Precal Standards: Difference between revisions

Revision as of 00:23, 25 July 2025

Item Type,Sequence,Human Code,Full Statement,Supp. Info Grade Level,17,MA19.PRE,Precalculus, Content Area,17.1,MA19.PRE.NQ,Number and Quantity, Focus Area,17.1.1,MA19.PRE.NQ.CN,The Complex Number System, Essential Concept,17.1.1.1,MA19.PRE.NQ.CN.A,Perform arithmetic operations with complex numbers.,

Standard,17.1.1.1.1,MA19.PRE.1,Define the constant _e_ in a variety of contexts.,"

Teacher Vocabulary

Applications
Behavior
Continuous
Explore

Knowledge

Students know:

Exponential forms y=a-b^x and y=A₀e^k-x.
b must be nonnegative.
a is the initial value.
If b>1, the function models exponential growth.
If 0, the function models exponential decay.

Skills

Students are able to:

Use natural exponential functions to describe the growth of natural phenomena.
Use natural logarithm models to describe the time needed for the growth of natural phenomena.

Understanding

Students understand that:

ln(x) gives the time needed to grow x.
e^x gives the amount of growth after the time x.

"

Sub-Standard,17.1.1.1.1.1,MA19.PRE.1a,Explore the behavior of the function $y=e^x$ and its applications., Sub-Standard,17.1.1.1.1.2,MA19.PRE.1b,"Explore the behavior of _ln(x)_, the logarithmic function with base _e_, and its applications.",

Standard,17.1.1.1.2,MA19.PRE.2,Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.,"

Teacher Vocabulary

Complex number
Conjugate
Modulus/moduli

Knowledge

Students know:

The definition of the conjugate of a complex number.
A complex number divided by itself equals 1.
The product of a complex number and its conjugate is a real number (the square of the modulus).

Skills

Students are able to:

Find the conjugate of a complex number.
Find the modulus of a complex number
Find the product of two complex numbers.
Find (simplify) the quotient of complex numbers.

Understanding

Students understand that:

The conjugate of a complex number differs by the sign of its imaginary part and has the same modulus.
The modulus of a complex number corresponds to the magnitude of a vector and, therefore, is useful in the geometric representation of complex numbers.
Mathematical convention is that radical expressions are not left in denominators to facilitate numerical approximations; therefore, since the i is equal to the square root of -1, the conventional form says that i does not appear in the denominator of a fraction.
Different forms of a complex number quotient (indicated quotient, single complex number) may be more useful for various purposes.

"

Essential Concept,17.1.1.2,MA19.PRE.NQ.CN.B,Represent complex numbers and their operations on the complex plane.,

Standard,17.1.1.2.1,MA19.PRE.3,"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.","

Teacher Vocabulary

Complex plane
Polar form

Knowledge

Students know:

In the complex plane the horizontal axis is the real axis (a) and the vertical axis is the imaginary axis (b).
Trigonometric techniques for finding measures of angles and coordinates on the unit circle.
The characteristics of the polar coordinate system.
Techniques for plotting polar coordinates.

Skills

Students are able to:

Use trigonometry to find the measures of angles and coordinates on the unit circle.
Use the Pythagorean Theorem to find the lengths of sides of a right triangle.
Convert between polar and rectangular forms.
Plot polar coordinates.

Understanding

Students understand that:

A complex number (a+bi) can be graphed in a rectangular coordinate system as (a, b).
A complex number may be represented in the plane using equivalent polar and rectangular coordinates.
Different representations of a complex number may be more useful for various purposes.

" Standard,17.1.1.2.2,MA19.PRE.4,"Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.","

Teacher Vocabulary

Argument
Complex plane
Conjugation of complex numbers

Knowledge

Students know:

Complex numbers are represented geometrically in the complex plane with the real part measured on the x-axis and the imaginary represented on the y-axis.
Complex numbers can be added or subtracted by combining the real parts and the imaginary parts or by using vector procedures geometrically (end-to-end, parallelogram rule).
The product of complex numbers in polar form may be found by multiplying the magnitudes and adding the arguments.

Skills

Students are able to:

Add, subtract, and multiply the component parts of complex numbers to find sums, differences, and products.
Identify the conjugate of a complex number and use this as a computational aid, e.g., to find a quotient of complex numbers.
Represent complex numbers in the complex plane.
Add and subtract complex numbers geometrically.
Multiply complex numbers in polar form.

Understanding

Students understand that:

Different representations of mathematical concepts (e.g., algebraic and geometric representations of complex numbers) reveal different features of the concept and each may facilitate computation and sense-making in different settings.
Mathematics is a coherent whole and structure within mathematics allows for procedures from one area to be used in another (e.g., coordinate geometry and the complex plane, vectors and complex numbers, or plotting of a conjugate of a complex number in transformational geometry).

" Standard,17.1.1.2.3,MA19.PRE.5,"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.","

Teacher Vocabulary

Complex plane
Distance
Midpoint
Modulus

Knowledge

Students know:

Complex numbers can be represented geometrically.
Complex numbers can be added and subtracted either geometrically or algebraically.

Skills

Students are able to:

Add and subtract complex numbers.
Find the modulus of a complex number.
Represent complex numbers geometrically.

Understanding

Students understand that:

Representing complex numbers on a rectangular coordinate system allows the use of techniques developed for real numbers to find distances and midpoints.

"

Essential Concept,17.1.1.3,MA19.PRE.NQ.CN.C,Use complex numbers in polynomial identities and equations.,

Standard,17.1.1.3.1,MA19.PRE.6,"Analyze possible zeros for a polynomial function over the complex numbers by applying the Fundamental Theorem of Algebra, using a graph of the function, or factoring with algebraic identities.","

Teacher Vocabulary

Fundamental Theorem of Algebra
Quadratic polynomial
Zeros

Knowledge

Students know:

The definition of the degree of a polynomial.
The difference between real and complex roots.

Skills

Students are able to:

Find roots of a polynomial algebraically and/or graphically.
Rewrite an imaginary number as a complex number.

Understanding

Students understand that:

The degree of a polynomial determines the number of roots, some of which may be real, complex, or used more than once.
Only real roots will be x-intercepts on a graph.

"

Focus Area,17.1.2,MA19.PRE.NQ.L,Limits, Essential Concept,17.1.2.1,MA19.PRE.NQ.L.A,Understand limits of functions.,

Standard,17.1.2.1.1,MA19.PRE.7,"Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.","

Teacher Vocabulary

Continuous function
Convergent
Discontinuity (infinite, jump, removable)
Discontinuous function
Divergent
Limit

Knowledge

Students know:

How to graph the families of functions.
How to factor, simplify, and rationalize functions.
How to evaluate functions at a given input value.

Skills

Students are able to:

Identify continuity and discontinuity (if discontinuous, identify the type).
Calculate limits.

Understanding

Students understand that:

Functions have various types of continuity and discontinuity.
Limits can be determined numerically, graphically, and algebraically.
Converging means that a limit exists while diverging means that a limit does not exist.

"

Sub-Standard,17.1.2.1.1.1,MA19.PRE.7a,Apply limits of functions at specific values and at infinity in problems involving convergence and divergence., Focus Area,17.1.3,MA19.PRE.NQ.VM,Vector and Matrix Quantities, Essential Concept,17.1.3.1,MA19.PRE.NQ.VM.A,Represent and model with vector quantities.,

Standard,17.1.3.1.1,MA19.PRE.8,"Explain that vector quantities have both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.","

Teacher Vocabulary

Component form
Directed line segment
Magnitude
Vector quantity

Knowledge

Students know:

The difference between a ray and a directed line segment.
When drawing a vector on the xy-plane, magnitude is represented by a distance.

Skills

Students are able to:

Locate vectors on the xy-plane.
Use the unit circle to find trigonometric ratios for values in all four quadrants.
Use the Pythagorean Theorem to find the lengths of sides of a right triangle.

Understanding

Students understand that:

Magnitude as the length of a vector and direction of a vector as the measure of the angle it makes with a horizontal line
When vectors are represented in component form, ensuing computations and applications can be accomplished.

" Standard,17.1.3.1.2,MA19.PRE.9,Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.,"

Teacher Vocabulary

Components
Initial point
Terminal point

Knowledge

Students know:

If a vector is transposed in the xy-plane, it retains its magnitude and direction.

Skills

Students are able to:

Transpose a vector from one position to another position in the xy-plane.
Find the component form of a vector.

Understanding

Students understand that:

Vectors having the same magnitude and direction are equivalent regardless of where they are in the xy-plane.
Vectors in standard position have a terminal point that is equal to the components of the vector.

" Standard,17.1.3.1.3,MA19.PRE.10,Solve problems involving velocity and other quantities that can be represented by vectors.,"

Teacher Vocabulary

Component form
Components
Direction
Initial point
Magnitude
Scalar
Terminal point
Vector
Velocity

Knowledge

Students know:

The interpretation of forces as vectors.
Vectors can be used to represent forces.
Vector arithmetic.
Trigonometric functions used to write vectors.
Vector formulas.
Graphing points.
Right triangle trigonometry.
Unit circle.

Skills

Students are able to:

Write and represent a given force as a vector.
Combine vector quantities.
Write vectors in component form.
Use a protractor.

Understanding

Students understand that:

The result of combining multiple vector forces creates a net magnitude and direction.
Certain situations can be represented using vectors.

" Standard,17.1.3.1.4,MA19.PRE.11,Find the scalar (dot) product of two vectors as the sum of the products of corresponding components and explain its relationship to the cosine of the angle formed by two vectors.,"

Teacher Vocabulary

Components
Decomposition
Dot product
Orthogonal
Parallel
Vector components
Vector projection

Knowledge

Students know:

The formula and alternative formula for the dot product.
The properties of the dot product.
The formula for the angle between two vectors.
The relationship between the dot product and orthogonal vectors.
Projection of a vector onto another vector.
Vector components of v.

Skills

Students are able to:

Find the dot product of two vectors.
Find the angle between two vectors.
Use the dot product to determine if two vectors are orthogonal.
Find the projection of a vector onto another vector.
Express a vector as the sum of two orthogonal vectors.

Understanding

Students understand that:

The dot product of two vectors is the sum of the products of their horizontal components and their vertical components.
If 𝑣 = 𝑎1𝑖 + 𝑏1𝑗 and 𝑤 = 𝑎2𝑖 + 𝑏2𝑗, the dot product of 𝑣 and 𝑤 is defined by 𝑣 ∙ 𝑤 = 𝑎1𝑎2 + 𝑏1𝑏2.
Alternative Formula for the Dot Product: 𝑣 ∙𝑤 = ‖𝑣‖ ‖𝑤‖ cos 𝜃, where 𝜃 is the smallest nonnegative angle between v and w.
Two vectors are orthogonal when the angle between them is 90°. To show that two vectors are orthogonal, show that their dot product is zero.
A vector may be expressed as the sum of two orthogonal vectors, called the vector components.

"

Essential Concept,17.1.3.2,MA19.PRE.NQ.VM.B,Perform operations on vectors.,

Standard,17.1.3.2.1,MA19.PRE.12,Add and subtract vectors.,"

Teacher Vocabulary

Additive Inverse
Component
End
Parallelogram Rule
Sum of Two Vectors
Vector Subtraction-end-to-wise

Knowledge

Students know:

The aspects of end-to-end, componentwise, and the parallelogram rule involving vectors.
The additive inverse of a vector has the same magnitude but the opposite direction.

Skills

Students are able to:

Draw and find the diagonal of a parallelogram.
Represent vectors on an xy-plane.
Find the components of a vector given the direction and magnitude.
Find the additive inverse of a vector.

Understanding

Students understand that:

There are multiple ways to find the sum and difference of a pair of vectors.
The magnitude of the sum of two vectors will not be the same as the sum of the magnitudes unless the vectors are in the same direction.
The vector with the larger magnitude will have the greatest effect on the result.

"

Sub-Standard,17.1.3.2.1.1,MA19.PRE.12a,"Add vectors end-to-end, component-wise, and by the parallelogram rule, understanding that the magnitude of a sum of two vectors is not always the sum of the magnitudes.", Sub-Standard,17.1.3.2.1.2,MA19.PRE.12b,"Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.", Sub-Standard,17.1.3.2.1.3,MA19.PRE.12c,"Explain 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.",

Standard,17.1.3.2.2,MA19.PRE.13,Multiply a vector by a scalar.,"

Teacher Vocabulary

Scalar multiple
Scalar multiplication
Scaling vectors

Knowledge

Students know:

The representation of vectors graphically on the xy-plane.

Skills

Students are able to:

Find the components of a vector.
Find the magnitude and direction of a vector.

Understanding

Students understand that:

Scalar multiplication results in dilation of the original vector where a scalar greater than 1 would increase the magnitude and a scalar from 0 to 1 would decrease the magnitude.
A negative scalar would reverse the direction of the vector.
The absolute value of the scalar has the resulting effect on the magnitude.

"

Sub-Standard,17.1.3.2.2.1,MA19.PRE.13a,Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise., Sub-Standard,17.1.3.2.2.2,MA19.PRE.13b,"Compute the magnitude of a scalar multiple _c_**v** using ||_c_**v**|| = |_c_|**v**. Compute the direction of _c_**v** knowing that when |_c_|**v** ≠ 0, the direction of _c_**v** is either along **v** (for _c_ > 0) or against **v** (for _c_ < 0).",

Standard,17.1.3.2.3,MA19.PRE.14,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.,"

Teacher Vocabulary

Transformation of vectors

Knowledge

Students know:

Conditions under which matrix multiplication is defined.
Techniques for adding and multiplying matrices.
Techniques for scalar multiplication.
Techniques for performing translations, rotations, reflections, and dilations.

Skills

Students are able to:

Write a vector in matrix notation.
Determine if matrix multiplication is defined for a given product of a matrix by a vector.
Multiply a vector by a matrix.
Add, multiply, and perform scalar multiplication on matrices.

Understanding

Students understand that:

Mathematical representations such as vectors and matrices may be used in a wide variety of settings to model and solve real-world problems and this modeling is facilitated by fluent use of techniques for working with these representations.
Transformations of multiple vectors may be accomplished by the use of matrix multiplication as a means of transforming all vectors at the same time in a similar manner.

"

Content Area,17.2,MA19.PRE.A,Algebra, Focus Area,17.2.1,MA19.PRE.A.SS,Seeing Structure in Expressions, Essential Concept,17.2.1.1,MA19.PRE.A.SS.A,Write expressions in equivalent forms to solve problems.,

Standard,17.2.1.1.1,MA19.PRE.15,"Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems, extending to infinite geometric series.","

Teacher Vocabulary

Common ratio
Geometric series (finite and infinite)

Knowledge

Students know:

Characteristics of a geometric series.
Techniques for performing algebraic manipulations and justifications for the equivalence of the resulting expressions.

Skills

Students are able to:

Identify the regularity that exists in a series as being that which defines it as a geometric series.
Accurately perform the procedures involved in using geometric series to solve contextual problem,
Explain with mathematical reasoning why each step in the derivation of the formula for the sum of a finite geometric series is legitimate, including explaining why the formula does not hold for a common ratio of 1.

Understanding

Students understand that:

When each term of a geometric series is multiplied by a value, the result is a new geometric series,
When many problems exist with the same mathematical structure, formulas are useful generalizations for efficient solutions of problems, (e.g., mortgage payment calculation with geometric series).

"

Focus Area,17.2.2,MA19.PRE.A.AP,Arithmetic With Polynomials and Rational Expressions, Essential Concept,17.2.2.1,MA19.PRE.A.AP.A,Understand the relationship between zeros and factors of polynomials.,

Standard,17.2.2.1.1,MA19.PRE.16,"Derive and apply the Remainder Theorem: For a polynomial _p(x)_ and a number _a_, the remainder on division by _x - a_ is _p(a)_, so _p(a)_ = 0 if and only if (_x - a_) is a factor of _p(x)._","

Teacher Vocabulary

If and only if
Remainder theorem

Knowledge

Students know:

Procedures for dividing a polynomial p(x) by a linear polynomial (x⋅a) (e.g., long division and synthetic division).

Skills

Students are able to:

Accurately perform procedures for dividing a polynomial p(x) by a linear polynomial (x⋅a).
Evaluate a polynomial p(x) for any value of x.

Understanding

Students understand that:

There is a structural relationship between the value of a in (x⋅a), as well as the remainder when p(x) is divided by (x⋅a).
If p(a)=0, then x-a if a factor of p(x).

"

Essential Concept,17.2.2.2,MA19.PRE.A.AP.B,Use polynomial identities to solve problems.,

Standard,17.2.2.2.1,MA19.PRE.17,"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.","

Teacher Vocabulary

Binomial Theorem
Combinatorial argument
Mathematical induction
Pascal's Triangle

Knowledge

Students know:

Distributive property of multiplication over addition for polynomials.
The generation pattern for Pascal's Triangle and which binomial expansion term has coefficients corresponding to each row.
Simplification procedures for expressions involving the number of combinations of n things taken r at a time.
The patterns of coefficients and exponents in a binomial expansion.

Skills

Students are able to:

Accurately perform algebraic manipulations on polynomial expressions.
Generate rows of Pascal's Triangle.
Accurately perform simplification procedures for expressions involving the number of combinations of n things taken r at a time.
Apply the patterns of coefficients and exponents to expand any binomial raised to a power.

Understanding

Students understand that:

Regularities noted in one part of mathematics may also be seen in very different areas of mathematics, (i.e., Pascal's Triangle from counting procedures and the Binomial Theorem). These regularities are useful in computing or manipulating mathematical expressions.
The regularities that are seen in exponents and coefficients in a binomial expansion will be generalized to all binomials to aid in identifying specific terms.

"

Essential Concept,17.2.2.3,MA19.PRE.A.AP.C,Rewrite rational expressions.,

Standard,17.2.2.3.1,MA19.PRE.18,"Rewrite simple rational expressions in different forms; write _a(x)/b(x)_ in the form _q(x) + r(x)/b(x)_, where _a(x), b(x), q(x)_, and _r(x)_ are polynomials with the degree of _r(x)_ less than the degree of _b(x)_, using inspection, long division, or, for the more complicated cases, a computer algebra system.","

Teacher Vocabulary

Degree of polynomial
Inspection
Long Division
Rational expression

Knowledge

Students know:

Techniques for long division of polynomials.
Techniques for utilizing a computer algebra system.

Skills

Students are able to:

Accurately perform polynomial long division.
Efficiently and accurately use a computer algebra system to divide polynomials.

Understanding

Students understand that:

The role of the remainder in polynomial division is analogous to that of the remainder in whole number division.
Different forms of rational expressions are useful to reveal important features of the expression.

" Standard,17.2.2.3.2,MA19.PRE.19,"Add, subtract, multiply, and divide rational expressions.","

Teacher Vocabulary

Analogous
Closed under an operation
Nonzero rational expression
Rational expression

Knowledge

Students know:

Techniques for performing operations on polynomials.

Skills

Students are able to:

Accurately perform addition, subtraction, multiplication, and division of rational expressions.

Understanding

Students understand that:

They can communicate a mathematical justification for all four operations on rational expressions being closed.
The structure of mathematics present in the system of rational numbers is also present in the system of rational expressions.

"

Sub-Standard,17.2.2.3.2.1,MA19.PRE.19a,"Explain why rational expressions form a system analogous to the rational numbers, which is closed under addition, subtraction, multiplication, and division by a non-zero rational expression.", Focus Area,17.2.3,MA19.PRE.A.RE,Reasoning With Equations and Inequalities, Essential Concept,17.2.3.1,MA19.PRE.A.RE.A,Understand solving equations as a process of reasoning and explain the reasoning.,

Standard,17.2.3.1.1,MA19.PRE.20,"Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a clear-cut solution. Construct a viable argument to justify a solution method. **Include equations that may involve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric functions, and their inverses.**","

Teacher Vocabulary

Equivalence
Viable

Knowledge

Students know:

How to solve equations using a reasoning process centered around inverse operations and order of operations.

Skills

Students are able to:

Solve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric equations (including their inverses) using multiple solution strategies and explain each step in the solution path.
Construct a viable argument to justify a chosen solution path used to solve a linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric equation (including their inverses).
Compare the steps in each and determine which solution path is most efficient, given an equation with multiple solution paths.
Explain when an equation has no solution or infinitely many solutions.

Understanding

Students understand that:

The process of solving equations is a reasoning process to determine a solution that satisfies the equation rather than a procedural list of steps.
An equation has no solution because there is no value that can maintain equivalency and an equation has infinitely many solutions because all values used for the variable create a true equivalency statement.

" Standard,17.2.3.1.2,MA19.PRE.21,"Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.","

Teacher Vocabulary

Extraneous solutions
Rational equations

Knowledge

Students know:

Algebraic rules for manipulating rational equations.
Conditions under which a solution is considered extraneous.

Skills

Students are able to:

Accurately rearrange rational equations to produce a set of values to test against the conditions of the original situation and equation, and determine whether or not the value is a solution.
Explain with mathematical reasoning from the context (when appropriate) why a particular solution is or is not extraneous.

Understanding

Students understand that:

Values that arise from solving equations may not satisfy the original equation.
Values that arise from solving the equations may not exist due to considerations in the context.

"

Essential Concept,17.2.3.2,MA19.PRE.A.RE.B,Solve systems of equations.,

Standard,17.2.3.2.1,MA19.PRE.22,Represent a system of linear equations as a single matrix equation in a vector variable.,"

Teacher Vocabulary

Matrix equation
Vector variable

Knowledge

Students know:

The structure of a matrix equation is the product of the coefficient matrix on the left and the vector variable on the right is equal to the constant vector.

Skills

Students are able to:

Transform a system of linear equations into a matrix equation.

Understanding

Students understand that:

Multiple representations of linear systems are needed to facilitate work with more complex systems.
Matrix representation allows for solutions with systems containing any number of variables.

" Standard,17.2.3.2.2,MA19.PRE.23,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 x 3 or greater).,"

Teacher Vocabulary

Dimension of a matrix
Inverse of a matrix

Knowledge

Students know:

Procedures for converting systems of equations to matrix equations.
Conditions that determine the inverse of a matrix exist.
Techniques for determining the inverse of a matrix (including using technology).
Process for using the inverse of a matrix to find the solution to a matrix equation (2 x 2).
Matrix multiplication is not commutative.

Skills

Students are able to:

Perform row operations on a matrix to find the inverse of the matrix.
Efficiently and accurately find the inverse of a matrix using technology.

Understanding

Students understand that:

The solutions to a matrix equation are the solutions to the system of equations that produced the matrix equation.
Solving a matrix equation is analogous to solving a linear equation.
Technology is a useful tool that facilitates investigation and, once the initial process is understood, helps find solutions to more complex problems.

"

Content Area,17.3,MA19.PRE.F,Functions, Focus Area,17.3.1,MA19.PRE.F.IF,Interpreting Functions, Essential Concept,17.3.1.1,MA19.PRE.F.IF.A,Interpret functions that arise in applications in terms of the context.,

Standard,17.3.1.1.1,MA19.PRE.24,"Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features. _Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity._ **Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.**","

Teacher Vocabulary

Absolute maximum
Absolute minimum
Asymptotes
End behavior
Function
Increasing/decreasing intervals
Intercepts
Periodicity
Relative maximum
Relative minimum
Symmetry (even/odd)

Knowledge

Students know:

Properties of functions and make connections between different representations of the same function.

Skills

Students are able to:

Compare properties of functions when represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions).

Understanding

Students understand that:

Each representation provides a unique perspective of the function.
Different representations are most appropriate for revealing certain key features of the function.

" Standard,17.3.1.1.2,MA19.PRE.25,"Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. **Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.**","

Teacher Vocabulary

Average rate of change
Difference quotient
Specified interval

Knowledge

Students know:

Techniques for graphing.
Techniques for finding a rate of change over an interval on a table or graph.
Techniques for estimating a rate of change over an interval on a graph.

Skills

Students are able to:

Calculate the average rate of change on a specified interval when given an equation or table of a polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.
Interpret the average rate of change of a polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions in the context of a problem when given symbolic representations, tables, graphs, or contextual situations.
Estimate the average rate of change for a specific interval of a polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions when given a graph.

Understanding

Students understand that:

The rate of change is the ratio of the change between the dependent and independent variables.

"

Sub-Standard,17.3.1.1.2.1,MA19.PRE.25a,Find the difference quotient $\frac {f(x+\Delta x) - f(x)}{\Delta x}$ of a function and use it to evaluate the average rate of change at a point., Sub-Standard,17.3.1.1.2.2,MA19.PRE.25b,Explore how the average rate of change of a function over an interval (presented symbolically or as a table) can be used to approximate the instantaneous rate of change at a point as the interval decreases., Essential Concept,17.3.1.2,MA19.PRE.F.IF.B,Analyze functions using different representations.,

Standard,17.3.1.2.1,MA19.PRE.26,"Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.","

Teacher Vocabulary

Amplitude
Domain
Frequency
Horizontal asymptote
Midline
Period
Phase shift
Range
Rational functions
Slant asymptote
Vertical asymptote

Knowledge

Students know:

Techniques for graphing,
Key features of graphs of functions.

Skills

Students are able to:

Determine horizontal, vertical, and slant asymptotes of rational functions, and use these to sketch the graphs.
Identify domains, ranges, and end behaviors.
Sketch the graphs, analyze, compare, and identify domains and ranges of the basic trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant).
Find the amplitude and period of a trigonometric function and use these characteristics to sketch its graph.
Identify and sketch translations of trigonometric graphs (vertical shifts and phase shifts).
Evaluate, graph, and identify the domains and ranges of inverse trigonometric functions.

Understanding

Students understand that:

A rational function is the ratio of two polynomial functions.
Rational functions contain restrictions on their domains and/or ranges. Therefore, their graphs contain asymptotes, holes, and/or discontinuity.
The graphs of rational functions vary, yielding various patterns.
Using algebraic methods to manipulate and/or solve the equation of a rational function can help determine important properties such as its zeroes, intercepts, asymptotes, domain, range, types of discontinuity, and end behavior.
Key characteristics (rational and trigonometric) of functions can help you visualize the sketch of its graph and can lead to more effective and efficient graphing methods.

"

Sub-Standard,17.3.1.2.1.1,MA19.PRE.26a,"Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.", Sub-Standard,17.3.1.2.1.2,MA19.PRE.26b,"Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.", Focus Area,17.3.2,MA19.PRE.F.BF,Building Functions, Essential Concept,17.3.2.1,MA19.PRE.F.BF.A,Build a function that models a relationship between two quantities.,

Standard,17.3.2.1.1,MA19.PRE.27,"Compose functions. **Extend to polynomial, trigonometric, radical, and rational functions.**","

Teacher Vocabulary

Compose functions
Explicit expression
Recursive process

Knowledge

Students know:

Techniques for expressing functional relationships (explicit expression, a recursive process, or steps for calculation) between two quantities.
Techniques to combine functions using arithmetic operations.
Techniques to compose functions using algebraic operations.
Notation for function composition.

Skills

Students are able to:

Accurately develop a model that shows the functional relationship between two quantities.
Accurately create a new function through arithmetic operations of other functions.
Accurately create a new function through the composition of other functions.

Understanding

Students understand that:

Relationships can be modeled by several methods (e.g., explicit expression or recursive process).
Arithmetic combinations and/or composition of functions may be used to improve the fit of a model.
When functions are combined to create a new function, present an argument to show how the function models the relationship between the quantities.

"

Essential Concept,17.3.2.2,MA19.PRE.F.BF.B,Build new functions from existing functions.,

Standard,17.3.2.2.1,MA19.PRE.28,Find inverse functions.,"

Teacher Vocabulary

Composition
Domain/output of a relation/function
Horizontal line test (one-to-one)
Inverse
Inverse function
Invertible function
Non-invertible function
Range/output of a relation/function
Restricting the domain

Knowledge

Students know:

The domain and range of a given relation or function.
Algebraic properties.
Symmetry about the line y = x.
Techniques for composing functions.
The composition of a function and its inverse is the identity function.
When (x,y) is a point on an invertible function, (y,x) is a point on the inverse.
In order for a function to have an inverse function, the original function must have a one-to-one correspondence.

Skills

Students are able to:

Find the inverse of a function.
Accurately perform algebraic properties to find the inverse.
Accurately identify restrictions on a non-invertible function that allow it to be invertible.
Accurately find the composition of two functions.

Understanding

Students understand that:

The graphical and algebraic relationship between a function and its inverse.
The process of finding the inverse of a function.
The inverse of a function interchanges the input and output values from the original function.
The inverse of a function must also be a function to exist and the domain may need to be restricted to make this occur.

"

Sub-Standard,17.3.2.2.1.1,MA19.PRE.28a,"Given that a function has an inverse, write an expression for the inverse of the function.", Sub-Standard,17.3.2.2.1.2,MA19.PRE.28b,Verify by composition that one function is the inverse of another., Sub-Standard,17.3.2.2.1.3,MA19.PRE.28c,"Read values of an inverse function from a graph or a table, given that the function has an inverse.", Sub-Standard,17.3.2.2.1.4,MA19.PRE.28d,Produce an invertible function from a non-invertible function by restricting the domain.,

Standard,17.3.2.2.2,MA19.PRE.29,Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents. **Extend from logarithms with base 2 and 10 to a base of _e_.**,"

Teacher Vocabulary

Inverse relationship
Logarithm
Natural logarithm

Knowledge

Students know:

Definition of logarithm: If b^x = y, then log_b y = x.
Restrictions on domain and range of both functions (b > 0, b not equal to 1, y > 0).
Techniques for creating graphs of exponential and logarithmic functions.
Situations that can be modeled by exponential and logarithmic functions.

Skills

Students are able to:

Re-write the equivalent inverse function from an exponential function or logarithm function in various forms (graphs, tables, or equations).
Model contextual situations using logarithmic and exponential functions.
Solve exponential and logarithmic equations by isolating the variable.

Understanding

Students understand that:

Equivalent forms of a function, specifically logarithmic and exponential, may be useful at different times to solve problems.
Although the algebraic representations look different, the logarithmic form and exponential form of the same relationship are equivalent.

" Standard,17.3.2.2.3,MA19.PRE.30,"Identify the effect on the graph of replacing $f(x)$ by $f(x) + k$, $k \cdot f(x)$, $f(k \cdot x)$, and $f(x + k)$ for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs. Extend the analysis to include all trigonometric, rational, and general piecewise-defined functions with and without technology.","

Teacher Vocabulary

Amplitude
Horizontal shift
Horizontal stretch/shrink
Midline
Phase shift
Reflection
Vertical shift
Vertical stretch/shrink

Knowledge

Students know:

Graphing techniques of functions.
Methods of using technology to graph functions.

Skills

Students are able to:

Accurately graph functions.
Check conjectures about how a parameter change in a function changes the graph and critique the reasoning of others about such shifts.
Identify shifts, stretches, or reflections between graphs.

Understanding

Students understand that:

Graphs of functions may be shifted, stretched, or reflected by adding or multiplying the input or output of a function by a constant value.

" Standard,17.3.2.2.4,MA19.PRE.31,"Graph conic sections from second-degree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.","

Teacher Vocabulary

Asymptote
Degenerate conic
Directrix
Eccentricity
Ellipse
Focus (foci)
Hyperbola
Latus rectum (focal distance)
Locus
Major axis (transverse axis)
Minor axis (conjugate axis)

Knowledge

Students know:

Vertex form of a parabola.
Standard form of a circle.
Vertex and axis of symmetry of a parabola.
Completing the square.
Factoring a quadratic function.

Skills

Students are able to:

Graph equations of parabolas.
Graph equations of circles.
Graph equations of ellipses.
Calculate eccentricities of ellipses.
Graph equations of hyperbolas.
Classify a conic section using its general equation and/or its discriminant.

Understanding

Students understand that:

A conic section is a graph of an equation of the form Ax² + Bxy + Cy² +Dx +Ey + F = 0.
The only conic sections that are functions are parabolas that open upward or downward, previously learned as quadratic functions and hyperbolas that are written in the form of a rational function.
Using algebra to manipulate the equation of a conic section, particularly the method of “completing the square” can be used to determine the parts and properties of its graph, and can result in the use of more effective and efficient graphing methods.

"

Sub-Standard,17.3.2.2.4.1,MA19.PRE.31a,Graph conic sections given their standard form., Sub-Standard,17.3.2.2.4.2,MA19.PRE.31b,"Identify the conic section that will be formed, given its equation in general form.", Focus Area,17.3.3,MA19.PRE.F.TF,Trigonometric Functions, Essential Concept,17.3.3.1,MA19.PRE.F.TF.A,Recognize attributes of trigonometric functions and solve problems involving trigonometry.,

Standard,17.3.3.1.1,MA19.PRE.32,Solve application-based problems involving parametric and polar equations.,"

Teacher Vocabulary

Orientation
Parameter
Parametric curve
Parametric equations
Polar equations
Rectangular form

Knowledge

Students know:

How to model motion using quadratic functions.
Algebraic manipulation of equations.
Situations parametric and polar modeling is appropriate.

Skills

Students are able to:

Develop parametric and polar equations for a given situation.
Graph the parametric or polar equations.
Answer questions in the context of the parametric or polar problem.

Understanding

Students understand that:

Parametric equations can be used to model and evaluate the trajectory of projectiles.
Parametric equations can be used to model and evaluate the range of projectiles.
Polar equation can be used to model motion for some mechanical systems.

"

Sub-Standard,17.3.3.1.1.1,MA19.PRE.32a,Graph parametric and polar equations., Sub-Standard,17.3.3.1.1.2,MA19.PRE.32b,Convert parametric and polar equations to rectangular form., Essential Concept,17.3.3.2,MA19.PRE.F.TF.B,Extend the domain of trigonometric functions using the unit circle.,

Standard,17.3.3.2.1,MA19.PRE.33,"Use special triangles to determine geometrically the values of sine, cosine, and tangent for $\frac {\pi}{3}$, $\frac {\pi}{4}$, and $\frac {\pi}{6}$, and use the unit circle to express the values of sine, cosine, and tangent for $\pi - x$, $\pi + x$, and $2\pi - x$ in terms of their values for $x$, where $x$ is any real number.","

Teacher Vocabulary

Special triangles
Unit circle

Knowledge

Students know:

The relationship between the lengths of the sides of a 45-45-90 and 30-60-90 triangle.
The basic trig ratios.

Skills

Students are able to:

Find the value of sine, cosine, and tangent of π/3, π/4, and π/6 using special triangles.
Locate an angle in the standard position in the unit circle.
Convert between degrees and radians.

Understanding

Students understand that:

For an angle in standard position, the point where the terminal ray intersects the unit circle has an x-coordinate which is the value of the cosine of the angle, and a y-coordinate which is the value of the sine of the angle.
Patterns that can be identified on the unit circle allow for the application of right triangle trigonometry to angles of all sizes.

" Standard,17.3.3.2.2,MA19.PRE.34,Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.,"

Teacher Vocabulary

Odd and even symmetry
Periodicity

Knowledge

Students know:

The characteristics of the unit circle.
The characteristics of even and odd functions.

Skills

Students are able to:

Transpose coordinates from the unit circle to a graph on the xy-plane.
Use the unit circle to verify even and odd trig identities.

Understanding

Students understand that:

Basic trig functions can be classified as either even or odd.
The repetitive nature of the unit circle creates periodic functions which is displayed in their graphs.

"

Essential Concept,17.3.3.3,MA19.PRE.F.TF.C,Model periodic phenomena with trigonometric functions.,

Standard,17.3.3.3.1,MA19.PRE.35,Demonstrate that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.,"

Teacher Vocabulary

Inverse functions
Restricting the domain

Knowledge

Students know:

Characteristics of an always increasing or always decreasing function.
When a function has an inverse that is also a function.

Skills

Students are able to:

Identify intervals where the function is increasing or decreasing.
Identify the domain of a function that will produce an inverse.
Produce different models of functions and their resulting inverse.

Understanding

Students understand that:

Due to the periodic nature of trig functions, domains have to be restricted to produce a one-to-one relationship.
While many different intervals are always increasing or decreasing, there are conventional choices for the restricted domain of the trig functions.

" Standard,17.3.3.3.2,MA19.PRE.36,"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.","

Teacher Vocabulary

Inverse functions
Trigonometric equations

Knowledge

Students know:

Periodic situations are best modeled by trig functions.
Solutions that are mathematically possible may not be physically possible in the context of the problem.
Determine when technology is appropriate for finding a solution.

Skills

Students are able to:

Solve a trig equation.
Interpret the meaning of the solution.
Interpret a domain in a contextual situation.
Use technology to solve a trig equation.
Translate a computed solution to an equivalent solution that fits in the physical domain (i.e., If -30 degrees is a solution, then another solution can be 330 degrees).

Understanding

Students understand that:

There are periodic phenomena in real life that can be modeled by trig functions.
From the many solutions that result from a periodic function, only some may be logical given the context.

"

Essential Concept,17.3.3.4,MA19.PRE.F.TF.D,Prove and apply trigonometric identities.,

Standard,17.3.3.4.1,MA19.PRE.37,Use trigonometric identities to solve problems.,"

Teacher Vocabulary

Cosecant
Cosine
Cotangent
Double
Even and odd function
Fundamental Identities
Half
Identity
Pythagorean identities
Quotient identities
Reciprocal identities
Secant
Sine
Sum and difference identities
Tangent
Angle identities
Angle identities

Knowledge

Students know:

Applications of the Pythagorean Theorem.
Operations with trigonometric ratios.
Operations with radians and degrees.
Even and odd functions .

Skills

Students are able to:

Use and transform the Pythagorean Identity.
Simplify trigonometric expressions.
Verify trigonometric identities.
Write the sum and difference identities for sine, cosine, and tangent.
Use sum and difference identities to find the exact values of a trig function.
Derive the double-angle and half-angle identities.

Understanding

Students understand that:

The fundamental identities allow functions to be written in terms of other functions, and algebraic methods can be applied to simplify expressions or to match them with another expression.
Given the trig values for a pair of angles, identities can be used to find the trig values of the sum or difference of the given angles.
Given the trig values for an angle, identities can be used to find the trig values for twice and half the angle.

"

Sub-Standard,17.3.3.4.1.1,MA19.PRE.37a,Use the Pythagorean identity $sin^2 (\theta) + cos^2(\theta) = 1$ to derive the other forms of the identity., Sub-Standard,17.3.3.4.1.2,MA19.PRE.37b,"Use the angle sum formulas for sine, cosine, and tangent to derive the double angle formulas.", Sub-Standard,17.3.3.4.1.3,MA19.PRE.37c,Use the Pythagorean and double angle identities to prove other simple identities.,