Mathematical Domains and Ranges of Nonlinear Functions - CoffeeCup Software

Chapter 9:

Nonlinear Functions

Mathematical Domains and Ranges of Nonlinear Functions

1. For the following problems:

?

Sketch a complete graph for the given function. Show the coordinates of any

intercepts.

?

Describe the domain and range for each mathematical situation.

Function

Graph

Process Area

(optional)

Domain and

Range

Domain:

A.

f (x ) =

1 2

x

2

Range:

Domain:

B.

y = x2 + 3

Range:

Domain:

C.

y = ¨C3x2

Range:

Domain:

D.

y = x(5 ¨C x)

Range:

The Charles A. Dana Center

Ch 9¨C1

at The University of Texas at Austin

Inverse Variations, Exponential Functions, and Other Functions

Algebra Assessments Through

the Common Core (grades 6-12), 2011

393

Chapter 9:

Nonlinear Functions

Domain:

E.

x

h(x) = 3

Range:

Domain:

F.

? 1?

m (x ) = ? ?

? 3?

x

Range:

Domain:

G.

g (x ) =

4

x

Range:

2. Write a summary comparing the domains, ranges, and graphs of the functions.

3. Describe a practical situation that the functions in problems D, E, and F might

represent. What restrictions will the situation place on the mathematical domain and

range of the function? How will the situation affect the graph of the mathematical

function?

The Charles A. Dana Center

at The University of Texas at Austin

394

Algebra Assessments Through

the Common Core (grades 6-12), 2011

Inverse Variations, Exponential Functions, and Other Functions

Ch 9¨C2

Chapter 9:

Nonlinear Functions

The Charles A. Dana Center

Ch 9¨C3

at The University of Texas at Austin

Inverse Variations, Exponential Functions, and Other Functions

Algebra Assessments Through

the Common Core (grades 6-12), 2011

395

Mathematical

Mathematical Domains

Domains

Teacher

Notes

Teacher Notes

Notes

Chapter 9:

Nonlinear Functions

Scaffolding Questions

CCSS

Content Task

Materials:

(F-IF) Interpret functions that

One in

graphing

calculator

per

arise

applications

in terms

of

the context

student

(optional)

4. For a function that models

TEKS

Focus:

aAlgebra

relationship

between

two

quantities, interpret key features

(A.9) Quadratic and other

of graphs and tables in terms

functions.

The

ofnonlinear

the quantities,

and sketch

student

understands

that

graphs showing key features

the graphs

quadraticof

given

a verbalofdescription

functions

are affected

the

relationship.

Key features

include:

intercepts; intervals

by the parameters

of the

where

the

function

is

increasing,

function and can interpret

decreasing,

positive,

or

and describe

the effects

of

negative;

relative

maximums

changes in the parameters of

and minimums; symmetries;

quadratic functions.

end behavior; and periodicity.*

?

What kind of function is this (linear, exponential, etc?)

?

What is the dependent variable?

?

What is the independent variable?

?

What are the constants in the function? What do they

mean?

?

What restrictions does the function place on the

independent variable?

?

What is a reasonable domain for the function?

?

What is a reasonable range for the function?

?

If the graph of a function approaches but does not

cross an x- or y-axis, what might that indicate about

either the domain or the range of that function?

The student is expected to:

(F-IF) Interpret functions that

(A) in

determine

the domain

arise

applications

in terms

range for quadratic

of theand

context

functions

in given

5. Relate

the domain

of a

situations;

function

to its graph and, where

applicable, to the quantitative

relationship

describes.

For

AdditionalitAlgebra

TEKS:

example, if the function h(n)

(A.2)the

Foundations

for

gives

number of personfunctions.

student nuses

hours

it takesThe

to assemble

engines

in a factory,

the

the properties

andthen

attributes

positive

integers

would

be

an

of functions.

appropriate domain for the

function.*

The student is expected to:

(F-IF)

functions using

(B) Analyze

identify mathematical

different

representations

domains and ranges

andfunctions

determine

7. Graph

expressed

reasonable

domain

symbolically

and show

key

features

of

the

graph,

by

and range values for

hand in

simple

cases andboth

given

situations,

using continuous

technology for

more

and

discrete;

complicated cases.*

(C)

interpret

situations

in

a. Graph

linear

and quadratic

terms and

of given

functions

show graphs or

intercepts,

maxima, and

create situations

that fit

minima.

given graphs; and

c. Graph polynomial

functions, identifying zeros

when suitable factorizations

The Charles A. Dana Center

at The University of Texas at Austin

388

396

Sample Solutions

Note: We recommend that you assign this assessment

to be completed without a graphing calculator. We

include the screen shots in the sample answers to

clarify the look of the graphs. Student graphs will be

hand drawn. Students can use the process column on

the chart to help them reason about the fuctions (with

scratch work, a table, initial ideas, etc.)

1. For the following problems:

?

Sketch a complete graph for the given function.

Show the coordinates of any intercepts.

?

Describe the domain and range for each

mathematical situation.

A.

f (x ) =

1 2

x

2

Algebra Assessments Through

the Common Core (grades 6-12), 2011

Inverse Variations, Exponential Functions, and Other Functions

Ch 9¨C4

Mathematical

Mathematical Domains

Domains

Teacher

Notes

Teacher Notes

Chapter 9:

Nonlinear Functions

The y-intercept and the x-intercept for this function

are at the origin (0, 0).

The domain for this function is the set of all numbers

greater than or equal to 0, since any value for x can

be squared and multiplied by 1 .

2

The range for this function is the set of all numbers

greater than or equal to 0, which is the result of

squaring any number and multiplying by 1 .

2

x ¡Ý0

2

1

2

x2 ¡Ý 0

B. y = x2 + 3

The y-intercept for this function is (0, 3).

There is no x-intercept because the graph does not

intersect the x-axis.

The domain for this function is the set of all real

numbers, since any value for x can be squared and

increased by 3.

The range for this function is the set of all numbers

greater than or equal to 3.

x ¡Ý0

2

C.

Texas

Assessment

of

are available,

and showing

end

behavior.

Knowledge and Skills:

d. (+) Graph rational

Objective

The student

functions,5:identifying

zeros

willand

demonstrate

asymptotesan

when

understanding

of the are

suitable factorizations

available,and

andother

showing end

quadratic

behavior.

nonlinear functions.

e. Graph exponential and

logarithmic functions,

showing intercepts and end

behavior, and trigonometric

functions, showing period,

midline, and amplitude.

(F-LE) Construct and compare

linear, quadratic, and

exponential models and solve

problems

1. Distinguish between

situations that can be modeled

with linear functions and with

exponential functions.

a. Prove that linear functions

grow by equal differences

over equal intervals, and that

exponential functions grow

by equal factors over equal

intervals.

b. Recognize situations in

which one quantity changes

at a constant rate per unit

interval relative to another.

Standards for

Mathematical Practice

x2 + 3 ¡Ý 3 ¡Ý 0

2. Reason abstractly and

quantitatively.

y = ¨C3x2

7. Look for and make use of

structure.

The y-intercept and the x-intercept for this function

are at the origin (0, 0).

The Charles A. Dana Center

Ch 9¨C5

at The University of Texas at Austin

Inverse Variations, Exponential Functions, and Other Functions

Algebra Assessments Through

the Common Core (grades 6-12), 2011

389

397

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