# AII.7abcdf and *AII.7abcdefgh - Polynomial Characteristics

AII.7    The student will investigate and analyze functions algebraically and graphically. Key concepts include

a)  domain and range, including limited and discontinuous domains and ranges;

b)  zeros;

c)  x- and y-intercepts;

d)  intervals in which a function is increasing or decreasing;

f)  end behavior;

Graphing calculators will be used as a tool to assist in investigation of functions.

AII.7  The student will investigate and analyze linear, quadratic, absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic function families algebraically and graphically. Key concepts include

a)  domain, range, and continuity;

b)  intervals in which a function is increasing or decreasing;

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;

### BIG IDEAS

• I will be able to model various kinds of mathematical relationships and express those relationships in different ways and I will be able to write symbolic representations of the way numbers behave and will know that in order to maintain equality, an operation performed on one side must also be performed on the other side.

### UNDERSTANDING THE STANDARD

• Functions may be used to model real-world situations.
• The domain and range of a function may be restricted algebraically or by the real-world situation modeled by the function.
• A function can be described on an interval as increasing, decreasing, or constant.
• Asymptotes may describe both local and global behavior of functions.
• End behavior describes a function as x approaches positive and negative infinity.
• A zero of a function is a value of x that makes f(x) equal zero.
• If (a, b) is an element of a function, then (b, a) is an element of the inverse of the function.
• Exponential (y = ax) and logarithmic (y = logax) functions are inverses of each other.
• Functions can be combined using composition of functions.

·  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 {x|0 £ 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 x-intercept 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 x-intercept for the graph of f(x) = 0; and

­  (xk) 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 if f(g(x)) = g(f(x)) = x.

### ESSENTIALS

Characteristics

All.7abc1  Identify the domain, range, zeros, and intercepts of a function presented algebraically and graphically.

·  AII.7ade1  Identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities.

·  AII.7a1  Describe a function as continuous or discontinuous.

·  AII.7c1  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.7c2  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.7f1  For any x value in the domain of f, determine f(x).

·  AII.7g1  Represent relations and functions using verbal descriptions, tables, equations, and graphs. Given one representation, represent the relation in another form.

·  AII.71  Investigate and analyze characteristics and multiple representations of functions with a graphing utility.

All.7d1  Given the graph of a function, identify intervals on which the function is increasing and decreasing.

·  AII.7b1  Given the graph of a function, identify intervals on which the function (linear, quadratic, absolute value, square root, cube root, polynomial, exponential, and logarithmic) is increasing or decreasing.

All.7f1  Describe the end behavior of a function.

·  AII.7h1  Describe the end behavior of a function.

### KEY VOCABULARY

function, algebraically, graphically, domain, range, limited functions, discontinuous domains, discontinuous ranges, zeros, x-intercept, y-intercept, 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

Updated: Oct 27, 2017