Math  201920
AII.7acdefghi  Rational Characteristics
AII.7 The student will investigate and analyze
linear, quadratic, absolute value, square root, cube root,rational,polynomial, exponential, and logarithmicfunction families algebraically and graphically. Key concepts includea) domain, range, and continuity;
c) extrema;
d) zeros;
e) intercepts;
f) values of a function for elements in its domain;
g) connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs;
h) end behavior;
i) vertical and horizontal asymptotes;
BIG IDEAS
 I can compare the cost of
pizza with number of toppings, find the correlation between number of hours
spent in the sun and development of skin cancer, match the amount of money put
into a vending machine with the kind of selection that comes out, and equate
the growth of plants with amount of food and water they receive.
 I will be able to model various kinds of mathematical relationships and express those
relationships in different ways.
UNDERSTANDING THE STANDARD
2016 VDOE Curriculum Framework  AII.7 Understanding
· Functions may be used to model practical situations.
· Functions describe the relationship between two variables where each input is paired to a unique output.
· Function families consist of a parent function and all transformations of the parent function.
· The domain of a function is the set of all possible values of the independent variable.
· The range of a function is the set of all possible values of the dependent variable.
·
For each x in the domain of f, x is a member of the
input of the function f, f(x) is a member of the output of f, and the ordered pair
(x, f(x)) is a member of f.
· A function is said to be continuous on an interval if its graph has no jumps or holes in that interval.
· The domain of a function may be restricted algebraically, graphically, or by the practical situation modeled by a function.
· Discontinuous domains and ranges include those with removable (holes) and nonremovable (asymptotes) discontinuities.
· A function can be described on an interval as increasing, decreasing, or constant over a specified interval or over the entire domain of the function.
· A function, f(x), is increasing over an interval if the values of f(x) consistently increase over the interval as the x values increase.
· A function, f(x), is decreasing over an interval if the values of f(x) consistently decrease over the interval as the x values increase.
· A function, f(x), is constant over an interval if the values of f(x) remain constant over the interval as the x values increase.
· Solutions and intervals may be expressed in different formats, including set notation, using equations and inequalities, or interval notation. Examples may include:
Equation/Inequality 
Set Notation 
Interval Notation 
x = 3 
{3} 

x = 3 or x = 5 
{3, 5} 

0 £ x £ 3 
{x0 £ x £ 3} 
[0, 3) 
y ≥ 3 
{y: y ≥ 3} 
[3, ¥) 
Empty (null) set ∅ 
{ } 
· A function, f, has an absolute maximum located at x = a if f(a) is the largest value of f over its domain.
· A function, f, has an absolute minimum located at x = a if f(a) is the smallest value of f over its domain.
· Relative maximum points occur where the function changes from increasing to decreasing.
· A function, f, has a relative maximum located at x = a over some interval of the domain if f(a) is the largest value of f on the interval.
· Relative minimum points occur where the function changes from decreasing to increasing.
· A function, f, has a relative minimum located at x = a over some interval of the domain if f(a) is the smallest value of f on the interval.
· A value x in the domain of f is an xintercept or a zero of a function f if and only if f(x) = 0.
· Given a polynomial function f(x), the following statements are equivalent for any real number, k, such that f(k) = 0:
k is a zero of the polynomial function f(x) located at (k, 0);
k is a solution or root of the polynomial equation f(x) = 0;
the point (k, 0) is an xintercept for the graph of f(x) = 0; and
(x – k) is a factor of f(x).
· Connections between multiple representations (graphs, tables, and equations) of a function can be made.
· End behavior describes the values of a function as x approaches positive or negative infinity.
· If (a, b) is an element of a function, then (b, a) is an element of the inverse of the function.
· The reflection of a function over the line represents the inverse of the reflected function.
· A function is invertible if its inverse is also a function. For an inverse of a function to be a function, the domain of the function may need to be restricted.
· Exponential and logarithmic functions are inverses of each other.
· Functions can be combined using composition of functions.
· Two functions, f(x) and g(x), are inverses of each other iff(g(x)) = g(f(x)) = x.
ESSENTIALS
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
· AII.7ade_{1} Identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities.
· AII.7c_{1} Identify the location and value of absolute maxima and absolute minima of a function over the domain of the function graphically or by using a graphing utility.
· AII.7c_{2} Identify the location and value of relative maxima or relative minima of a function over some interval of the domain graphically or by using a graphing utility.
· AII.7f_{1} For any x value in the domain of f, determine f(x).
· AII.7g_{1} Represent relations and functions using verbal descriptions, tables, equations, and graphs. Given one representation, represent the relation in another form.
· AII.7_{1} Investigate and analyze characteristics and multiple representations of functions with a graphing utility.
·
AII.7a_{1} Describe
a function as continuous or discontinuous.
··
AII.7i_{1} Determine
the equations of vertical and horizontal asymptotes of functions (rational, exponential,
and logarithmic).
·
AII.7h_{1} Describe
the end behavior of a function.
· AII.7_{1} Investigate and analyze characteristics and multiple representations of functions with a graphing utility.
KEY VOCABULARY
function,
algebraically, graphically, domain, range, limited functions, discontinuous
domains, discontinuous ranges, zeros, xintercept, yintercept, intervals,
increasing, decreasing, asymptotes, end behavior, inverse of a function,
composition of multiple functions, restricted domain, restricted range,
vertical asymptote, horizontal asymptote, reflections, exponential function,
logarithmic function, verify, natural numbers