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).