Download : Precalculus Essential Learning Objectives Foundations for Functions

Precalculus Essential Learning Objectives
Foundations for Functions:
The Student Will:
• Be able to find slope, distance and midpoint
• Graph and write equations of lines (include function notation)
• Know relationships of parallel lines and perpendicular lines
• Make a mathematical model for a linear function given a word problem or real-life
data
• Solve quadratic equations by factoring and quadratic formula
• Graph quadratic function by vertex form and general form
• Know how to complete the square to write vertex form
• Write a quadratic function from three points (matrices or quadratic regression on
calculator)
• Model real-world situations as a quadratic function (CBL Basketball lab, throwing
object in the air, etc.)
Functions:
The Student Will:
• Identify functions by definition and graph and state their domain and range
• Discuss continuity; determine intervals of increase, decrease or constant; describe
end behavior
• Find the sum, difference, product, and quotient of two functions
• Find the composition of two functions given their equations and be able to determine
functions given their composition.
• Be able to reflect graphs of functions over x-axis, y-axis and line y = x.
• Be able to test a function for symmetry and determine if it is odd, even or neither
algebraically and from the graph.
• Given the graph of y = f(x), be able to graph a f(x), f(x) + d, f(x - c), f(bx), f( x ) , and
f(|x|).
• Given a graph that has a transformation, write the equation for the graph.
• (Optional to address this in this unit) Determine if a function is periodic and, if so,
give its period and amplitude and graph it using shifts and stretches.
• Find the inverse of a function and, given two functions, determine if they are inverses
of each other algebraically and graphically.
• Solve application problems using composite functions.
Trigonometry of the Unit Circle:
The Student Will:
• Unit Circle: know definition of radians, degree measure, standard position, coterminal
angles, quadrantal angles, coordinates, and the six trig functions
• Understand the concept of radian measure as a location on the Unit Circle)
• Define sine, cosine and tangent in terms of x, y and r
•
Convert between radian measure and degree measure of special angles (multiples
π π π π
, , , )
6 4 3 2
π π π π
•
Know the sine, cosine and tangent values of special angles (multiples
•
•
•
•
•
•
•
•
Find the values of cotangent, secant, and cosecant from sine, cosine, and tangent
Find the reference angle of any given angle
Recognize equivalent fractions created by rationalized denominators
Find arc length and area of a sector (may be taught in another unit)
Solve right triangles using sin, cos, and tan
Find area of triangles
Use Law of Sines and Law of Cosines
Solve application problems involving navigation and surveying, including angles of
elevation and depression, area of parallelogram using area of triangle.
, , , )
6 4 3 2
Graphs of Trigonometric Functions:
The Student Will:
• Graph all six trig functions and identify their period, amplitude, domain, range and
zeros
• Graph all six trig functions with parameter changes (i.e.
y = a sin b(x – c) + d; f( x ) , and f(|x|) are optional)
•
•
•
•
•
•
•
•
Write/identify the equation of a trig function from its graph
State whether a trig function graph is odd or even
Investigate continuity, end behavior, vertical asymptotes, increasing/decreasing
(especially tan, cot, sec, and csc)
Model real-life data using sine and cosine functions with and without a regression
equation
Define and graph principle trig functions (1 to 1 functions) and their inverses (sine,
cosine, and tangent).
Make connections between graphs of inverse trig functions and the Unit Circle).
Find values of inverse trig functions with and without the calculator (sine, cosine, and
tangent)
Sketch the graphs of the functions Sin-1, Cos-1, Tan-1 and state the domain and
range. (Emphasize the range of inverse sine and tangent)
Trigonometric Identities:
The Student Will:
• Know and apply the following identities:
→ Pythagorean
→ Reciprocal
→ Quotient
→ Negative angle
→ Cofunctions
→ Sum and difference
•
•
•
→ Double angle
→ Half angle (optional)
Prove trig identities
Simplify and expand trig expressions
Solve trig equations
Polar, Vector and Parametric Equations:
The Student Will:
•
•
•
•
•
•
•
•
•
•
•
Using geometric and algebraic representation of vectors, add, subtract, and multiply
by a scalar and find the magnitude (length) and direction
Show, analyze, and solve navigation and force problems using vectors and trig
Use vectors, vector equations, and parametric equations to show an object’s motion.
Use polar coordinates to plot points on a graph
Determine if two vectors are parallel or perpendicular using dot product
Calculate the angle between two vectors
Work with 3-dimensional vectors to find their magnitude and midpoint
Find equations of planes parallel and perpendicular to another plane
Write complex numbers and find products in polar form
Find powers and roots of complex numbers
Convert polar coordinates to rectangular coordinates and vice versa
Conic Sections:
The Student Will:
• Know the geometric and algebraic descriptions of circle, parabola, hyperbola, and
ellipse
• Represent conic sections in standard form (graphing form) and in parametric form
and graph
• Determine the equation of a conic given the graph
• Determine the equation and graph of a conic from given information
• Apply properties of conic sections to solve real-life problems (both as parametric and
rectangular)
¾ orbits of planets
¾ LORAN
• Convert from general to standard form
• Describe how properties of conics are used in the physical world, such as
¾ reflective properties (flashlight, whispering room, lithotripsy. etc.)
¾ elliptipool
¾ cooling towers for nuclear reactors
• (optional) graphs of rotated conics
Exponential and Logarithmic Functions:
The Student Will:
•
•
•
Know the definition of an exponential function, identify restrictions and graph
Define the logarithmic function as the inverse of an exponential function and graph
Know and apply laws of integral and rational exponents
•
•
•
•
Apply transformations to exponential and logarithmic functions
Know restrictions on the log function
Identify intervals where graph of an exponential function is increasing or decreasing,
as well as its domain, range, and zeros
Use exponential growth and decay formulas to solve application problems:
A = A0 (1 + r )
t
t
A = A0 b k
r⎞
⎛
A = A0 ⎜ 1 + ⎟
n⎠
⎝
nt
A = A0 e rt
•
•
•
Use rational exponents to simplify expressions and solve equations
Write an exponential function from a set of data
Know the definition of e and the function f(x) = ex and use it in application problems
Sequences and Series:
The Student Will:
•
•
•
Identify a sequence as arithmetic (linear), geometric (exponential when r > 0), or
other
Define a sequence both explicitly and recursively
Explore representations of a sequence on a graph and table
Use sequences defined recursively to solve problems
Find the sum of the first n terms and the value of a specific term of an arithmetic and
geometric series
Find or estimate the limit of an infinite sequence or determine that the limit does not
exist
Find the sum of an infinite geometric series when possible
Use sigma notation to represent series
•
Expand a binomial of the form ( a + b ) and find the nth term of a binomial expansion
•
Use sequences and series to solve real-life problems.
•
•
•
•
•
n
Polynomial Functions:
The Student Will:
•
•
•
•
•
•
•
Identify and classify polynomials by degree
Find zeros (roots) and factors of polynomial functions (equations)
Use synthetic substitution/division
Apply Theorems: Remainder, Factor, Rational Root, Complex Conjugate, and
Fundamental Theorem of Algebra
Graph a polynomial function given a factorable equation
Write the equation for a polynomial function given its graph or a set of data
Find maximum and minimum values with or without a graphing calculator and apply
to situations (area, volume, cost, distance, etc.)
•
•
•
•
•
•
•
•
•
Solve and graph (on a number line) polynomial inequalities with answer in interval
notation
Sketch the graph of a polynomial function using properties of end behavior, zeros,
and multiplicity of zeros
Graph rational functions, including those containing more than one vertical
asymptote, an oblique asymptote, removable discontinuities, and a crossover of the
horizontal or oblique asymptote by locating the asymptotes, “holes”, and intercepts
1
1
as transformations of one of the parent functions f ( x ) = or f ( x ) = 2 .
x
x
Manipulate the function rule algebraically to obtain graphing form. (Example: Use
x+3
1
as 1 +
long division to rewrite the
x+2
x+2
Describe the continuity and limits of the function.
State the domain and range of the function and write the equations of its asymptotes.
Write a rational function given the graph, or description of characteristics such as a
verbal description of the transformations of the parent function, the asymptote
equations, or function notation showing the transformations.
Make connections between the graphical, tabular, verbal, and symbolic
representations of the function.
Describe the symmetry of the function by identifying it as odd or even.