Tennessee: Precalculus | P Math Standards
89 standards · 5 domains
P.A
- P.A.C.A.1 Display all of the conic sections as portions of a cone.
- P.A.C.A.2 Know and write the equation of a circle of given center and radius using the Pythagorean Theorem.
- P.A.C.A.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
- P.A.C.A.4 From an equation in standard form, graph the appropriate conic section: ellipses, hyperbolas, circles, and parabolas. Demonstrate an understanding of the relationship between their standard algebraic form and the graphical characteristics.
- P.A.C.A.5 Transform equations of conic sections to convert between general and standard form.
- P.A.PE.A.1 Graph curves parametrically (by hand and with appropriate technology).
- P.A.PE.A.2 Eliminate parameters by rewriting parametric equations as a single equation.
- P.A.REI.A.1 Represent a system of linear equations as a single matrix equation in a vector variable.
- P.A.REI.A.2 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
- P.A.REI.A.3 Solve rational and radical equations in one variable, and identify extraneous solutions when they exist.
- P.A.REI.A.4 Solve nonlinear inequalities (quadratic, trigonometric, conic, exponential, logarithmic, and rational) by graphing (solutions in interval notation if one-variable), by hand and with appropriate technology.
- P.A.REI.A.5 Solve systems of nonlinear inequalities by graphing.
- P.A.S.A.1 Demonstrate an understanding of sequences by representing them recursively and explicitly.
- P.A.S.A.2 Use sigma notation to represent a series; expand and collect expressions in both finite and infinite settings.
- P.A.S.A.3.a Determine whether a given arithmetic or geometric series converges or diverges.
- P.A.S.A.3.b Find the sum of a given geometric series (both infinite and finite).
- P.A.S.A.3.c Find the sum of a finite arithmetic series.
- P.A.S.A.4 Understand that series represent the approximation of a number when truncated; estimate truncation error in specific examples.
- P.A.S.A.5 Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined, for example, by Pascal’s Triangle.
P.F
- P.F.BF.A.1 Understand how the algebraic properties of an equation transform the geometric properties of its graph (for example, given a function, describe the transformation of the graph resulting from the manipulation of the algebraic properties of the equation such as translations, stretches, reflections, and changes in periodicity and amplitude).
- P.F.BF.A.2 Develop an understanding of functions as elements that can be operated upon to get new functions: addition, subtraction, multiplication, division, and composition of functions.
- P.F.BF.A.3 Compose functions (for example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time).
- P.F.BF.A.4 Construct the difference quotient for a given function and simplify the resulting expression.
- P.F.BF.A.5.a Calculate the inverse of a function, f (x), with respect to each of the functional operations; in other words, the additive inverse, − f(x), the multiplicative inverse, 1/f(x), and the inverse with respect to composition, f^−1(x). Understand the algebraic and graphical implications of each type.
- P.F.BF.A.5.b Verify by composition that one function is the inverse of another.
- P.F.BF.A.5.c Read values of an inverse function from a graph or a table, given that the function has an inverse.
- P.F.BF.A.5.d Recognize a function is invertible if and only if it is one-to-one. Produce an invertible function from a non-invertible function by restricting the domain.
- P.F.BF.A.6 Explain why the graph of a function and its inverse are reflections of one another over the line y = x.
- P.F.GT.A.1 Interpret transformations of trigonometric functions.
- P.F.GT.A.2 Determine the difference made by choice of units for angle measurement when graphing a trigonometric function.
- P.F.GT.A.3 Graph the six trigonometric functions and identify characteristics such as period, amplitude, phase shift, and asymptotes.
- P.F.GT.A.4 Find values of inverse trigonometric expressions (including compositions), applying appropriate domain and range restrictions.
- P.F.GT.A.5 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
- P.F.GT.A.6 Determine the appropriate domain and corresponding range for each of the inverse trigonometric functions.
- P.F.GT.A.7 Graph the inverse trigonometric functions and identify their key characteristics.
- P.F.GT.A.8 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology and interpret them in terms of the context.
- P.F.IF.A.1 Determine whether a function is even, odd, or neither.
- P.F.IF.A.2 Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real-world problems that can be modeled with these functions (by hand and with appropriate technology).
- P.F.IF.A.3 Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential, polynomial, logarithmic, trigonometric, and rational).
- P.F.IF.A.4 Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax^2 + c.
- P.F.IF.A.5 Visually locate critical points on the graphs of functions and determine if each critical point is a minimum, a maximum, or point of inflection. Describe intervals where the function is increasing or decreasing and where different types of concavity occur.
- P.F.IF.A.6 Graph rational functions, identifying zeros, asymptotes (including slant), and holes (when suitable factorizations are available) and showing end-behavior.
- P.F.IF.A.7 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers (for example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1) for n ≥ 1).
- P.F.TA.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and explain how to use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
- P.F.TA.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
- P.F.TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
- P.F.TF.A.2 Convert from radians to degrees and from degrees to radians.
- P.F.TF.A.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
P.G
- P.G.AT.A.1 Use the definitions of the six trigonometric ratios as ratios of sides in a right triangle to solve problems about lengths of sides and measures of angles.
- P.G.AT.A.2 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
- P.G.AT.A.3 Derive and apply the formulas for the area of sector of a circle.
- P.G.AT.A.4 Calculate the arc length of a circle subtended by a central angle.
- P.G.AT.A.5 Prove the Laws of Sines and Cosines and use them to solve problems.
- P.G.AT.A.6 Understand and apply the Law of Sines (including the ambiguous case) and the Law of Cosines to find unknown measurements in right and non-right triangles (such as surveying problems and resultant forces).
- P.G.PC.A.1 Graph functions in polar coordinates.
- P.G.PC.A.2 Convert between rectangular and polar coordinates.
- P.G.PC.A.3 Represent situations and solve problems involving polar coordinates.
- P.G.TI.A.1 Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, reciprocal, quotient, sum/difference, double-angle, and half-angle.
- P.G.TI.A.2 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
P.N
- P.N.CN.A.1 Know there is a complex number i such that i^2 = –1, and every complex number has the form a + bi with a and b real.
- P.N.CN.A.2 Perform arithmetic operations with complex numbers expressing answers in the form a + bi.
- P.N.CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- P.N.CN.A.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
- P.N.CN.A.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation (for example, (–1 + 3i)^3 = 8 because (–1 + 3i) has modulus 2 and argument 120°).
- P.N.CN.A.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
- P.N.CN.B.7 Extend polynomial identities to the complex numbers (for example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
- P.N.CN.B.8 Solve quadratic equations with real coefficients that have complex solutions.
- P.N.CN.B.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
- P.N.NE.A.1 Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or compare them.
- P.N.NE.A.2 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
- P.N.NE.A.3 Classify real numbers and order real numbers that include transcendental expressions, including roots and fractions of π and e.
- P.N.NE.A.4 Simplify complex radical and rational expressions; discuss and display understanding that rational numbers are dense in the real numbers and the integers are not.
- P.N.NE.A.5 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
- P.N.VM.A.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, V→).
- P.N.VM.A.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- P.N.VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.
- P.N.VM.B.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
- P.N.VM.B.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
- P.N.VM.B.4.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
- P.N.VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise (e.g., as c(v_x , v_y) = (cv_x , cv_y).
- P.N.VM.B.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
- P.N.VM.B.6 Calculate and interpret the dot product of two vectors.
- P.N.VM.C.7 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- P.N.VM.C.8 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
- P.N.VM.C.9 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
- P.N.VM.C.10 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
P.S
- P.S.MD.A.1 Create scatter plots, analyze patterns, and describe relationships for bivariate data (linear, polynomial, trigonometric, or exponential)to model real-world phenomena and to make predictions.
- P.S.MD.A.2 Determine a regression equation to model a set of bivariate data. Justify why this equation best fits the data.
- P.S.MD.A.3 Use a regression equation, modeling bivariate data, to make predictions. Identify possible considerations regarding the accuracy of predictions when interpolating or extrapolating.