Hartfield MATH Notes

Hartfield – College Algebra (Version 2016b - © Thomas Hartfield) Unit ONE | Page - 1 - of 50

Topic 0: Basic Equations and Inequalities Definition: An equation is a statement that the

values of two expressions are equal. Ex. 1 Show that x = 5 is a solution to 3x – 4 = 2x + 1. Show that x = 2 is not.

Two primary rules can be used when solving a basic equation:

1. Simplify each expression on either side of the relationship independent of the other side.

2. An action which changes an expression on one side of a relationship must be matched by an equivalent change on the other side of the relationship.

Ex. 3 Solve.

( )− = −3 2 1 4x x


In some environments, the solutions to an equation may be expressed as a set. Definition: A set is a well-defined collection of

distinct objects. Sets are usually expressed using braces, also referred to as curly brackets. If x = 5 is the only solution to 3x – 4 = 2x + 1, then we can refer to {5} as the solution set of the equation.

Some basic equations can be solved by every real number. A short hand way of expressing the set of every real number is which is called blackboard r. Definition: An equation where every real number

solves the equation is called an identity statement.

Some basic equations cannot be solved by any real number. A set without any object in it is called an empty set and is expressed by ∅. Definition: An equation where no real number

exists to solve the equation is called a contradiction.


Definition: An inequality is a statement that the values of two expressions are not equal.

Examples of inequality relationships include: a < b a is (strictly) less than b a > b a is (strictly) greater than b a ≠ b a is not equal to b Inequality relationships that include the possibility for equality as well include: a < b a is less than or equal to b a > b a is greater than or equal to b Note that the use of the word or implies two equally possible relationships.

Ex. 2 Determine whether 5 is a solution to each inequality below.

a. 2x + 4 ≠ 4x + 2

b. x – 1 > 6x

c. 4x – 7 < x + 8


The primary rules for solving basic equations can be applied to inequalities if one more rule is added:

3. When multiplying or dividing by a negative number, reverse the direction of the inequality symbol.

Ex. 4 Solve. ( )− >3 5 2 1x Ex. 5 Solve.

24x

− ≤

A compound inequality, also called a three-part inequality, is an inequality which relates three expressions using two (like-directed) inequality symbols. The simplest compound inequalities will involve a single variable expression between two numeric expressions can be solved using the same rules of basic inequalities. Ex. 6 Solve. 5 2 1 13x− < − ≤


Typically inequalities have many solutions, a significant difference from the usually limited number of solutions an equation has. While a special form of a set can be created to express the solutions within a set, typically the solutions to inequalities are expressed using interval notation. Interval Notation ( ) parentheses – boundary excluded or , or brackets – [ ] boundary included

left end

boundaryor −∞

right endboundary

or ∞

Ex. 7 Express the solution inequalities from previous examples in interval notation.

a. Ex. 4

b. Ex. 5

c. Ex. 6


Symbols/terminology relevant to intervals & interval notation: Closed Interval: [ ]2,7 , [ )5,∞ any and all numeric boundaries are included Open Interval: ( )4,3− , ( ),4−∞ any and all numeric boundaries are excluded Half-open Interval: [ )5, 1− − , ( ]0,4 one end open, one end closed Infinite Interval: ( ),4−∞ , [ )5,∞ , ( ),−∞ ∞

unbounded on one or both ends Union Symbol: <means or> each entire interval appropriate Intersection Symbol: <means and> only common (i.e. overlapping) interval appropriate

Ex. 8 Classify the intervals found in previous examples.

a. Ex. 4

b. Ex. 5

c. Ex. 6


Another way of expressing the solutions to an inequality is to sketch a graph of the solutions on a number line. We graph inequalities so that we can “see” the solutions. To graph the solution set of an inequality:

1. Identify the boundaries of the solution interval. 2. Use an appropriate symbol to express

whether the boundary is included or excluded a. Use a closed dot if the boundary is

included. b. Use an open dot if the boundary is

excluded. 3. Shade appropriately, using test values as

necessary to identify the intervals.

Ex. 9 Sketch a graph of the solution set of each inequality from previous examples.

a. Ex. 4

b. Ex. 5

c. Ex. 6


Topic 1: The Coordinate Plane; Symmetry Overview of properties and terminology associated with the coordinate plane:

Definition: Symmetry is the property of having exact correspondence of form and configuration when an image is reflected about an axis or rotated about a point.

Two primary forms of symmetry: Reflective Symmetry: Rotational Symmetry:


On the coordinate plane:

The point (c, b) is symmetric about the x-axis to the point (a, b) if c = −a. or in other words, a point (x, y) can be reflected about the x-axis by replacing y with –y. The point (a, d) is symmetric about the y-axis to the point (a, b) if d = −b. or in other words, a point (x, y) can be reflected about the y-axis by replacing x with –x. The point (c, d) is symmetric about the origin to the point (a, b) if c = −a and d = −b. or in other words, a point (x, y) can be rotated about the x-axis by replacing both x with –x and y with –y.

Ex. 1 Plot points on the coordinate plane that satisfy the condition of symmetry given.

a. Point D is symmetric about the x-axis to A.

b. Point E is symmetric about the y-axis to B.

c. Point F is symmetric about the origin to C.

A ●

B ●

C ●


Topic 2: Distance and Midpoint The distance d across the coordinate plane from point ( )1 1,x y to point ( )2 2,x y is determined by

( ) ( )2 22 1 2 1 .d x x y y= − + −

Ex. 1a Find the distance between ( )3, 2− and

( )1,4 .−

The midpoint M of a line segment connecting point ( )1 1,x y to point ( )2 2,x y is determined by

( )1 2 1 2

, , .2 2x y

x x y yM + + =

Ex. 1b Find the midpoint between ( )3, 2− and ( )1,4 .−


Ex. 2 Find the distance from A to B and the location of the midpoint between A & B.

( ) ( )2, 5 1,4A B− −


Topic 3: Linear Equations of Two Variables; Graphs of Equations of Two Variables Definition: A linear equation of two variables is

an equation which can be expressed in the form Ax + By = C, where A, B, and C are real numbers and both A and B are not zero. Solutions to an equation of two variables are expressed as ordered pairs (x, y).

The graph of a linear equation of two variables produces a line on the coordinate plane. Linear equations are examples of relations. Definition: A relation is a set of points in the

plane. Note that relations may or may not be defined by equations at all.

Other important forms of linear equations include: Slope-intercept form: y = mx + b, m is the slope includes the point (0, b) Point-slope form: y – y1 = m(x – x1) m is the slope includes the point (x1, y1) Vertical line: x = a, a is a real number Horizontal line: y = b, b is a real number Recall that the slope of a line is the value found by the ratio of vertical change to horizontal change of two points on the line. Horizontal lines have a slope of zero while vertical lines do not have slope.

Slope formula: −=

−2 1

2 1

y ymx x


Ex. 1 Find the slope-intercept form of the line which passes through (−5, 1) & (3, −3).

Ex. 2 Restate the equation in slope-intercept form, identify the slope, and sketch a graph of the line on the coordinate plane.

3x + 4y = 12


Although we will use strategies in the future for producing graphs that occur in common or important situations, it is worth noting that a sketch can be approximated by creating a table of values, assigning values for one variable and calculating the value of the other variable. Ex. 3 Create a table of values to sketch a

graph of the equation. 2 3y x x= −


Topic 4: Linear Systems of Two Variables Definition: A system of equations is a set of

equations involving the same variables simultaneously.

A system of equations is said to be solved when a set of values for each variable in the system satisfies every equation in the system.

Systems of linear equations are those systems where every equation is linear. In college algebra we will want to solve systems of linear equations. A system of nonlinear equations is a system with one or more nonlinear equation and will not be covered here.

Solving a linear system of two variables can be accomplished in four ways: 1. Finding the intersection point of the graphs of

each equation. 2. Substituting one equation into the other to find

the value of one variable and then using that value to find the value of the second variable.

3. Adding equations together, after appropriate multiplicative operations to one or both equations, to find the value of one variable. Then using that value to find the value of the second variable.

4. Corresponding the coefficients of the system to a matrix and applying elementary row operations to create an identity matrix and a solution matrix.

We will not apply matrices to solve systems in this class. Our primary approaches will be methods 2 and 3, the Substitution Method and the Elimination method, respectively, with supplemental graphing for confirmation.


The Substitution Method requires the following set of steps:

1. Solve either equation for x or y. 2. Plug that equation in for the variable in the

other equation. 3. Solve the created equation involving one

variable. 4. Plug the solution found in step 3 into any

equation involving two variables & solve. 5. Write your solution as an ordered pair.

The Addition/Elimination Method requires the following set of steps:

1. As necessary, multiply one or both coefficients by a positive or negative whole number so that a column of coefficients have the same absolute value but different signs.

2. Add the equations to eliminate a variable and create a new equation.

3. Plug the solution found in step 2 into any equation involving two variables & solve.

4. Write your solution as an ordered pair.

Ex. Solve the system of equations using the Substitution Method.

+ = −

− =

4 4

3 2 16

x y

x y


Ex. Solve the system of equations using the Elimination Method.

+ =

+ =

2 1

3 2 4

x y

x y

Observe that if each system is graphed, the solution corresponds with the intersection point of the equations in the system.


Ex. 3 Solve the system of equations.

+ =

− =

5 3 2

2 4 6

x y

x y

Most systems of linear equations have exactly one solution. A system that has exactly one solution is classified as being consistent and independent. A system of linear equation can have no solution if the graphs of the equations are parallel (as thus lack an intersection point). Such a system is classified as being inconsistent.

When solving a system through either method, if a contradiction is reached (such as two different numbers supposedly being equal to each other), the system must be inconsistent.

A system of linear equations can have infinitely-many solutions if the equations are not unique (and thus share all their points along a coincident line). Such a system is classified as being dependent.

When solving a system through either method, if an identity statement is reached (such as a number being equal to itself), the system must be dependent.


Topic 5: Linear Systems of Three Variables To solve a linear system of three variables, we will follow the following set of steps:

1. Write each equation in standard form Ax + By + Cz = D

2. Choose a pair of equations and eliminate one of the variables by the Addition/Elimination method.

3. Choose a different pair of equations and eliminate the same variable.

4. Utilize the equations in steps 2 and 3 to solve for one variable by either the Substitution or Addition/Elimination method. Then find the value of the other variable in those equations.

5. Plug the values found in step 4 into any of the original equations to find the value of the third variable.

6. Write your solution as an ordered triple.

Observation: If a contradiction or an identity statement is found through the process of rewriting into triangular form, then the system has no solution or infinitely-many solutions, respectively.



− + =

+ + = − − = −

2 2

3 5 8

2 2 7

x y z

x y z

x y z



+ + =

− =− + + =

0

3 6

2 5 3

x y z

x y

x y z



+ + =

+ + =− + − = −

2 3 8

5

2 4 5 1

x y z

x y z

x y z


Topic 6: Intercepts; Symmetry of a Graph Definition: An intercept is a point of a graph on

an axis. If you have x and y axes, the any points on these axes would be x-intercepts and y-intercepts, respectively. To find intercepts of an equation for one variable, set the other variable equal to zero and solve. Thus, for an equation involving x and y variables: To find x-intercepts, To find y-intercepts, set y = 0 and set x= 0 and solve for x solve for y x = h1, x = h2, x = h3, … y = k1, y = k 2, y = k 3, … (h1, 0), (h2, 0), (h3, 0), … (0, k1), (0, k2), (0, k3), …

Ex. 1a Find the intercepts of + =212 9 36.x y


To determine if the graph of an equation displays symmetry, apply a test to see if the equation satisfies the definition of symmetry. To test for symmetry about the x-axis:

Replace y with −y in the equation and simplify. If the original equation is returned, then (the graph of) the equation is symmetric about the x-axis.

To test for symmetry about the y-axis:

Replace x with −x in the equation and simplify. If the original equation is returned, then (the graph of) the equation is symmetric about the y-axis.

To test for symmetry about the origin:

Replace (x, y) with (−x, −y) in the equation and simplify. If the original equation is returned, then the equation is symmetric about the origin.

Ex. 1b Determine if the graph of + =212 9 36x y displays any type of symmetry.


Ex. 2 Find any intercepts for the equation − =22 4 8x y and determine if its graph

displays any type of symmetry.


Topic 7: An Introduction to Functions Definition: A function is a relation defined by a

rule that assigns each element in one set to exactly one element in a second set.

The first set is called the domain.

The domain of a function is the collection of values that can be acceptable inputs for a function. The variable of the domain is the independent variable of the function.

The second set is called the range.

The range of a function is the collection of values that can result from an input into the function. The variable of the range is the dependent variable of the function.

Visualizations of Functions


Functions can be expressed in many ways: Verbally A verbal rule that satisfies the

definition

As Data A collection of points that satisfies the definition

Visually A graph representing a collection

of points that satisfy the defintion As Equations A symbolic statement

representing a rule satisfying the definition

Often it is obvious if a relation is a function when it is expressed verbally or conveyed as data.

More often we run into relations expressed as equations or expressed visually and strategies are helpful to determine when a function exists. When a relation is expressed in graphical form, the following concept can be applied: Concept: The Vertical Line Test states that a

graph represents a function if every possible vertical line intersects the graph at most once.


Ex. 1 Determine if each graph of a relation below represents a function. Then determine the domain and range of the functions.

a.

b.

Ex. 2 Determine if each graph of a relation below represents a function. Then determine the domain and range of the functions.

a.

b.


When a relation is expressed as an equation, the following concept can be applied: Concept: An equation represents a function if

it can be uniquely solved for the variable of the range (often y).

Ex. 3 Determine if each equation below

represents a function.

a. = +3y x x

b. = −2 1x y

Ex. 4 Determine if each equation below represents a function.

a. =2 8xy

b. + =3 3 1x y


Topic 8: Function Notation Ex. 1 Identify the parts of the functions given.

a. ( ) 1 5 , 0g t t t= − <

b. ( ) =H x x

Ex. 2 Express the verbally defined function using function notation.

a. Given x, multiply by 2, then add 4, and

then take the square root.

b. Given x, where x is greater than or equal to 1, square and then subtract 2.

c. Given x, take the square root, subtract

1, and then divide by 3.

independent variable variable of the domain

dependent variable variable of the range

name of the function

rule of the function

conditions of the function (may not always be present)

( ) = − ≥ −2 3, 2f x x x


Evaluating versus Solving Definition: To solve an equation, you want to

find values of the variable(s) which make the equation true.

Definition: To evaluate a function, you need a

value for the independent variable and by applying the rule of the function, you find the value of the dependent variable associated with it.

Solve: 6 7 10x + = Evaluate: ( ) = −2 3f x x for x = −1

Functions can be evaluated for numerical values, constant values represented by letters, variables and variable expressions, or some other combination thereof. Ex. 3 Evaluate the function at the given values

of x. ( ) = +5 2f x x if x = – x or if x = 2 .a


Ex. 4a Evaluate the function at the given values of x.

( ) = −2 3f x x if x = a – 2.

Ex. 4b Evaluate the function at the given values of x.

( ) = −2 3f x x if x = x + h.


A special kind of function which involves multiple rules and conditions for using each rule is called a piecewise-defined function.

Example − ≤ −

= − > −

4 , 2( )

2 3, 2x x

g xx x

Ex. 5 Evaluate function g above if:

a. x = −4

b. x = 4

Ex. 6 Evaluate the piecewise-defined function given below at each value of x.

<

= − ≥

4 , 3( )

5 , 3x x

f xx x

a. x = 3

b. x = 1


Topic 9: Domains of Functions The domain of a function can either be explicitly or implicitly defined and it can either be limited or unlimited over the set of all real numbers. Most functions have an implicitly unlimited domain; that is, the domain of the function is the set of all real numbers. If a function has a condition attached to its rule, then its domain is explicitly defined and may be limited or unlimited. Example ( ) = − ≥2 3, 2f x x x

Some functions have implicitly limited domains. Despite lacking a condition stated with the rule, these functions are not defined for all real values of the independent variable. Two families of functions that are often implicitly limited: Radical Functions (with an even radical index) Rational Functions Functions which involve radicals and rational expressions can often be implicitly limited in multiple ways.


Ex. 1 Find the domain of ( ) = −5 2f x x .

Ex. 2 Find the domain of ( ) +=

−264

xF xx

.


Ex. 3 Find the domain of ( ) −=

−41

xg xx

.

Ex. 4 Find the domain of ( ) −=

−1

4xG x

x.


Topic 10: Even and Odd Functions A function f is even if ( ) ( ).f x f x− = A function f is odd if ( ) ( ).f x f x− = − Many functions are neither even nor odd. A basic example of an even function is f(x) = x² and a basic example of an odd function is g(x) = x³.

Even functions display symmetry with respect to the y-axis (assuming f(x) = y) and odd functions display symmetry with respect to the origin. Because of the relationship between evenness/oddness and symmetry, the processes for determining each are somewhat similar. To determine if a function is either even or odd, evaluate the function at –x and then simplify. 1. If it is possible to simplify back to the original

function, then the function is even. 2. If it is possible to simplify to the opposite of

the original function, then the function is odd. 3. If it not possible to return to either the original

function or the opposite of the original, then the function is neither even nor odd.


Ex. 1 Determine if the function below is even, odd, or neither even nor odd.

( ) = −34f x x x


( ) =+3

64

xg xx x



( ) = −5 66 5h x x x


( ) = +24 5j x x


Topic 11: Graphs of Functions, a Summary Recall that the graph of a function is the collection of all the ordered pairs formed by evaluating the function. For the purposes of the following definitions, assume that the first variable is x, the function is f, and the second variable f(x) = y:

Recall that the domain is the collection of all possible values of x for the function.

Recall that the range is the collection of all possible values of y for the function.

Recall that the y-intercept is the point of a graph of a function on the y-axis, if it exists.

Recall that the x-intercept(s) are the point(s) of a graph of a function on the x-axis.

The zeroes of the function are the values of x such that f(x) = 0. x-intercepts occur at zeroes.

A function is said to be increasing over an interval if for all x1 < x2 in the interval, f(x1) < f(x2).

A function is said to be decreasing over an interval if for all x1 < x2 in the interval, f(x1) > f(x2).


A point where a function changes from increasing to decreasing, or vice versa, is called a turning point. Turning points are locations where extreme points occur.

A point of a function (h, k) where k < f(x) for all x in an open interval containing h is called a relative minimum point of f. If k < f(x) for all x in the domain of f, then (h, k) is simply called the minimum point.

A point of a function (h, k) where k > f(x) for all x in an open interval containing h is called a relative maximum point of f. If k > f(x) for all x in the domain of f, then (h, k) is simply called the maximum point.

You may also see/hear relative extreme points referred to as local extreme points. A true extreme point is often referred to as an absolute extreme point or a global extreme point.

Ex. Analyze the graph of f below.


Ex. 1 Identify and/or determine the properties of the function represented in the graph.

Identify the domain and range of f.

Identify the intercepts of f. Evaluate f(2). Solve f(x) = 5. Identify the intervals over which f is increasing or decreasing. Identify the extreme points of f.


Ex. 2 Identify and/or determine the properties of the function represented in the graph.

Identify the domain and range of f.

Identify the intercepts of f. Evaluate f(2). Solve f(x) = 2. Identify the intervals over which f is increasing or decreasing. Identify the extreme points of f.


Topic 12: Graphing Piecewise-Defined Functions Here is a summary of our approach: 1. Set each rule equal to y.

2. Identify any boundary implied by the conditions and graph a dotted vertical line.

3. Evaluate each rule at any boundary that applies. Appropriately mark an open or closed dot with respect to whether the rule is used at the boundary.

4. Use appropriate techniques to graph each rule within its conditions.

a. Rules that are constant will be horizontal segments or rays.

b. Rules that are linear can be sketched using the slope starting at the point on the boundary

c. Rules that are nonlinear can be sketched using a table of values or other appropriate methods.

Evaluate the piecewise-defined function at x = 3 and then sketch a graph of the function.

Ex. 1 < −

= + ≥ −

4, 1( )

2, 1x

f xx x



Ex. 2 + ≤ −

= − > −

12 4, 2

( )3 2, 2

x xg x

x x


Ex. 3 + <

= − ≥

2 3, 2( )

5 , 2x x

h xx x


Topic 13: Average Rate of Change Definition: The average rate of change of a

function = ( )y f x between x = a and x = b is

−=

−change in ( ) ( )avg rate of change = change in

y f b f ax b a

The average rate of change is equal to the slope of the secant line between x = a and x = b on the graph of f; that is, the line passes through ( ) ( ), ( ) and , ( ) .a f a b f b

Ex. 1 Find the average rate of change of f over the given interval.

( ) [ ]= − −37 1,2f x x x



( ) [ ]= − −37 2,3f x x x


( ) [ ]= + 3 1,4g x x x


Topic 14: The Difference Quotient The concept of a derivative in calculus is based upon an algebraic concept called the difference

quotient: ( ) ( ).

f x h f xh

+ −

In algebra, we can associate the difference quotient with the idea of average rate of change. x represents the value of a while x + h represents b (and thus the the difference in b and a is h). The difference quotient allows you to quickly find an average rate of change for many different a and b. The process of finding a difference quotient involves finding f(x + h), setting up the overall expression, and then simplifying the expression.

Ex. 1 Find the difference quotient for ( ) 3 4.f x x= −


Ex. 2 Find the difference quotient for ( ) = − 22 .f x x x

Ex. 3 Find the difference quotient for ( ) 2 4 .f x x x= −


Ex. 4 Find the difference quotient

for ( ) 1.1

xf xx+

=−

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