Functions Involving Parameters, Vectors, and Matrices Additional Topics Available to Schools (not included on AP Precalculus Exam)
Unit 4: Functions Involving Parameters, Vectors, and Matrices
- 4.1 — Parametric Functions Additional Topic Available to Schools A parametric function given by f t( ) = (x t( ), y t( )) can be used to model particle motion in the plane. The graph of this function indicates the position of a particle at time t. 4.2.A.2 The horizontal and vertical extrema of a particle’s motion can be determined by identifying the maximum and minimum values of the functions x(t) and y t( ), respectively. 4.2.A.3 The real zeros of the function x(t) correspond to y-intercepts, and the real zeros of y t( ) correspond to x-intercepts.
- 4.2 — Parametric Functions Modeling Planar Motion Additional Topic Available to Schools
- 4.3 — Parametric Functions and Rates of Change Additional Topic Available to Schools Express motion around a circle or along a line segment
- 4.4 — Parametrically Defined Circles and Lines Additional Topic Available to Schools
- 4.5 — Implicitly Defined Functions Construct a graph of an equation involving two
- 4.6 — Conic Sections Additional Topic Available to Schools
- 4.7 — Parametrization of Implicitly Defined Functions Represent a curve in the
- 4.8 — Vectors Identify characteristics of a vector. A vector is a directed line segment. When a vector is placed in the plane, the point at the beginning of the line segment is called the tail, and the point at the end of the line segment is called the head. The length of the line segment is the magnitude of the vector. 4.8.A.2 A vector
- 4.9 — Vector-Valued Functions INSTRUCTIONAL
- 4.10 — Matrices Additional Topic Available to Schools
- 4.11 — The Inverse and Determinant of a Matrix Determine the inverse of a 2 2 × matrix. The identity matrix, I, is a square matrix consisting of 1s on the diagonal from the top left to bottom right and 0s everywhere else. 4.11.A.2 Multiplying a square matrix by its corresponding identity matrix results in the original square matrix. 4.11.A.3 The product of a square matrix and its inverse, when it exists, is the identity matrix of the same size. 4.11.A.4 The inverse of a 2 2 × matrix, when it exists, can be calculated with or without technology.
- 4.12 — Linear Transformations and Matrices a x + a y, a x a y 11 12 21 22
- 4.13 — Matrices as Functions Additional Topic Available to Schools Determine the association between a linear
- 4.14 — Matrices Modeling Contexts Additional Topic Available to Schools A contextual scenario can indicate the rate of transitions between states as percent changes. A matrix can be constructed based on these rates to model how states change over discrete intervals. 4.14.B 4.14.B.1 Apply matrix models to predict future and past states for n transition steps.
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