Chapter 1: Equations and Inequalities Section 1.6: Equations ......Chapter 1: Equations and Inequalities Section 1.6: Equations and Inequalities Involving the Absolute Value Exploration

Section 1.6 Equations and Inequalities Involving the Absolute Value 51

Copyright © 2016 Pearson Education, Inc.

Chapter 1: Equations and Inequalities Section 1.6: Equations and Inequalities Involving the Absolute Value

Exploration 1*: Solve Absolute Value Equations Recall that the absolute value of a number is the distance from that number to 0 on the real number line. (a)* First, verbally describe what solving 5x = actually means based on the definition of

absolute value. Next, solve the equation 5x = . Show your solution graphically by graphing

( )f x x= and ( ) 5f x = on the same axis.

(b) Solve 7x = − .

(c) Can there ever be one solution to an absolute value equation? Explain.

Equations Involving Absolute Value: if a is a positive real number and if u is any algebraic expression, then u a= is equivalent to _______ or _______.

52 Chapter 1 Equations and Inequalities


Steps for Solving Absolute Value Equations with One Absolute Value Let u represent the algebraic expression inside the absolute value bars and a any positive real number. Step 1: Isolate the expression containing the absolute value so that you have u a= . Step 2: Rewrite the absolute value equation as two equations u a= − or u a= , where u

is the algebraic expression in the absolute value symbol. Step 3: Solve each equation. Step 4: Verify your solution.

Example 1*: Solve Absolute Value Equations Solve the equation 15 23c − = . Example 2*: Solve Absolute Value Equations Solve the equation 5 8 1x− + =

Exploration 2*: Solve Absolute Value Inequalities (a) Verbally describe what 3x < actually means based on the definition of absolute value.

(b) Solve the equation 3x < . Express your solution using set– builder and interval notation.

Section 1.6 Equations and Inequalities Involving the Absolute Value 53


Inequalities of the Form < or ≤ Involving Absolute Values: If a is positive real number and if u represents an algebraic expression then:

u a< is equivalent to ________________________________

u a≤ is equivalent to ________________________________

Note: In other words, u a< is equivalent to _________ and ________. Example 3*: Solve Absolute Value Inequalities Solve the inequality 15 23x − < . Express your solution using set– builder and interval notation. Graph the solution set.

Exploration 3: Solve Absolute Value Inequalities (a) Verbally describe what 4x > actually means based on the definition of absolute value.

(b) Solve the equation 4x > . Express your solution using set– builder and interval notation.

Inequalities of the Form > or ≥ Involving Absolute Values: If a is a positive real number and if u represents an algebraic expression then

u a> is equivalent to ________________________________

u a≥ is equivalent to ________________________________

54 Chapter 1 Equations and Inequalities


Example 5: Solve Absolute Value Inequalities Solve the inequality 2 1 9x− + ≥ . Express your solution using set– builder and interval notation. Graph the solution set. . Example 6: Solve Absolute Value Inequalities Solve the inequality 3 5 2w− + − ≥ . Express your solution using set– builder and interval notation. Graph the solution set.

Section 3.1 Functions 77


Chapter 3: Functions and Their Graphs Section 3.1: Functions

A relation is a _________________________________. If x and y are two elements in these sets and if a relation exists between x and y, then we say that x __________________ to y or that y _____________ x, and we write________.

What are the four ways to express relations between two sets? Definition: Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X __________________________ of Y.

In other words, for a function, no input has more than one output.

The domain of a function is:

The Range of a function is:

Can an element in the range be repeated in a function? For example, can a function contain the points (2,4) and (3,4) ? Why or why not? Example 1*: Determine Whether a Relation Represents a Function Determine which of the following relations represent a function. If the relation is a function, then state its domain and range. (a) (b) ( ) ( ) ( ) ( ){ }2,3 , 4,1 , 3, 2 , 2, 1− − (c) ( ) ( ) ( ) ( ){ }2,3 , 4,1 , 3, 2 , 2, 1− − − (d) ( ) ( ) ( ) ( ){ }2,3 , 4,3 , 3,3 , 2, 1−

Level of Education

Unemployment Rate

No High School Diploma

High School Diploma

Some College

College Graduate

7.7%

5.9%

5.4%

3.4%

78 Chapter 3 Functions and Their Graphs


Example 2*: Determine Whether a Relation Represents a Function Determine whether the following are functions, with y as a function of x.

(a) 1 32

y x= − − (b) 22 1x y= +

In general, the idea behind a function is its predictability. If the input is known, we can use the function to determine the output. With “nonfunctions,” we don’t have this predictability. Sometimes it is helpful to think of a function f as a machine that receives as input a number from the domain, manipulates it, and outputs the value with certain restrictions:

1. It only accepts numbers from _______________.

2. For each input, there is _____________________. For a function ( ) y f x= , the variable x is called the ______________________because it can be assigned to any of the numbers in the domain. The variable y is called the _________________ because its value depends x.

Example 3: Illustrate Language Used with Functions State the name, independent variable, and dependent variable of the functions below.

( )(a) 6 y g x x= = + ( )(b) 2 3y f x x= = + ( ) 2(c) p r n n= =



Example 4*: Find the Value of a Function For the function f defined by ( ) 23 2f x x x= − + , evaluate:

( )(a) 3f ( ) ( )3(b) f x f+ ( )(c) f x−

( )(d) f x− ( )(e) 3f x + In general, when a function f is defined by an equation x and y, we say that f is given ___________. If it is possible to solve the equation for y in terms of x, then we write

( )y f x= and say that the function is given _______________. Example 5*: Implicit Form of a Function Circle the functions below that are written in their implicit form.

2

3 5

( ) 6

4

x y

y f x x

xy

+ =

= = −

=



The difference quotient of a function f at x is given by:

_________________________________ The difference quotient is used in calculus to find derivatives. In this class, we will only work on simplification. Let’s practice a few more of these.

Example 5: Find the Difference Quotient of a Function

Find the simplified difference quotient of f; that is, find( ) ( )

(f) , 0f x h f x

hh

+ −≠ for each of

the functions below. ( )(a) 2 1f x x= − + ( ) 2(b) 2 5 1f x x x−= −

The domain is the largest set of real numbers for which ( )f x is a real number. In other words, the domain of a function is the largest set of real numbers that produce real outputs.

Steps for Finding the Domain of a Function Defined by an Equation 1. Start with the domain as the set of _____________.

2. If the equation has a denominator, exclude any numbers that ___________________.

Why do we have to exclude these numbers? 3. If the equation has a radical of even index, exclude any numbers that cause the expression

inside the radical to be _________________. Why do we have to exclude these numbers?

Example 7*: Find the Domain of a Function Defined by an Equation Find the domain of each of the following functions using interval or set-builder notation:

( ) 2

4(a) 2 3

xf xx x

+=− −

2(b) ( ) 9g x x= − ( )(c) 3 2h x x= −



Tip: When finding the domain of application problems, we must take into account the context of the problem. For example, when looking at the formula for the area of a circle,

2A rπ= , it does not make sense to have a negative radius or a radius of 0. The domain would be{ }| 0r r > .

Example 8*: Find the Domain of an Application A rectangular garden has a perimeter of 100 feet. Express the area, A, of the garden as a function of the width, w. Find the domain. Definition: Function Operations The difference, f g− , is defined by ______________________ Domain of f g− : __________________________ The sum, f g+ , is defined by _________________ Domain of f g+ : __________________________ The product, f g⋅ , is defined by ____________________

Domain of f g⋅ :___________________________

The quotient, fg

, is defined by _____________________

Domain of fg

:_____________________________

Example 9: Form the Sum, Difference, Product, and Quotient of Two Functions For the functions ( ) ( )2 4 and 3 6,f x x g x x= − = − + find the following. For parts a – d also find the domain.

( )( )(a) f g x+ ( )( )(b) f g x− ( )( )(c) f xg⋅ ( )(d) f xg

( )( )(e) 1f g+ ( )( )1(f) f g− ( )( )(g) 1f g −⋅ ( )(h) 1fg



Example 10*: Form the Sum, Difference, Product, and Quotient of Two Functions

For the functions ( ) ( )1 and ,2 4

xf x g xx x

= =− +

find the following and determine the

domain in each case.

( )( )(a) f g x+ ( ) ( )(b) f g x− ( )( )(c) f xg⋅ ( )(d) f xg

Section 3.2 The Graph of a Function 83


Chapter 3: Functions and Their Graphs Section 3.2: The Graph of a Function

Vertical-Line Test Theorem: A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in ____________________

Example 1*: Identify the Graph of a Function Which of the following are graphs of functions? In either case, state the domain and range of the relation.

Using the definition of a function from Section 3.1, state why the vertical – line test works.



Obtaining information from the graph of a function: If ( ),x y is a point on the graph of a function f, then y is the value of f at x; that is _________. Also, if ( ) y f x= , then ( ),x y is a point on the graph of f. For example, if (1, 9) is on the graph of f, then ( )f =__ __ and if (5) 4f = then the point _____ is on the graph if f.

Example 2*: Obtain Information from or About the Graph of a Function

The graph of ( )y f x=

(a) What are ( ) ( )30 32

f f fπ π

, , ? (b) What is the domain of f?

(c) What is the range of f? (d) List the intercepts. (e) How often does the line 2y = intersect the graph? (f) For what values of x is ( ) 4f x = ? (g) For what values of x is ( ) 0f x > ?

Section 3.2 The Graph of a Function 85


Example 3*: Obtain Information from or about the Graph of a Function

Consider the function ( )1

xf xx

=+

.

(a*) Is the point 11,2

on the graph of f?

(b*) For 2x = what is ( )f x ? What is this coordinate point on the graph of f?

(c*) If ( ) 2f x = , what is x? What are these coordinate points on the graph of f?

(d) What is the domain of f?

(e) What are the x-intercepts of the graph of f (if any)? What are these coordinate points on the graph of f?

(f) List the y-intercept, if there is one, of the graph of f.



Example 5: Application Problem A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by:

( ) 3 6001000100

,xC xx

= + +

where x is the ground speed (airspeed ± wind). (a) Determine the cost if the ground speed is 200 miles per hour; 350 miles per hour.

(b) Find the domain of C.

(c) Use a graphing utility to graph the function ( ).C C x=

(d) Create a TABLE with Table start 0= and Tbl 50Δ = .

(e) To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

Section 3.3 Properties of Functions 87


Chapter 3: Functions and Their Graphs Section 3.3: Properties of Functions

All functions are either even, odd, or neither even or odd. Classifying functions as such can provide us with information about the graph of a function including specific points.

Definition: A function is even if, for every number x in its domain, the number – x is also in the domain and

( ) ____f x− = So, in an even function, for every point (x, y) on the graph, the point ( , ) is also on the graph.

Theorem: A function is even if and only if its graph is ____________ with respect to the ____ – axis

Definition: A function is odd if, for every number x in its domain, the number – x is also in the domain and

( ) ____f x− = So, in an odd function, for every point (x, y) on the graph, the point ( , ) is also on the graph.

Theorem: A function is odd if and only if its graph is symmetric with respect to the ________.

Example 1*: Determine Even and Odd Functions from a Graph Determine whether each graph shown below is an even function, an odd function, or a function that is neither even nor odd.



Example 2*: Identify Even and Odd Functions from the Equation Use the definitions of even and odd functions to determine algebraically whether each of the following functions is even, odd, or neither. Then graph to see symmetry. (a) ( ) 2 5f x x= − (b) ( ) 3 1g x x= −

(c) ( ) 35h x x x= − (d) ( )F x x=



Definitions: A function f is increasing on an open interval I if, for any choice of 1x and 2xin I, with 1 2x x< we have _______________.

A function f is decreasing on an open interval I if, for any choice of 1x and 2x in I, with

1 2x x< we have _______________ .

A function f is constant on an open interval I if, for all choices of x in I, the values of f(x) are ________.

In other words, if a function is increasing, then, as the values of x get bigger, the values of the function also get bigger.

Using this same idea, what happens when a function is decreasing?

What happens when a function is constant?

Example 3*: Use a Graph to Determine Where a Function is Increasing, Decreasing, or Constant Determine where the function is increasing, decreasing, or constant from its graph.

WARNING: We describe the behavior of a graph in terms of its x – values. Do not say the graph above is increasing from the point ( -4, -2) to the point (0, 4). Rather, say it is increasing on the interval (-4, 0).

(a) Where is the function increasing?

(b) Where is the function decreasing?

(c) Where is the function constant?



When a function changes from increasing to decreasing or decreasing to increasing, it creates maximums and minimums.

Definitions: Local Maximum and Minimums A function f has a local maximum at c if there is an open interval I containing c so that, for all x c≠ in I, ______________. We call ( )f c a local maximum of f. A function f has a local minimum at c if there is an open interval I containing c so that, for all x c≠ in I, ______________. We call ( )f c a local minimum of f.

Example 4*: Use a Graph to Locate Local Maxima and Local Minima

(a)* At what number(s), if any, does f have a local maximum? (b)* What are the local maxima?

(c) At what number(s), if any, does f have a local minimum?

(c) What are the local minima?

WARNING: The y –value is the local maximum and minimum and it occurs at some x – value. For the graph above, we say the local maximum is 2 and that the local maximums occurs at x = 1.

Example 5*: Use a Graphing Utility to Approximate Local Maxima and Minima Use a graphing utility to find the local maxima or local minima of ( ) 32 3 1 f x x x= − + for

2 2x− < < . Then determine where f is increasing and where it is decreasing.



While it is possible to have multiple local maximums or minimums there is only one “absolute” or “most” maximum or minimum.

Definition: Let f denote a function defined on some interval I. If there is a number u in I for which ____________ for all x in I, then ( )f u is the absolute maximum of f on I and we say the absolute maximum of f occurs at __. If there is a number v in I for which ____________ for all x in I, then ( )f v is the absolute minimum of f on I and we say the absolute minimum of f occurs at ____. Note: The absolute maximum and minimum values of a function f are sometimes called the extreme values of f on I.

Example 6: Use a Graph to Locate the Absolute Maximum and the Absolute Minimum For each graph of a function = ( ), find the absolute maximum and minimum, if it exists.



Extreme Value Theorem: If f is a continuous function whose domain is a closed interval [ ],a b , then f has an ____________ and an ____________ on [ ],a b

With the Extreme Value Theorem the absolute maximum (or minimum) can be found by selecting the largest (or smallest) value of f from the following list:

1. The value of f at any local maxima and local minima of f in [ ],a b ,

2. The value of f at each endpoint of [ ],a b .

3. The value of f on any interval in[ ],a b on which f is constant.

Example 7: Extreme Value Theorem Notice that the fourth graph of Example 6 is continuous on the closed interval [ ]2,3− . Therefore, the Extreme Value Theorem guarantees f has an absolute maximum and minimum on [ ]2,3− . Is this what you found when you completed the problem?

Definition: If a and b, a b≠ , are in the domain of a function ( )y f x= , the average rate of change of f from a to b is defined as:

Average rate of change = =

What other algebra concept is defined byyx

ΔΔ

?



Example 8*: Find the Average Rate of Change

Find the average rate of change of 21( )2

f x x= from:

(a) 0 to 1 (b) 0 to 3 (c) 0 to 5

Slope of the Secant Line: The average rate of change of a function from a to b equals the slope of the secant line containing the two points _____________ and ____________ on its graph.

Example 9*: Find the Average Rate of Change and the Equation of a Secant Line Suppose that ( ) 22 4 3.g x x x= − + − (a) Find the average rate of change of g from 2− to x.

(b) Use the result of part (a) to find the average rate of change of g from -2 to 1. Interpret the result.

(c) Find an equation of the secant line containing ( 2, ( 2))g− − and (1, (1))g .

(d) Using a graphing utility, draw the graph of g and the secant line obtained in part (c) on the same screen.



Chapter 3: Functions and Their Graphs Section 3.4: Library of Functions; Piecewise-defined Functions

In the real world, seldom can data be represented by just one function. More often, data changes over time and must be represented by many different functions. When a function is defined by different equations on different parts of its domain, it is called a piecewise-defined function. Before we begin looking at piecewise-defined functions, we must make sure we are familiar with a lot of different functions, which we will call our “library of functions.” It’s important that you know the properties of each of the key functions here. Being able to visualize basic functions will be helpful in later topics such as nonlinear systems and graphing using transformations. Exploration 1: Graph the Functions Listed in the Library of Functions For each of the functions in our library, fill in the table of values, graph the function, then complete the table of properties. Identity Function: y x=

x y 2− 1−

0 1 2

Properties Domain: Range: x-intercept(s): y-intercept: Symmetry: Interval the function is decreasing: Interval the function is increasing: Local maxima/Minima: Absolute maxima/Minima:

Section 3.4 Library of Functions; Piecewise – defined Functions 95


Square Function: 2y x=

Cube Function: 3y x=

x y 2− 1−

0 1 2


x y 2− 1−

0 1 2




Square Root Function: y x=

Cube Root Function: 3y x=

x y 4− 1−

0 1 4


x y 8− 1−

0 1 8




Reciprocal Function: 1yx

=

Absolute Value Function: y x=

x y 2− 1−

0 1 2


x y 2− 1−

0 1 2




Some other important functions to know: The Constant Function: ( )f x b= where b is a real number.

These graphs will always be a ___________________ line with a y – intercept of ________.

Greatest Integer Function:

( ) ( )int greatest integer less than or equal to f x x x= = For example, (1.3) int(1.3) 1f = = since 1 is the greatest integer less than or equal to 1.3. Sometimes the greatest integer function is called the Step Function. Why do you think this is? Is the greatest integer function continuous or discontinuous? Explain.

x y 1− 12

−

14

−

0 14

12

34

1 1.5 2

2.7 3

3.2



Now that we have our “library of functions,” let’s look at piecewise – defined functions. Definition: When a function is defined by different __________ on different parts of its ________, it is called a piecewise-defined function. Example 1*: Graph Piecewise – Defined Functions The function f is defined as

( )2 if 0

2 if 02 if 0

x xf x x

x x

<= = + >

(a) Find ( ) ( ) ( )2 , 0 , and 3f f f−

(b) Determine the domain of f.

(c) Graph f.

(d) Use the graph to find the range of f.

(e) Is f continuous on its domain?



Example 2: Graph Piecewise – Defined Functions The function f is defined as

( )2

2 1 3 12 1

1

x if xf x if x

x if x

− + − ≤ <= = >

(a) Find ( ) ( ) ( )2 , 1 , and 2f f f−

(b) Determine the domain of f.

(c) Locate any intercepts

(d) Graph f.

(e) Use the graph to find the range of f.

(f) Is f continuous on its domain?



Example 3: Piecewise – Defined Application Problem In May 2006, Commonwealth Edison Company supplied electricity to residences for a monthly customer charge of $7.58 plus $0.08275 per kilowatt – hour (kWhr) for the first 400 kWhr supplied in the month, and $0.06208 per kWhr for all usage over 400 kWhr in the month. (a) What is the charge for using 300 kWhr in a month? (b) What is the charge for using 700 kWhr in a month? (c) If C is the monthly charge for x kWhr, develop a model relating the monthly charge and

kilowatt – hours used. That is, express C as a function of x.



Chapter 3: Functions and Their Graphs Section 3.5: Graphing Techniques: Transformations

In the last section we looked at our “library of functions” and made sure we could recognize any of them. In most applications however, we are often asked to graph or recognize a function that is “almost” like one that we already know. In this section, we will look at some of these functions and develop techniques for graphing them. Collectively, these techniques are referred to as transformations.

Exploration 1: Graph Functions Using Vertical Shifts

(a) Sketch the graph of 2y x=

(b) On the same set of axes sketch the graphs

of 2 2y x= + . (c) What is the connection between the

graphs? (d) Test your connection by predicting the graph of 2 3y x= − and then graphing it. If we were asked to write the transformations 2 2y x= + and 2 3y x= − in terms of the original function, we could say that 2 2 ( ) 2y x f x= + = + and 2 3 ( ) 3y x f x= − = − . Summary of Vertical Shifts

• If a positive real number k is added to the output of a function, ( )y f x= , the graph of the new function, ( )y f x k= + , is the graph of f shifted _______________ k units.

• If a positive real number k is subtracted from the output of a function, ( )y f x= , the graph of the new function, ( )y f x k= − is the graph of f shifted __________ k units.

Section 3.5 Graphing Techniques: Transformations 103


Exploration 2: Graph Functions Using Horizontal Shifts Two functions f and g are related by the function, ( ) ( 1)g x f x= − , where the graph of ( )f x is given along with its table of values. Graph ( ) ( 1)g x f x= − after first completing its table.

Two functions f and g are related by the function ( ) ( 2)g x f x= + , where the graph of ( )f x is given along with its table of values. Graph ( ) ( 2)g x f x= + after first completing its table.

x ( ) ( 2)g x f x= + -4 -3 -2 -1 0

Summary of Horizontal Shifts

• If the argument x of a function f is replaced by , 0x h h− > , the graph of the new function ( )y f x h= − is the graph of f___________________________ h units.

• If the argument x of a function f is replaced by , 0x h h+ > , the graph of the new function ( )y f x h= + is the graph of f___________________________ h units.

x ( )f x -2 2 -1 0 0 -2 1 -1 2 0

x ( ) ( 1)g x f x= − -1 0 1 2 3

x ( )f x -2 2 -1 0 0 -2 1 -1 2 0

Observation:

Observation:



Exploration 3: Graph Functions Using Vertical Stretches and Compressions

On the same set of axes, sketch the graphs of 2( ) , 2 ( ),f x x y f x= = and ( )12

y f x=

Summary of Vertical Stretches and Compressions The graph of ( )y af x= when 0a > is a transformation of the graph ( )y f x= .

• If 1a > , then the graph of the original function is ______________ vertically by a factor of a units.

• If 0 1a< < , then the graph of the original function is ______________ vertically by a factor of a units.



Exploration 4: Graph Functions Using Horizontal Stretches and Compressions

Two functions f and g are related by the function 1( )2

g x f x =

, where the graph of ( )f x

is given along with its table of values. Graph 1( )2

g x f x =

after first completing its table.

x 1( )

2g x f x =

-4 -2 0 2 4

Two functions f and g are related by the function ( ) (2 )g x f x= , where the graph of ( )f x is given along with its table of values. Graph ( ) (2 )g x f x= after first completing its table.

x ( ) (2 )g x f x= -2 -1 0 1 2

x ( )f x -2 2 -1 0 0 -2 1 -1 2 0

x ( )f x -2 2 -1 0 0 -2 1 -1 2 0

Observation:

Observation:



Summary of Horizontal Stretches and Compressions: The graph of ( )y f ax= when 0a > is a transformation of the graph ( )y f x= .

• If 1a > , then the graph of the original function is ____________ horizontally by a

factor of 1a

units.

• If 0 1a< < , then the graph of the original function is ____________ horizontally by

a factor of 1a

units.

Exploration 5: Graph Functions Using Reflections about the x – axis and the y – axis

Now graph the following functions in your graphing calculator on the same window as its basic function (for example, the basic function of ( )1g x x= − is y x= ). Compare each function to its basic function as we did in the earlier activities. What are the differences?

( )1g x x= −

( )2g x x= −

( )3g x x= −

( ) 34g x x= −

Summary of Horizontal and Vertical Reflections: The graph of ( )y f x= − is a transformation of the graph ( )y f x= . It reflects the graph of the original function over the __________ axis. The graph of ( )y f x= − is a transformation of the graph ( )y f x= . It reflects the graph of the original function over the __________ axis.



Summary of Transformations* To Graph: Draw the Graph of f and : Functional Change to ( )f x : Vertical Shifts

( ) , 0y f x k k= + >

( ) , 0y f x k k= − >

Horizontal Shifts

( ) , 0y f x h h= + >

( ) , 0y f x h h= − >

Compressing or Stretching

( ) , 0y af x a= >

( ) , 0y f ax a= >

Reflection about the x – axis

( )y f x= −

Reflection about the y – axis

( )y f x= −



Example 1*: Summary of Transformations Use the graph of ( )f x x= to obtain the graph of ( ) 2 1 3g x x= − + − .

Example 2: Graph Using Transformations

Use the graph of ( )f x x= to obtain the graph of ( ) 1 12

g x x= +



Example 3: Graph Using Transformations Use the graph of ( ) 2f x x= to obtain the graph of ( ) ( )23 2 1g x x= − +

Example 4: Graph Using Transformations Use the graph of ( ) 3f x x= to obtain the graph of ( ) ( )3 2g x x= − +

118 Chapter 4 Linear and Quadratic Functions


Chapter 4: Linear and Quadratic Functions Section 4.3: Quadratic Functions and Their Properties

A quadratic function is a function of the form _________________ where a, b, and c are real numbers and _____. The domain of a quadratic function consists of ___________.

Why does the definition above say that 0a ≠ ? What type of function would

2( )f x ax bx c= + + be if 0a = ? We know how to graph a square function 2( )f x x= and we also know how to transform it by the properties learned in Section 3.5. Let’s use these properties and some new terminology to explore quadratic functions.

Terminology

The graph of a quadratic function is called a ____________________________________.

If 0a > the graph of the quadratic opens _______________________________________.

If 0a < the graph of the quadratic opens _______________________________________.

The lowest or highest point of a parabola is called the ____________________________.

The vertical line passing through the vertex of a parabola is called the _______________. Why is this vertical line called the axis of symmetry?

There are two options for graphing parabolas. The first is using transformations: Option 1 for graphing a quadratic function 2( ) , 0f x ax bx c a= + + ≠

Step 1: Complete the square in x to write the quadratic function in the form 2( ) ( )f x a x h k= − +

Step 2: Graph the function in stages using transformations. Example 1*: Graph a Quadratic Function Using Transformations Graph 2( ) 2 6 2f x x x= − + + onto the grid on the following page using transformations.

Section 4.3 Quadratic Functions and Their Properties 119


Example 2: Graph a Quadratic Function Using Transformations Graph 2( ) 2 4 1f x x x= − + using transformations.

Properties of the Graph of a Quadratic Function 2( ) , 0f x ax bx c a= + + ≠

Vertex:___________________ Axis of Symmetry:__________________ Parabola opens up if ___________; the vertex is the ____________________________. Parabola opens down if_________; the vertex is the ____________________________.



Example 3*: Identify the Vertex and Axis of Symmetry of a Quadratic Function Without graphing, locate the vertex and axis of symmetry of the parabola defined by

2( ) 3 12 5f x x x= + − . Does it open up or down? Example 4: Identify the Vertex and Axis of Symmetry of a Quadratic Function Without graphing, locate the vertex and axis of symmetry of the parabola defined by

2( ) 3 6 1f x x x= − + + . Does it open up or down?

Option 2 for Graphing a Quadratic Function 2( ) , 0f x ax bx c a= + + ≠ . Step 1: Determine whether the graph of f opens up or down.

Step 2: Determine the vertex ,2 2b bfa a

− −

and the axis of symmetry, 2bxa

= − .

Step 3: Determine the y – intercept, (0)f . Determine the x – intercept(s), if any. (a) If 2 4 0b ac− > , then the graph of the quadratic function has _______ distinct

x – intercepts, which are found by solving the equation 2 0ax bx c+ + = . (b) If 2 4 0b ac− = , then the graph has _____ x – intercept so it touches the x –

axis at its ________. (c) If 2 4 0b ac− < , there are _______ x – intercepts.

Step 4: Determine an additional point by using the y – intercept and the axis of symmetry. Plot the points and draw the graph.

Example 5*: Graph a Quadratic Function Using its Vertex, Axis, and Intercepts. (a) Graph 2( ) 3 12 5f x x x= + − by determining whether the graph opens up or down and by

finding its vertex, axis of symmetry, y – intercept, and x – intercepts, if any.

Section 4.3 Quadratic Functions and Their Properties 121


Example 6*: Graph a Quadratic Function Using its Vertex, Axis, and Intercepts II (a) Graph 2( ) 4 7f x x x= − + − by determining whether the graph opens up or down and by

finding its vertex, axis of symmetry, y – intercept, and x – intercepts, if any.

Given the vertex ( , )h k and one additional point on the graph of a quadratic function

2( ) , 0f x ax bx c a= + + ≠ , are known, then _______________________ can be used to obtain the quadratic function.

Example 7*: Find a Quadratic Function Given its Vertex and One Other Point Determine the quadratic function whose graph has a vertex of ( 2,5)− and passes through the point (0, 7)− .

Note: If 0a > , the vertex is the ___________ point on the parabola since the parabola opens up. This means that ______ is the minimum value of f. If 0a < , the vertex is the ___________ point on the parabola since the parabola opens up. This means that ______ is the maximum value of f.

Example 8*: Find the Maximum or Minimum Value of a Quadratic Function Determine whether the quadratic function ( ) 2 4 5f x x x= − + + has a maximum or minimum value. Then find the maximum or minimum value.



Chapter 4: Linear and Quadratic Functions Section 4.4: Build Quadratic Models from Verbal Descriptions and from Data

When a mathematical model is in the form of a quadratic function, the properties of the graph of the quadratic function can provide important information about the model. In particular, we can use the quadratic function to determine the maximum or minimum values of the function by using the vertex.

Note: In economics, revenue R, in dollars, is defined as the amount of money received from the sale of an item and is equal to the unit selling price p, in dollars of the item times the number x of units actually sold. That is:

________R = In economics, the Law of Demand states that p and x are related: as one increases, the other decreases. The equation that relates p and x is called the demand equation.

Example 1*: Build Quadratic Models from Verbal Descriptions (Maximizing Revenue) The prince p (in dollars) and the quantity x sold of a certain product obey the demand equation

1 100 0 6006

p x x= − + ≤ ≤

(a) Find a model that expresses the revenue R as a function of the price x. (Remember, .)R xp=

(b) What is the revenue if 200 units are sold?

(c) What is the quantity x that maximizes revenue? What is the maximum revenue?

(d) What price should the company charge to maximize revenue?

Section 4.4 Build Quadratic Models from Verbal Descriptions and from Data 123


Example 2*: Build Quadratic Models from Verbal Descriptions (Analyzing the Motion of a Projectile) A projectile is fired from a cliff 200 feet above the water at an inclination of 45° to the horizontal, with a muzzle velocity of 50 feet per second. The height h of the projectile above the water is given by

2

2

32( ) 200(50)

xh x x−= + +

where x is the horizontal distance of the projectile from the base of the cliff. (a) How far from the base of the cliff is the height of the projectile a maximum?

(b) Find the maximum height of the projectile.

(c) How far from the base of the cliff will the projectile strike the water?

(d) Using a graphing utility, graph the function h, 0 200.h≤ ≤

(e) When the height of the projectile is 100 feet above the water, how far is it from the cliff?



Example 3*: Build Quadratic Models from Verbal Descriptions (Maximizing the Area Enclosed by a Fence) David has available 400 yards of fencing and wishes to enclose a rectangular area. (a) Express the area A of the rectangle as a function of the width w of the rectangle.

(b) For what value of w is the area the largest?

(c) What is the maximum area?

Example 4*: Build Quadratic Models from Verbal Descriptions (Suspension Bridge) A suspension bridge with weight uniformly distributed along its length has twin towers that extend 75 meters above the road surface and are 400 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 100 meters from the center. (Assume that the road is level.)

Section 4.4 Build Quadratic Models from Verbal Descriptions and from Data 125


Example 5*: Build Quadratic Models from Data An individual’s income varies with his or her age. The following table shows the median income I of individuals of different age groups within the United States for 2003. For each age group, let the class midpoint represent the independent variable, x. For the class “65 years and older,” we will assume that the class midpoint is 69.5. (a) Draw a scatter diagram of the data.

Comment on the type of relation that may exist between the two variables.

(b) The quadratic function of best fit to these data is 2( ) 34.3 3157 39,115I x x x= − + − . Use the function to determine the age at which an individual can expect to earn the most income.

(c) Use the function to predict the peak income earned.

(d) Use a graphing utility to verify that the function given in part (b) is the quadratic function of best fit.

(e) With a graphing utility, draw a scatter diagram of the data and then graph the quadratic function of best fit on the scatter diagram.

Age (years) Class

Midpoint, x

Median Income ($), I

15 – 24 19.5 8514 25 – 34 29.5 26,212 35 – 44 39.5 30,914 45 – 54 49.5 32,563 55 – 64 59.5 29,068

65 and older 69.5 14,664



Chapter 4: Linear and Quadratic Functions Section 4.5: Inequalities Involving Quadratic Functions

There are two options for solving inequalities involving quadratic functions: Option 1: Graphical Method

1. Rearrange the inequality so that 0 is on the right hand side. 2. Graph 2( )f x ax bx c= + + using the vertex, axis of symmetry, x – intercepts, and y -

intercepts. 3. Use the graph to determine the solution set.

(a) If you’re solving the inequality 2 0ax bx c+ + > the solution set is the values where the function is above the x – axis, that is, where ( ) 0f x > .

(b) If you’re solving the inequality 2 0ax bx c+ + < the solution set is the values where the function is below the x – axis, that is, where ( ) 0f x < .

Option 2: Algebraic Method 1. Rearrange the inequality so that 0 is on the right hand side. 2. Solve 2 0ax bx c+ + = . 3. Use these solutions to divide the real number line into intervals. 4. Determine the sign of 2ax bx c+ + in each interval. 5. Determine your solution set.

(a) If you’re solving the inequality 2 0ax bx c+ + > the solution is the intervals in which the function is positive, where ( ) 0f x > .

(b) If you’re solving the inequality 2 0ax bx c+ + < the solution is the intervals in which the function is negative where ( ) 0f x < .

Example 1*: Solve Inequalities Involving a Quadratic Function I Solve the inequality 2 8 2x x− + > using both methods above.

Section 4.5 Inequalities Involving Quadratic Functions 127


Example 2*: Solve Inequalities Involving a Quadratic Function II Solve the inequality 2 3 28 0x x+ − ≥ .

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Chapter 1: Equations and Inequalities Section 1.6: Equations ......Chapter 1: Equations and Inequalities Section 1.6: Equations and Inequalities Involving the Absolute Value Exploration