binder chapter1

7/30/2019 Binder Chapter1

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Copyright 2008 by Farimah Fazeli

College Algebra

Section 1.1 Equations in One Variable

Definitions:

Equation:

Solution or Root:

Solution Set:

Linear Equation in One Variable:

Properties of Equalities:Addition /SubtractionMultiplicationDivision

Ex 1: Solve 32

34

+=xx

Identities:

Conditional Equations:

Inconsistent Equations:

Ex 2: Solve and Identify.4(y - 1) = 4y 4

Equations Involving Rational Expressions:

Ex 3: Solve and Identify

6

613

6 +=

+ xx

x


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Section 1.1 Equations in One Variable page 2


Equations Involving Absolute Value


| 3x + 4 | = 12


5 + 3| x 4| = 0

Ex 6: Corporate Taxes. For a class C corporation in Louisiana, the amount of stat income tax S isdeductible on the federal return and the amount of federal income tax F is deductible on the state return.

With $200,000 taxable income and a 30% federal tax rate, the federal tax is 0.30(200,000 S). If thestate tax rate is 6% them the state tax satisfies S = 0.06(200,000 0.30(200,000 S)).Find the state tax S and the federal tax F.


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College Algebra

Section 1.2 Constructing Models to Solve Problems

Applied problems do not come in the form Solve the equation. In this section we will review how totranslate verbal descriptions into the language of mathematics. Sometimes we use well-known formulas

to model real situations, but we must often construct our own models.

A Formula is:

Solve a formula for a specified variable: when a variable is solved for a specified variable, thatvariable is isolated on one side of the equal sign and must not occur on the other side.

Ex 1: Solve for F: )32(9

5= FC

Solve a Problem involving sales tax

Ex 2: To be able to afford the house of their dreams, Dave and Leslie must clear $128,000 form the daleof their first house. If they must pay $780 in closing costs and 6% of the selling price for the salescommission, them what is the minimum selling price for which they will get $128,000?

Solve a Geometric problem

Ex 3: Julias soybean field is 3 m longer than it is wide. To increase her production, she plans toincrease both the length and width by 2m. If the new field is 46m2 larger than the old field, then whatare the dimensions of the old field?


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Section 1.2 Constructing Models to Solve Problems Page 2


Solve Uniform Motion Problems

Uniform Motion Formula: If an object moves at an average velocity r, the distance d covered in time t isgiven by the formula d=rt. That is, Distance = Velocity Time.

Ex 4: An air rescue plane averages 300 miles per hour in still air. It carries enough fuel for 5 hours offlying time. If, upon takeoff, it encounters a head wind of 30 mi/hr, how far can it fly and returnsafely? (Assume that the wind remains constant.)

Solve Mixture Problems

In mixture problems, two or more quantities are combined to form a mixture. To solve these types ofproblems, set up two equations; one for quantity and one for value.

Ex 5: A candy store sells boxes of candy containing white and dark chocolates. Each box sells for$12.50 and holds 30 pieces of candy (all pieces are the same size). If the whites cost $0.25 toproduce and the darks cost $0.45 to produce, how many of each should be in a box to make a profitof $3?


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Section 1.2 Constructing Models to Solve Problems Page 3


Solve Interest Problems

Simple Interest Formula: If a principal ofP dollars is borrowed for a period oft years at a per annuminterest rate r, expressed as a decimal, the interest I charged is I Prt= .

Ex 6: Suppose you borrow $1000 for 6 months at the simple interest rate of 6% per annum. What is the

interest you will be charged on this loan? If you pay the loan back at the end of six months, what isthe amount you must pay?

Solve Constant Rate Job Problems

Some problems involve jobs that are performed at a constant rate. Our assumption is that, if a job can be

done in t units of time,1

tof the job is done in 1 unit of time.

Ex 7: A large pump can empty a pool in 5 hours. A smaller pump empties the same pool in 8 hours. If

the pumps are used together, how long will it take them to empty this pool?


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College Algebra

Section 1.3 Equations and Graphs in Two Variables

The Cartesian PlaneOrdered pairs of real numbers can be represented graphically using the Cartesian plane, or therectangular coordinate system. The Cartesian plane is formed by using two real lines, called axes,

intersecting at right angles. The names of the axes depend on the variables being used, however, thevertical axis is usually called they-axis, and the horizontal axis is usually called thex-axis. Theintersection of the axes is called the origin, and the axes divide the plane into four regions calledquadrants. Each point in the plane is represented by the ordered pairx,y( ), wherex is thex-coordinateandy is they-coordinate.The rectangular coordinate system allows us to see the relationship between data at a glance. There are

different types of graphical representations that can be illustrated in the plane, for example scatter plots,bar graphs, and line graphs. Which representation is best depends upon certain characteristics, such aswhat the variables represent and the size of the variables. In a scatter plot, a discreet number of datapoints are plotted which allows us to see the general distribution of the data. In a line graph the datapoints are joined by lines, determined by the independent variable, which allows us to see the change ofthe dependent variable with respect to the independent variable.

Example: Plot the points 2,3( ), 0,1( ), and 5, 4( ).

Horizontal axis

Verticalaxis

Origin (0, 0)

Quadrant 1Quadrant II

Quadrant III Quadrant IV

1

2

3

4

5

5

4

3

2

1

1 2 3 4 55 4 3 2 1

y

x


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Section 1.3 Equations and Graphs in Two Variables Page 2


The Distance FormulaThe Pythagorean Theorem tells us that for any right triangle with hypotenuse of length c and sides of

lengths a and b that a2 + b2 = c2. It is also true that ifa2 + b2 = c2 for some triangle, then the trianglemust be a right triangle. We can determine the distance dbetween two points x1,y1( )and x2,y2( ) in theplane using the Pythagorean Theorem.

Using the right triangle in the picturewe see that

d2 = x2 x1

2+ y2 y1

2

d= x2 x12

+ y2 y12

d= x2 x1( )2

+ y2 y1( )2

Distance Formula

The distance between the points x1,y1( )and x2,y2( ) in the plane is

d= x2 x1( )2

+ y2 y1( )2

.

Example: Find the distance between the points 7, 4( )and 2,8( ).

x2 x1

y2 y1

x1,y1( )

x2,y2( )

x1 x2

y2

y1


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Example: Verify that the triangle with vertices 1,5( ), 5, 2( ), and 1, 2( ) is a right triangle.

The distance between 1,5( ) and 5, 2( ) is 5 1( )2 + 2 5( )2 = 42 + 7( )2 = 16 + 49 = 65

The distance between 1,5( ) and 1, 2( ) is 1 1( )2 + 2 5( )2 = 02 + 7( )2 = 49 = 7

The distance between 1, 2( ) and 5, 2( ) is 5 1( )2 + 2 2( )( )2 = 42 + 02 = 16 = 4

To find the midpoint of a line segment that joins two points in the coordinate plane, we can find theaverage values of the respective coordinates of the two endpoints.

Midpoint FormulaThe midpoint of the line segment joining the points x1,y1( )and x2,y2( ) is

M=x1 +x2

2,y1 + y2

2

.

Example: Find the midpoint of the line segment joining the points 7, 4( )and 2, 8( ).


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Calculator Functions

Graphing; y =TableTablesetWindowx- andy-intercepts; value and zero

Example: Graph 23 19y x= + . Use a graphing utility to approximate the intercepts rounded to twodecimal places. Use the TABLE feature to help establish the viewing window.

CirclesA circle is a set of points in thexy-plane that are a fixed distance rfrom a fixed point ( ),h k . The

distance ris called the radius and the point ( ),h k is called the center of the circle.

A circle centered at the origin has the equation 2 2 2x y r+ = . Why?

A circle centered at the origin with radius 1r= is called a unit circle and has the equation 2 2 1x y+ = .

Example: Write the standard form of the equation of the circle with radius 4 and center ( )2, 4 .

Standard Form of theEquation of a Circle

( ) ( )2 2 2x h y k r + =


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Graphing Circles by Hand and Using a Graphing Utility

Example: Graph the equation ( ) ( )2 2

4 3 25x y + + = .

By hand: By calculator:

The General Form of the Equation of a Circle is 2 2 0x y ax by c+ + + + = . In this form its difficult, ifnot impossible, to recognize the center and radius of the circle. Convert the equation of a circle fromgeneral form to standard form by completing the square in bothx andy to find the center and radius.

Example: Graph the equation 2 2 2 4 17 0x y x y+ + = .



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Example: Find the standard equation of the circle with center at ( )2,3 and whose graph contains the

point ( )1,4 .



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College Algebra

Section 1.4 Linear Equations in Two Variables

The ratio of the dimensions of the width and height of your calculator screen is approximately 3:2.Keeping this in mind as we discuss lines, be aware of the Zoom Square function that will adjust your

viewing window such that the length of a unit in thex-direction is equal to a unit in the y-direction.Compare the graph of y x= in a standard window to the graph in a square window.

Equations of Vertical and Horizontal Lines

A vertical line is given by an equation of the form x a= where a is the x-intercept.

Example: Graph the equation 3x = .

A horizontal line is given by an equation of the form y b= where b is the y-intercept.

Example: Graph the equation 2y =

Point-Slope Form of a Line

An equation of a nonvertical line of slope m that contains the point ( )1 1,x y is ( )1 1y y m x x = .

Example: Find the equation of the line with slope4

5

and containing the point ( )3,2 .


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Section 1.4 Linear equations in Two Variables


Finding the Equation of a Line Given Two Points.

Example: Find an equation of the line L containing the points

( )1,4 and( )3, 1 . Graph the line L.

Slope-Intercept and Standard Form of the Equation of a Line

The Slope-Intercept Form of an equation of line L is y mx b= + . In this equation, m is the slope andb

is the y-intercept.The equation of a line L is in Standard Form when it is written as Ax By C+ = whereA,B, andCare

real numbers andA andB are not both 0. (Usually A andB are whole numbers with 0A .)

Example: Find the slope-intercept and Standard forms of the equation of line L given by2

23

x y= + .

Then find the x- andy-intercepts and use them to graph the line.

Equations of Parallel and Perpendicular Lines

Example: Find the equations of the lines that contain the point ( )4, 3 and are (a) parallel and (b)

perpendicular to the line 3 6x y+ = . Graph the || and lines.


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College Algebra

Section 1.5 Scatter Diagrams and Curve Fitting

Linear functions have a constant rate of change. For linear functions of the form ( )f x mx b= + , theconstant rate of change is equal to m, the slope.

Notice that3

31

ym

x

= = =

and that this ratio is

a constant rate of change from any ( )( )1 1,x f x toany ( )( )2 2,x f x , as long as 1 2x x .

Example: The total private health expenditures H, in billions of dollars, is given by the function

( ) 26 411H t t= + , where tis the number of years since 1990.

(a)What was the total private health expenditures in2000 ( 10t = ) ?

(b)In what year will total private health expenditures be$879 billion?

(c)In what year will total private health expendituresexceed $1 trillion ($1000 billion) ?

Usually, data isnt neatly packaged and perfectly fitted to an equation. But we can often approximate thebehavior of data by fitting a graph to it. Well start by drawing a scatter diagram with the independentvariable plotted on the horizontal axis and the dependent variable plotted on the vertical axis. Lets doan example

Finding the line of best fit

A pediatrician wanted to estimate a linearfunction that relates a childs height, H(theindependent variable), to their headcircumference, C(the dependent variable). Sherandomly selects nine children from herpractice, measures their height and headcircumference, and obtains the data shown.

Height, H(inches)

25.2525.75

2527.7526.527

26.7526.7527.5

Head circumference, C(inches)

16.416.916.917.617.317.517.317.517.5


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Section 1.5 Linear Functions and Models Page 2


(a)Use a graphing utility to draw a scatter diagram.(b)Find the line of best fit to the data. Express the

equation in function notation.

(c)Interpret the slope.

(d)Predict the head circumference of a child that is 26 inches tall.(e)What is the height of a child whose head circumference is 17.4 inches?

Its probably not surprising that data is not always linear. Data, and their resulting scatter diagrams,form a variety of different relationships. For now, well learn to distinguish linear from nonlinearrelations.

0 .0

1 . 0

2 . 0

3 . 0

4 . 0

5 . 0

6 . 0

7 . 0

-2 . 5 -2 -1 . 5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

0 .0

1 . 0

2 . 0

3 . 0

4 . 0

5 . 0

6 . 0

7 . 0

8 . 0

9 . 0

1 0 . 0

-2 . 5 -2 -1 . 5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 0 . 0

-8 . 0

-6 . 0

-4 . 0

-2 . 0

0 .0

2 . 0

4 . 0

6 . 0

8 . 0

1 0 . 0

-2 . 5 -2 -1 . 5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 2 .0

-1 0 . 0

-8 . 0

-6 . 0

-4 . 0

-2 . 0

0 .0

2 . 0

4 . 0

6 . 0

8 . 0

-2 . 5 -2 -1 . 5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 5 .0

-1 0 . 0

-5 . 0

0 .0

5 . 0

1 0 . 0

1 5 .0

-2 . 5 -2 -1 . 5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 5 .0

-1 0 . 0

-5 . 0

0 .0

5 . 0

1 0 . 0

1 5 .0

-2 . 5 -2 -1 . 5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

Determine whether the relationship between the two variables is linear or nonlinear.


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College Algebra

Section 1.6 Quadratic Equations

A quadratic equation is an equation equivalent to one of the form 2 0ax bx c+ + = . There are a variety of

ways to solve quadratic equations. Following are four algebraic methods for solving quadratic

equations. All solutions will be verified using a graphing utility.

Solving Quadratic Equations by Factoring

When a quadratic equation is written in the form 2 0ax bx c+ + = , it is often possible to factor the leftside as the product of two linear factors. By setting each factor equal to zero and solving, we obtain theexactsolutions of the quadratic equation.

Example: Solve 215 29 12 0x x+ + =

Solving Quadratic Equations Using the Square Root Method

Recall, if 2x p= and 0p , then x p= orx p= . If 0p > , we usually abbreviate these solutions

as x p= .

Example: Solve 225 16 40x x+ =

Solving Quadratic Equations by Completing the SquareThe idea behind the method of completing the square is to modify the left side of a quadratic equation sothat it becomes a perfect square. We do this by adding the same constant to both sides of the equation.When the left side is a perfect square, we can use the square root method to solve the quadratic equation.

Example: Solve 22 3 1 0x x =


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Section 1.6 Quadratic Equations Page 2


Solving Quadratic Equations Using the Quadratic Formula

If 0a > , we can use the method of completing the square to obtain a general formula for solving the

quadratic equation 2 0ax bx c+ + = . This derivation is presented in your book.

The solutions of 2 0ax bx c+ + = are2 4

2

b b acx

a

= . 2 4b ac is called the discriminant and yields

information concerning the number and type of solutions to the quadratic equation. Do you remember?

If 2 4 0b ac > , then

If 2 4 0b ac = , then

If 2 4 0b ac < , then

Example: Solve 24 6 9 0u u + =

Example: An object is propelled vertically upward with an initial velocity of 20 meters per second. The

distance s (in meters) of the object from the ground aftertseconds is 24.9 20s t t= + .

a) When will the object be 15 meters above the ground?

b) When will it strike the ground?

c) Will the object reach a height of 100 meters?


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College Algebra

Section 1.7 Linear and Absolute Value Inequalities

Remember, when solving inequalities, multiplying (or dividing) by a negative number reverses thedirection of the inequality. In this section we will solve inequalities both algebraically and graphically.

Solving Linear Inequalities

Example: Solve 5 7 3 1x x + and graph the solution set.

Solving Compound Inequalities

Example: Solve 4 5 1 11x < + < and graph the solution set.


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Section 1.7 Linear and Absolute Value Inequalities Page 2


Solving Absolute Value Inequalities

Ifa is any positive number and ifu is any algebraic expression,

then u a< is equivalent to a u a < < . (This relationship also holds for .)

Example: Solve 2 4 3x + and graph the solution set.

Ifa is any positive number and ifu is any algebraic expression,

then u a> is equivalent to oru a u a< > . (This relationship also holds for .)

Example: Solve 2 5 3x > and graph the solution set.

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