ENRICHMENT MATH CLASSES
FINANCIAL ANALYSIS / BUSINESS CONSULTING
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents
to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the
cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational
number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Reason quantitatively and use units to solve problems. [Foundation for work with expressions, equations and functions]
1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Interpret the structure of expressions. [Linear, exponential, and quadratic]
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret
P(1 + r)n as the product of P and a factor not depending on P.
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by
the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression
1.15t can be rewritten as (1.151/12) 12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
Perform arithmetic operations on polynomials. [Linear and quadratic]
1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Create equations that describe numbers or relationships. [Linear, quadratic, and exponential (integer inputs only);
for A.CED.3 linear only]
1. Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CA
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions
as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost
constraints on combinations of different foods.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example,
rearrange Ohm’s law V = IR to highlight resistance R
Understand solving equations as a process of reasoning and explain the reasoning.
[Master linear; learn as general principle.]
1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Solve equations and inequalities in one variable. [Linear inequalities; literal equations that are linear in the variables being
solved for; quadratics with real solutions]
4-b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic
formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a and b
Solve systems of equations. [Linear-linear and linear-quadratic]
5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a
multiple of the other produces a system with the same solutions.
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions
of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of
values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.
4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of
the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include:
intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example,
if the function h gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers
would be an appropriate domain for the function.
6. 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.
Build new functions from existing functions. [Linear, exponential, quadratic, and absolute value; for F.BF.4a, linear only]
3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive
and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the
graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature
of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
Interpret expressions for functions in terms of the situation they model.
5. Interpret the parameters in a linear or exponential function in terms of a context. [Linear and exponential of form
f(x) = bx + k]
6. Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity. CA
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal
factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a
relationship, or two input-output pairs (include reading these from a table).
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
Summarize, represent, and interpret data on two categorical and quantitative variables.
[Linear focus; discuss general principle.]
5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of
the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the
data.
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions
or choose a function suggested
Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of
functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of
convergence and divergence of a function as the domain variable approaches either a number or infinity.
1.1 Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.
1.2 Students use graphical calculators to verify and estimate limits.
1.3 Students prove and use special limits, such as the limits of (sin(x))/x and (1−cos(x))/x as x tends to 0.
2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.
3.0 Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value
theorem.
4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph
of the function.
4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change.
Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that
involve the rate of change of a function.
4.3 Students understand the relation between differentiability and continuity.
4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse
trigonometric, exponential, and logarithmic functions.
5.0 Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite
functions.
6.0 Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of
problems in physics, chemistry, economics, and so forth.
7.0 Students compute derivatives of higher orders.
8.0 Students know and can apply Rolle’s Theorem, the mean value theorem, and L’Hôpital’s rule.
9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points,
and intervals in which the function is increasing and decreasing.
10.0 Students know Newton’s method for approximating the zeros of a function.
11.0 Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.
12.0 Students use differentiation to solve related rate problems in a variety of pure and applied contexts.
13.0 Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate
integrals.
14.0 Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in
terms of integrals.
15.0 Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as
antiderivatives.
16.0 Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of
revolution, length of a curve, and work.
17.0 Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate.
18.0 Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as
indefinite integrals.
19.0 Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the
algebraic techniques of partial fractions and completing the square.
20.0 Students compute the integrals of trigonometric functions by using the techniques noted above.
21.0 Students understand the algorithms involved in Simpson’s rule and Newton’s method. They use calculators or computers
or both to approximate integrals numerically.
22.0 Students understand improper integrals as limits of definite integrals.
23.0 Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of
real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine
whether a series converges.
24.0 Students understand and can compute the radius (interval) of the convergence of power series.
25.0 Students differentiate and integrate the terms of a power series in order to form new series from known ones.
26.0 Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
27.0 Students know the techniques of solution of selected elementary differential equations and their applications to a wide
variety of situations, including growth-and-decay problems.
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents
to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the
cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational
number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Reason quantitatively and use units to solve problems. [Foundation for work with expressions, equations and functions]
1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Interpret the structure of expressions. [Linear, exponential, and quadratic]
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret
P(1 + r)n as the product of P and a factor not depending on P.
2. Use the structure of an expression to identify ways to rewrite it. Write expressions in equivalent forms to solve problems. [Quadratic and exponential]
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by
the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t can be rewritten as (1.151/12) 12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual
rate is 15%.
Perform arithmetic operations on polynomials. [Linear and quadratic]
1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Create equations that describe numbers or relationships. [Linear, quadratic, and exponential (integer inputs only);
for A.CED.3 linear only]
1. Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CA
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions
as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost
constraints on combinations of different foods.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example,
rearrange Ohm’s law V = IR to highlight resistance R.
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions
of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of
values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.
12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict
inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the
corresponding half-planes.
Solve equations and inequalities in one variable. [Linear inequalities; literal equations that are linear in the variables being
solved for; quadratics with real solutions]
3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
3.1 Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in
context. CA
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form
(x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Interpret functions that arise in applications in terms of the context. [Linear, exponential, and quadratic]
4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of
the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include:
intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example,
if the function h gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers
would be an appropriate domain for the function.
Understand the concept of a function and use function notation. [Learn as general principle; focus on linear and
exponential and on arithmetic and geometric sequences.]
1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element
of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the
output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation
in terms of a context.
3. 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.
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the
function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and
symmetry of the graph, and interpret these in terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of
change in functions such as
y = (1.02)t, y = (0.97), y = (1.01)12t, and y = (1.2)t/10, and classify them as representing exponential growth or decay
Analyze functions using different representations. [Linear, exponential, quadratic, absolute value, step, piecewise-defined]
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology
for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.
Build a function that models a relationship between two quantities. [For F.BF.1, 2, linear, exponential, and quadratic]
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature
of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and
translate between the two forms.
Build new functions from existing functions. [Linear, exponential, quadratic, and absolute value; for F.BF.4a, linear only]
3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive
and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the
graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
Interpret expressions for functions in terms of the situation they model.
5. Interpret the parameters in a linear or exponential function in terms of a context. [Linear and exponential of form
f(x) = bx + k]
6. Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity. CA
Summarize, represent, and interpret data on two categorical and quantitative variables.
[Linear focus; discuss general principle.]
5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of
the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the
data.
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions
or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Summarize, represent, and interpret data on a single count or measurement variable.
1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile
range, standard deviation) of two or more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme
data points (outliers).
I. Perform arithmetic operations with complex numbers.
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.
2. Use the relation( i)^2 = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply
complex numbers.
II. Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients]
7. Solve quadratic equations with real coefficients that have complex solutions.
8. (+) Extend polynomial identities to the complex numbers. For example, rewrite
(x)^2 + 4 as (x + 2i)(x – 2i).
9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
I. Interpret the structure of expressions. [Polynomial and rational]
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret
P(1 + r)n as the product of P and a factor not depending on P.
2. Use the structure of an expression to identify ways to rewrite it.
II. Write expressions in equivalent forms to solve problems.
4. 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. For example, calculate mortgage payments.
I. Perform arithmetic operations on polynomials. [Beyond quadratic]
1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
II. Understand the relationship between zeros and factors of polynomials.
2. Know 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).
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of
the function defined by the polynomial.
III. Use polynomial identities to solve problems.
4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity
(x2 + y2)^2= (x2 – y2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
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.
IV. Rewrite rational expressions. [Linear and quadratic denominators]
6. 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 examples, a computer algebra system.
7. (+) 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.
I. Create equations that describe numbers or relationships. [Equations using all available types of expressions, including simple
root functions]
1. Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions
as viable or non-viable options in a modeling context.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
I) Understand solving equations as a process of reasoning and explain the reasoning. [Simple radical and rational]
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
II) Solve equations and inequalities in one variable.
3.1 Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in
context.
III) Represent and solve equations and inequalities graphically. [Combine polynomial, rational, radical, absolute value, and
exponential functions.]
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions
of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of
values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.
I) Build a function that models a relationship between two quantities. [Include all types of functions studied.]
1. Write a function that describes a relationship between two quantities.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature
of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
II) Build new functions from existing functions. [Include simple radical, rational, and exponential functions; emphasize common
effect of each transformation across function types.]
3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and
f(x + k) for specific values of k (both positive
and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the
graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
For example, f(x) =2x^3 or f(x) = (x + 1)/(x − 1) for x ≠ 1.
Construct and compare linear, quadratic, and exponential models and solve problems.
4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2,
10, or e; evaluate the logarithm using technology. [Logarithms as solutions for exponentials]
4.1 Prove simple laws of logarithms. CA
4.2 Use the definition of logarithms to translate between logarithms in any base. CA
4.3 Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their
approximate values. CA
I. Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed counterclockwise around the unit circle.
2.1 Graph all 6 basic trigonometric functions. CA
II. Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
III. Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ ) + cos2(θ ) = 1 and use it to find sin(θ ), cos(θ ), or tan(θ ) given sin(θ ), cos(θ ),or tan(θ ) and the quadrant of the angle.
Interpret functions that arise in applications in terms of the context.
4. For a function that models a relationship between two quantities, interpret key features of graphs and
tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the
relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive,
or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it
describes. For example, if the function gives the number of person-hours it takes to assemble engines in
a factory, then the positive integers would be an appropriate domain for the function.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
10. (+) Demonstrate an understanding of functions and equations defined parametrically and graph
them. CA
11. (+) Graph polar coordinates and curves. Convert between polar and rectangular coordinate systems. CA
Build new functions from existing functions.
3. Identify the effect on the graph of replacing by , and for specific
values of (both positive and negative); find the value of given the graphs. Experiment with cases and
illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them.
4. Find inverse functions.
b. (+) Verify by composition that one function is the inverse of another.
c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
d. (+) Produce an invertible function from a non-invertible function by restricting the domain.
Perform arithmetic operations with complex numbers.
3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex
numbers.
Represent complex numbers and their operations on the complex plane.
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.
5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically
on the complex plane; use properties of this representation for computation.
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.
Represent and model with vector quantities.
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., )
2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
3. (+) Solve problems involving velocity and other quantities that can be represented by vectors.
Perform operations on vectors.
4. (+) Add and subtract vectors.
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.
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their
sum.
c. Understand vector subtraction as , where is the additive inverse of , with the same
magnitude as and pointing in the opposite direction. Represent vector subtraction graphically by
connecting the tips in the appropriate order, and perform vector subtraction component-wise.
5. (+) Multiply a vector by a scalar.
a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction;
perform scalar multiplication component-wise.
b. Compute the magnitude of a scalar multiple.
Perform operations on matrices and use matrices in applications.
6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in
a network.
7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are
doubled.
8. (+) Add, subtract, and multiply matrices of appropriate dimensions.
9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a
commutative operation, but still satisfies the associative and distributive properties.
10. (+) 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 non-zero if and only
if the matrix has a multiplicative inverse.
11. (+) 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.
12. (+) Work with 2x2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
Interpret the structure of expressions.
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret as the product of and a factor not depending on .
2. Use the structure of an expression to identify ways to rewrite it.
Rewrite rational expressions.
6. Rewrite simple rational expressions in different forms; using inspection, long division, or, for the more complicated examples, a computer algebra system.
7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under
addition, subtraction, multiplication, and division by a non-zero rational expression; add, subtract, multiply, and divide rational expressions.
Solve systems of equations.
8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.
9. (+) 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).
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