## Carnegie Learning Algebra I Student Text Volume 1 3rd Edition Chapter 1 Quantities And Relationships

**Page 38 Problem 1 Answer**

Given increasing function and decreasing function To fill the table

Method used is increasing and decreasing function

We are given that

A function is described as increasing when the dependent variable increases as the independent variable increases.

If a function increases across the entire domain, then the function is called an increasing function.

A function is described as decreasing when the dependent variable decreases as the independent variable increases.

If a function decreases across the entire domain, then the function is called a decreasing function.

A graph is increasing if it rises from left to right, decreasing if it is falling from left to right, and constant if it is horizontal.

All the increasing and decreasing functions, constants are grouped in the table

Increasing function | Decreasing function | constant function | combination of increasing,decreasing or constant |

G | H | U | A,B,C |

K | L | D,F,I | |

O | M,Q,S | ||

P | T,V |

**Page 39 Problem 2 Answer**

Given: Some of the functions are given

To determine the shape of its graph.

Method: Graphing calculator

The graph of the function f(x)=x is

So the graph off(x)=(1/2)x−5

The shape of its graph is

**Page 39 Problem 3 Answer**

Given functions To sort the graphs into two groups based on the equations representing the functions and record the function letter in the table.

Method used is the concept of functions

The seven graphs from exercise 2 can be grouped as linear or exponential

A linear function is of the form y=mx+c

so the constant function are a linear function when m=0

An exponential function is of the formy=bx−k

The exponential function are then f(x)=(1/2)x−5

Where x is an integer (graph H)

f(x)=2x (graph K), and f(x)=(1/2)x (graph P)

The linear function aref(x)=x(graph G),f(x)=−2/3x+5(graph L)

f(x)=−x+3 where x is an integer (graph O), andf(x)=2

Wherex is an integer(graph U)

The table can be filled as

Group 1 | Group 2 |

P | G |

H | L |

K | O |

U |

All the data s are arranged in the table

**Page 39 Problem 4 Answer**

Given seven graphs and functions that are increasing functions, decreasing functions, or constant functions.

To find same about all the functions in each group Method used is the concept of function

All the in Group 1 are exponential functions

Also, all the functions in group 2 are linear functions

Functions in group 1 are exponential functions and group 2 are linear functions

**Page 40 Problem 5 Answer**

Given functions To identify which group represents linear and constant functions and which group represents exponential functions.

Method used is the concept of functions

All the functions in group 1 are exponential functions

Since they are of the form f(x)=ab x

All the functions in group 2 are linear functions

Since they are of the form f(x)=mx+b

All the functions in group 1 are exponential functions and in group 2 are linear functions

**Page 40 Problem 6 Answer**

Given linear function f(x)=mx+b

To describe the m and b values that produce a constant function. The method

used is the concept of functions

We are given the function f(x)=mx+b

f(x)=mx+b is the linear equation

The graph of the linear equation is a line.

The value of m tell us the steepness of the line that is the slope of the line b is the initial value

Whenx=0,f(x)=a(0)+b

That is, f(0)=b

So b is the initial value or y-intercept

The value of m steepness of the line and b is the initial value

**Page 40 Problem 7 Answer**

Given different functions To sort the graphs and record the function letter in the appropriate column of the table shown

Method used is the concept of absolute minimum and absolute maximum value

GraphA has no absolute minimum or maximum since it doesn’t have a highest or lowest point

GraphB has an absolute maximum at(2,2) since that is the highest point in the graph.

GraphC has no absolute maximum or minimum since it doesn’t have a highest or lowest point

GraphD has an absolute minimum at(0,0) since that is the lowest point in the graph.

GraphF has an absolute maximum at(0,4) since that is the highest point in the graph

GraphI has an absolute maximum at(0,0) since that is the highest point in the graph

Graph M has an absolute minimum at(0,0) since that is the lowest point in the graph

Graph Q has an absolute maximum at(2,4) since that is the highest point in the graph

Graph S does not have an absolute maximum or minimum since it doesn’t have a highest of lowest point

Graph T has an absolute minimum at(−4,−4) since that is the lowest point in the graph

Graph V has an absolute minimum at(3,−2) since that is the lowest point in the graph.

The table is then

absolute minimum | absolute maximum | no absolute minimum and maximum |

B,F,I | D,M,T | A,C,S |

Q,V |

All the data s are arranged in the table

**Page 41 Problem 8 Answer**

Given: Consider the graphs of functions that have an absolute minimum or an absolute maximum.

To sort the graphs into two groups based on the equations representing the functions and record the function letter in the table.

The method used: Functional method.

Group1 equations in the form of f(x)=a∣x+b∣+c,

Group 2 be the equations in the form of f(x)=ax^{2}+bx+c,

f(x)=a∣x+b∣+c,

f(x)=∣x−3∣−2 (graphV),

f(x)=∣x∣ (graph D),

f(x)=−∣x∣ (graphI),

f(x)=−2∣x+2∣+4 (graphQ).

f(x)=ax^{2}+bx+c,

f(x)=x^{2}+8x+12 (graph T),

f(x)=x^{2} (graph M),

f(x)=−3x^{2}+4 (graph F),

f(x)=−1/2x^{2}+2x (graphB).

The function letter is shown in the table.

**Page 41 Problem 9 Answer**

Given: Consider the graphs of functions that have an absolute minimum or an absolute maximum.

To find what is the same about all the functions in each group?

Method used: Functional method.

To find the same about all the functions in each group;

The functions in Group1 is in the form off(x)=a∣x+b∣+c ,

The functions in Group 2 is of the form off(x)=ax^{2}+bx+c.

The group1 and 2 in the form of

f(x)=a∣x+b∣+c,

f(x)=ax^{2}+bx+c.

Group1 | Group2 |

D,I,Q,V | B,F,M,T |

**Page 42 Problem 10 Answer**

Given: which group represents quadratic functions and which group represents linear absolute value functions.

To identify which group represents quadratic functions and which group represents linear absolute value functions.

The method used: Functional method.

To identify which group represents;

Group 1 is in the form off(x)=a∣x+b∣+c ,

Group 1 are linear absolute value functions.

Group 2 is in the form of f(x)=ax^{2}+bx+c

Group 2 is quadratic functions.

Group 2 represents quadratic functions and group1 represents linear absolute value functions.

**Page 43 Problem 11 Answer**

Given: Enter the remaining functions into your graphing calculator.To determine the shapes of their graphs.Method used: Graphical method.

To determine the shapes of their graphs;

Graph

f(x)={ −2x+10,−∞≤x<3

{ 4, 3≤x<7

{ −2x+8, 7≤x≤+∞

Y1=(−2X+10)/(X<3)

Y2=(4)/((3≤X)(X<7))

Y3=(−2X+18)/(7≤X)

The shape of the graph is

Graph

f(x)= { −2, x−2,

{ 1 −∞<x<0

{ 2 0≤x<+∞

Y1=(−2)/(X<0)

Y2=(1/2X−2)/(0≤X)

The shape of the given function is

Graph

f(x)= {1/2 x+4, −∞≤x<2

{−3x+11, 2≤x<3

{ 1/ 2x+1/2, 3≤x≤+∞

Y1=(1/2X+4)/(X<2)

Y2=(−3x+11)/((2≤X)(X<3))

Y3=(1/2X+1/2)/(3≤X)

The graph of the given function

The shape of the given functions are

**Page 45 Problem 12 Answer**

Given: Glue your sorted graphs and functions to the appropriate function family Graphic Organizer on the pages that follow.

To write a description of the graphical behavior for each function family.

Method used: Functional method.

Linear Functions: GraphsG,L,O,U,

Increasing/Decreasing: increasing, decreasing, or constant,

Maximum/Minimum: none,

Curve/Line: Line.

Exponential Functions:

Graphs H,K,P,

Increasing/Decreasing: Only one,

Maximum /Minimum: None,

Curve/Line: Curve.

Quadratic Functions:

GraphsB,F,M,T,

Increasing/Decreasing: Both,

Maximum/Minimum: One,

Curve/Line: Curve.

Linear Absolute Value Functions:

Graphs D,I,Q,U,

Increasing/Decreasing: Both,

Maximum/Minimum: One,

Curve/Line: Two lines.

Linear Piecewise Functions:

Graphs A,C,S,

Increasing/Decreasing: Can be both,

Maximum/Minimum: None or one,

Curve/Line: Two or more lines.

The graphical behavior for each function family was explained.