# A.7 and *A.7 - Relations

A.7    The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including

a)  determining whether a relation is a function;

b)  domain and range;

c)  zeros of a function;

d)  x- and y-intercepts;

e)  finding the values of a function for elements in its domain; and

f)  making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.

A.7  The student will investigate and analyze linear and quadratic function families and their characteristics both algebraically and graphically, including

a)  determining whether a relation is a function;

b)  domain and range;

c)  zeros;

d)  intercepts;

e)  values of a function for elements in its domain;

f)  connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs.

Bloom's Level:  Analyze

### 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

• A set of data may be characterized by patterns, and those patterns can be represented in multiple ways.
• Graphs can be used as visual representations to investigate relationships between quantitative data.
• Inductive reasoning may be used to make conjectures about characteristics of function families.
• Each element in the domain of a relation is the abscissa of a point of the graph of the relation.
• Each element in the range of a relation is the ordinate of a point of the graph of the relation.
• A relation is a function if and only if each element in the domain is paired with a unique element of the range.
• The values of f(x) are the ordinates of the points of the graph of f.
• The object f(x) is the unique object in the range of the function f that is associated with the object x in the domain of f.
• 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.
• An object x in the domain of f is an x-intercept or a zero of a function f if and only if f(x) = 0.
• Set builder notation may be used to represent domain and range of a relation.

·  A relation is a function if and only if each element in the domain is paired with a unique element of the range.

·  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 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);

­  (xk)is a factor of f(x);

­  k is a solution or root of the polynomial equation f(x) =0; and

­  the point (k, 0) is an x-intercept for the graph of y = f(x).

·  The x-intercept is the point at which the graph of a relation or function intersects with the x-axis.  It can be expressed as a value or a coordinate.

·  The y-intercept is the point at which the graph of a relation or function intersects with the y-axis.  It can be expressed as a value or a coordinate.

·  The domain of a function may be restricted by the practical situation modeled by a function.

·  Solutions and intervals may be expressed in different formats, including set notation or using equations and inequalities.

-  Examples may include:

 Equation/ Inequality Set Notation x = 3 {3} x = 3 or x = 5 {3, 5} y≥ 3 {y: y ≥ 3} Empty (null) set ∅ { }

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

A.7a1  Determine whether a relation, represented by a set of ordered pairs, a table, or a graph is a function.

·  A.7aDetermine whether a relation, represented by a set of ordered pairs, a table, a mapping, or a graph is a function.

A.7f1  Represent relations and functions using concrete, verbal, numeric, graphic and algebraic forms. Given one representation, students will be able to represent the relation in another form.

·  A.7fRepresent relations and functions using verbal descriptions, tables, equations, and graph. Given one representation, represent the relation in another form.

A.7bcd1  Identify the domain, range, zeros and intercepts of a function presented algebraically or graphically.

·  A.7bcdIdentify the domain, range, zeros, and intercepts of a function presented algebraically or graphically.

A.7c1  For each x in the domain of f, find f(x).

·  A.7eFor any value, x, in the domain of f, determine f(x).

A.7f2  Detect patterns in data and represent arithmetic and geometric patterns algebraically.

·  A.7abcdefInvestigate and analyze characteristics and multiple representations of functions with a graphing utility.

### KEY VOCABULARY

function, relation, ordered pairs, table, graph, domain, range, zeros, intercepts, patterns, arithmetic, geometric, quantitative data, inductive reasoning, conjectures, abscissa, ordinate, element,  f(x), input, output, member, set builder notation

Updated: Oct 27, 2017