Finite Mathematics Chapter 1 ' y y yfaculty.southwest.tn.edu/hprovinc/content/materials... · ·...

1 | P a g e Province – Mathematics Department – Southwest Tennessee Community College

Finite Mathematics – Chapter 1

Section 1.2 – Straight Lines

The equation of a horizontal line is of the form #y (namely b ), since 0m .

The equation of a vertical line is of the form #x (namely the x -intercept of the line).

Slope of a Vertical Line

Let L denote the unique straight line that passes through the two distinct points (x1, y1) and (x2,

y2).

If x1 = x2, then L is a vertical line, and the slope is undefined.

Slope of a Horizontal Line

Let L denote the unique horizontal line that passes through the two distinct points (x1, y1) and

(x2, y2).

If y1 = y2, then L is a horizontal line, and the slope is zero.

Slope of a Nonvertical Line

If (x1, y1) and (x2, y2) are two distinct points on a nonvertical line L, then the slope m of L is given

by

2 1

2 1

y y ym

x x x


If the graph of a function rises from left to right, it is said to be increasing.

If m > 0, the line slants upward from left to right.

If the graph of a function falls from left to right, it is said to be decreasing.

• If m < 0, the line slants downward from left to right.

Example - Sketch the straight line that passes through the point (2, 5) and has slope – 4/3.


Rectangular Coordinate System

The horizontal line is called the x-axis.

The vertical line is called the y-axis.

The point of intersection is the origin. (x,y) =(0,0)

Plotting Points

Each point in the xy-plane corresponds to a unique

ordered pair (a, b).

Plot the point (2, 4). Starting from the origin:

Move 2 units right

Move 4 units up

Plot means to show the location of a point on the rectangular coordinate system.

Ordered Pair: ( x , y )

x: is the x-coordinate (move on the x-axis) 1st coordinate

y: is the y-coordinate (move on the y-axis) 2nd coordinate

Example: Plot the points

A: (0,4)

B: (3,2)

C: (-2,5)

D: (-4, -5)

E: (2, -4)


Slope is a measure of the steepness of a line and is denoted by the letter m .

If a nonvertical line passes through two distinct points 11 , yx and 22 , yx , then the slope of the

line is given by 12

12

xx

yy

run

rise

xinchangethe

yinchangethem

.

The ratio of the vertical change to the horizontal change for any two points on the line.

Types of Slope

Positive slope rises from left to right.

Negative slope falls from left to right.

The slope of a vertical line is undefined.

(i.e. A vertical line has no slope.)

The slope of a horizontal line is zero.

The slope-intercept form of the equation of a nonvertical line is given by bmxy , where m is

the slope of the line, b is the y -intercept, and yx , represents any point on the line.

Notice that the slope-intercept form is solved for y .

Example - Find the slope of (3, 7) and (5, 1)

Example - What is the slope and y-intercept of y = 3x + 8

Example - Find the slope of (0, 1) and (6, 8)

2 1

2 1

vertical changeSlope =

horizontal change

y ym

x x


Note that vertical lines are parallel to vertical lines and perpendicular to horizontal lines.

Note that horizontal lines are parallel to horizontal lines and perpendicular to vertical lines.

Parallel Lines

Two distinct lines are parallel if and only if their slopes are equal or their slopes are undefined

Example - Let L1 be a line that passes through the points (–2, 9) and (1, 3), and let L2 be the line that

passes through the points (– 4, 10) and (3, – 4). Determine whether L1 and L2 are parallel

Point-Slope Form

You will be given at least one point that the line passes through as well as enough information

to find the slope of the line (if it is not also given). You can then substitute this information into the

point- slope form and finally solve for y in order to get the equation in slope-intercept form.

Point Slope Form of a Linear Equation

y – y1 = m(x – x1)

where x1 and y1 are coordinates of the known point, m is the slope of the line and x and y are the

variables of the equation

To Solve:

1. Use point slope form of a linear equation y – y1 = m(x – x1) 2. Substitute known values for x, y and m 3. Rearrange the equation to be in standard form (ax +by = c) or slope intercept form

(y = mx + b)


Example - Find the equation of the line in slope-intercept form.

Find an equation of the line that passes through the point (1, 3) and has slope 2

The line passes through )3,1( and )2,2( .

Find an equation of the line that passes through the points (–3, 2) and (4, –1).


Perpendicular Lines

If L1 and L2 are two distinct nonvertical lines that have slopes m1 and m2, respectively, then L1 is

perpendicular to L2 (written L1 ┴ L2) if and only if

Example - The line passes through )2,3( and is perpendicular to the line 84 yx .

Example - Find the equation of the line that passes through the point (3, 1) and is perpendicular to the

line described by

1

2

1m

m 1

2

1m

m

3 2( 1)y x


Crossing the Axis

A straight line L that is neither horizontal nor vertical cuts the x-axis and the y-axis at, say, points (a, 0)

and (0, b), respectively.

The numbers a and b are called the x-intercept and y-intercept, respectively, of L.

Example: Graph 3x + 2y = 6

Find the x-intercept.

Find the y-intercept.

X-Intercepts: Let y=0 then solve the point it

will be (#, 0)

Y-intercepts: Let x=0 then solve the point it

will be (0, #)


Slope-Intercept Form

The slope-intercept form of the equation of a nonvertical line is given by bmxy , where m is

the slope of the line, b is the y -intercept, and yx , represents any point on the line.

Notice that the slope-intercept form is solved for y .

Graphing Equations by Using the Slope and y-Intercept

Solve the equation for y to place the equation in slope-intercept form.

Determine the slope and y-intercept from the equation.

Plot the y-intercept.

Obtain a second point using the slope. Beginning at the point that you plotted, use the slope of

the line

run

rise to locate another point on the line.

Draw a straight line through the points.

Example - Find the equation of the line that has slope 3 and y-intercept of – 4

Example - Determine the slope and y-intercept of the line whose equation is 3x – 4y = 8.


Applied Example

Suppose an art object purchased for $50,000 is expected to appreciate in value at a constant rate of

$5000 per year for the next 5 years.

Write an equation predicting the value of the art object for any given year.

What will be its value 3 years after the purchase?

General Form of a Linear Equation

The equation Ax + By + C = 0 where A, B, and C are constants and A and B are not both zero, is called

the general form of a linear equation in the variables x and y.

An equation of a straight line is a linear equation; conversely, every linear equation represents a straight

line.

Example - Sketch the straight line represented by the equation 3x – 4y – 12 = 0


Equations of Straight Lines

Vertical line: x = a

Horizontal line: y = b

Point-slope form: y – y1 = m(x – x1)

Slope-intercept form: y = mx + b

General Form: Ax + By + C = 0


Section 1.3 - Linear Functions and Mathematical Models

Mathematical Modeling

Mathematics can be used to solve real-world problems.

Regardless of the field from which the real-world problem is drawn, the problem is analyzed using a

process called mathematical modeling.

Domain and Range

The domain, D, of a relation is the set of all first coordinates of the ordered pairs in the relation (the x’s).

The range, R, of a relation is the set of all second coordinates of the ordered pairs in the relation (the

y’s).

In graphing relations, the horizontal axis is called the domain axis and the vertical axis is called the range

axis.

The domain and range of a relation can often be determined from the graph of the relation.

**If the domain or range consists of a finite number of points, use braces and set notation.

**If the domain or range consists of intervals of real numbers, use interval (or inequality) notation.

Ex: State the domain and range of the relation. {(-1,1), (1,5), (0,3)}

Domain:

Range:


Functions

• A function f is a rule that assigns to each value of x one and only one value of y.

• The value y is normally denoted by f(x), emphasizing the dependency of y on x.

• A function is a special kind of relation that pairs each element of the domain with one and only one element of the range. (For every x there is exactly one y.) A function is a correspondent between a first set, domain, and a second set, range.

• In a function no two ordered pairs have the same first coordinate. That is, each first coordinate appears only once.

• Although every function is by definition a relation, not every relation is a function.

Example – Which of the following relations are functions?

)5,1(),0,3(),8,2( ,

)5,1(),5,3(),5,2( ,

)0,2(),0,3(),5,2(

To determine whether or not the graph of a relation represents a function, we apply the vertical line

test which states that if any vertical line intersects the graph of a relation in more than one point, then

the relation graphed is not a function.

Ex. Is the relation a function?


Function notation and evaluating functions (finding range values) . . .

EXAMPLE: 32)( xxf is read “ f of x is equal to 32 x ”.

f is the name of the function.

x is representative of an element in the domain of f .

)(xf is representative of an element of the range of f , and means the same as y .

32 x is the function rule.

EVALUATE: )5(f

($)f

)1(xf

Mathematical Modeling

Mathematical modeling is the process of using mathematics to solve real-world problems. This process

can be broken down into three steps:

1. Construct the mathematical model, a problem whose solution will provide information about the

real-world problem.

2. Solve the mathematical model.

3. Interpret the solution to the mathematical model in terms of the original real-world problem.

In this section we will discuss one of the simplest mathematical models, a linear equation.


Example - Let x and y denote the radius and area of a circle, respectively. Where x = radius=r and y=area

of circle=

From elementary geometry we have y = x2

• This equation defines y as a function of x, since for each admissible value of x there corresponds

precisely one number y = πx2 giving the area of the circle.

• The area function may be written as f(x) = πx2

To compute the area of a circle with a radius of 5 inches, we simply replace x in the equation by the

number 5: f(5) = π(52)= 25π

Linear Function

The function f defined by y=mx+b where m and b are constants, is called a linear function. Linear

because the exponent on the variable is 1.

A first degree, or linear, equation in one variable is any equation that can be written in the form

y=mx+b where m is not equal to zero.

To graph functions using a graphing calculator.

Step 1: Hit the “y=” button (purple) located under the screen on the left

Step2: You will see

y1=

y2=

…

You can enter your equation now

For instance if we wanted to graph y=2x+3 then you would enter 2x+3 on this screen.

Step 3: Hit enter

Step 4: Hit the “Graph” button (purple) located under the screen on the right. This step will graph the

function for you.

Note if you cannot see your graph then your window settings are not set correctly. You need to hit the

“window” button (purple) located under the window. You should have the x and y max be 10 and the x

and y min be -10, the increment should be 1.

You can also evaluate function values after you have entered your function into the “y1=”.

Let’s say your function is f(x) = 2x+3 and you have this saved in “y1=” then you can determine f(30) by

simply choosing “y1” hit enter open parenthesis then 30 then close parenthesis then enter and your

calculator will calculate this for you. the answer it will give you is 63.


Example - Applied Example: U.S. Health-Care Expenditures

Because the over-65 population will be growing more rapidly in the next few decades, health-care

spending is expected to increase significantly in the coming decades.

The following table gives the projected U.S. health-care expenditures (in trillions of dollars) from 2005

through 2010:

Year 2005 2006 2007 2008 2009 2010

Expenditure 2.00 2.17 2.34 2.50 2.69 2.90

A mathematical model giving the approximate U.S. health-care expenditures over the period in question

is given by where t is measured in years, with t = 0 corresponding to 2005.

a. Sketch the graph of the function S and the given data on the same set of axes.

b. Assuming that the trend continues, how much will U.S. health-care expenditures be in

2011?

c. What is the projected rate of increase of U.S. health-care expenditures over the period

in question?

( ) 0.178 1.989S t t ( ) 0.178 1.989S t t


Cost, Revenue, and Profit Functions

• Let x denote the number of units of a product manufactured or sold.

• Then, the total cost function is

C(x) = Total cost of manufacturing x units of the product

• The revenue function is

R(x) = Total revenue realized from the sale of x units of the product

• The profit function is

P(x) = Total profit realized from manufacturing and selling x units of the product

Example - Applied Example: Profit Function

Puritron, a manufacturer of water filters, has a monthly fixed cost of $20,000, a production cost of $20

per unit, and a selling price of $30 per unit.

Find the cost function, the revenue function, and the profit function for Puritron.

Example: The price-demand function for a company is given by

where x represents the number of items and P(x) represents the price of the item. Determine the

revenue function and find the revenue generated if 50 items are sold.


Section 1.4 - Intersections of Straight Lines

Finding the Point of Intersection

• Suppose we are given two straight lines L1 and L2 with equations

y = m1x + b1 and y = m2x + b2

(where m1, b1, m2, and b2 are constants) that intersect at the point P(x0, y0).

• The point P(x0, y0) lies on the line L1 and so satisfies the equation y = m1x + b1.

• The point P(x0, y0) also lies on the line L2 and so satisfies y = m2x + b2 as well.

• Therefore, to find the point of intersection P(x0, y0) of the lines L1 and L2, we solve for x and y the

system composed of the two equations

y = m1x + b1 and y = m2x + b2

Technology Help: Page 40

Example - Find the point of intersection of the straight lines that have equations y = x + 1 and y = – 2x + 4


Definition: The Break-even point P(x, y) is just the point of intersection of the straight lines representing

the cost and revenue functions. Page 41

Example - Applied Example: Break-Even Level

• Prescott manufactures its products at a cost of $4 per unit and sells them for $10 per unit.

• If the firm’s fixed cost is $12,000 per month, determine the firm’s break-even point.


Definition: The market equilibrium is a situation in which the supply of an item is exactly equal to its

demand. Since there is neither surplus nor shortage in the market, price tends to remain stable in this

situation. Page 41

Example - Applied Example: Market Equilibrium

The management of ThermoMaster, which manufactures an indoor-outdoor thermometer at its Mexico

subsidiary, has determined that the demand equation for its product is where p is

the price of a thermometer in dollars and x is the quantity demanded in units of a thousand.

The supply equation of these thermometers is where x (in thousands) is the

quantity that ThermoMaster will make available in the market at p dollars each.

Find the equilibrium quantity and price.

5 3 30 0x p 5 3 30 0x p

52 30 45 0x p 52 30 45 0x p


Break-Even and Profit-Loss Analysis

Any manufacturing company has costs C and revenues R.

The company will have a loss if R < C, will break even if R = C, and will have a profit if R > C.

Costs include fixed costs such as plant overhead, etc. and variable costs, which are dependent

on the number of items produced. C = a + bx (x is the number of items produced)

Price-demand functions, usually determined by financial departments, play an important role in

profit-loss analysis. p = m – nx

(x is the number of items than can be sold at $p per item.)

The revenue function is R = (number of items sold) ∙ (price per item)

= xp = x(m – nx) The profit function is P = R – C = x(m – nx) – (a + bx)

Example of Profit-Loss Analysis

A company manufactures notebook computers. Its marketing research department has determined

that the data is modeled by the price-demand function p(x) = 2,000 – 60x, when 1 < x < 25, (x is in

thousands). What is the company’s revenue function and what is its domain?


Profit Problem

The financial department for the company in the preceding problem has established the following cost

function for producing and selling x thousand notebook computers:

C(x) = 4,000 + 500x (x is in thousand dollars).

Write a profit function for producing and selling x thousand notebook computers, and indicate the

domain of this function.


Section 1.5 - The Method of Least Squares

In this section, we describe a general method known as the method for least squares for determining a

straight line that, in a sense, best fits a set of data points when the points are scattered about a straight

line.

Suppose we are given five data points P1(x1, y1), P2(x2, y2), P3(x3, y3), P4(x4, y4), and P5(x5, y5) describing the

relationship between two variables x and y.

By plotting these data points, we obtain a scatter diagram:

Suppose the plot of y vs. x shows a straight line (linear) relationship.


Suppose we try to fit a straight line, “best” fit , L to the data points P1, P2, P3, P4, and P5.

The line will miss these points by the amounts d1, d2, d3, d4, and d5 respectively.

The principle of least squares states that the straight line L that fits the data points best is the one

chosen by requiring that the sum of the squares of d1, d2, d3, d4, and d5, that is

be made as small as possible.

Suppose we are given n data points:

P1(x1, y1), P2(x2, y2), P3(x3, y3), . . . , Pn(xn, yn)

Then, the least-squares (regression) line for the data is given by the linear equation

y = f(x) = mx + b where the constants m and b satisfy the equations

and

simultaneously.

These last two equations are called normal equations.

Linear Regression

• Linear Regression - a mathematical technique for creating a linear model for paired data.

• Based on the “least-squares” criterion of best fit

In real world applications we often encounter numerical data in the form of a table. The powerful

mathematical tool, regression analysis, can be used to analyze numerical data. In general, regression

analysis is a process for finding a function that best fits a set of data points. In the next example, we use

a linear model obtained by using linear regression on a graphing calculator.

2 2 2 2 2

1 2 3 4 5d d d d d 2 2 2 2 2

1 2 3 4 5d d d d d

1 2 3 1 2 3( )n nx x x x m nb y y y y 1 2 3 1 2 3( )n nx x x x m nb y y y y

2 2 2 2

1 2 3 1 2 3

1 1 2 2 3 3

( ) ( )n n

n n

x x x x m x x x x b

y x y x y x y x

2 2 2 2

1 2 3 1 2 3

1 1 2 2 3 3

( ) ( )n n

n n

x x x x m x x x x b

y x y x y x y x

Use first

Use second


Example - Find the equation of the least-squares line for the data

P1(1, 1), P2(2, 3), P3(3, 4), P4(4, 3), and P5(5, 6)


A regression line is a straight line that describes how the average response value varies as the

explanatory variable (x) changes. We can use the regression line to predict the value of y at a given x.

Equation of a straight line

where b= the y-intercept (values of y when x=0) anda=slope of the line (change in y for an event change

in x)

If we know a and b we can predict y for a given value of x. How accurate the prediction is depends on

how much scatter there is in the data about the line.

Calculator Linear Regression:

2nd zero (Catalog) scroll down to Diagnostic On then press Enter, then Enter

Stat⟶ Edit⟶ then enter your data into L1, and L2

Stat⟶ Calc ⟶ LinReg(ax+b) Option #4.

Follow this set of steps to enter your data:

1. Press [STAT].

2. EDIT should be highlighted.

3. Press [ENTER].

4. You should be looking a screen that will allow you to put the data into a list.

5. The x-variable into L1 and the y-variable into L2. Pressing enter after each piece will take you to

the next position in the list.

6. QUIT ([2nd] [MODE]) when you have entered all the data values.

Follow this set of steps to find the Linear Regression line

1. To display the linear regression information, press [STAT].

2. Arrow over to CALC.

3. Arrow down to 4:LinReg(ax+b) and press [ENTER], or press 4 and then press 2nd [1] [,] 2nd [2] (i.e.

where you put your data.

4. The screen will display LinReg and under that y=ax+b. This is so that you will recognize which

variable is the slope and which is the y-intercept.

5. Underneath y=ax+b will be the values for a, b, r (the sample correlation coefficient), and (the

sample coefficient of determination).


Example - Applied Example: U.S. Health-Care Expenditures

Because the over-65 population will be growing more rapidly in the next few decades, health-care

spending is expected to increase significantly in the coming decades.

The following table gives the U.S. health expenditures (in trillions of dollars) from 2005 through 2010:

Year, t 0 1 2 3 4 5

Expenditure, y 2.00 2.17 2.34 2.50 2.69 2.90

Find a function giving the U.S. health-care spending between 2005 and 2010, using the least-squares

technique.

Example of Linear Regression

Prices for emerald-shaped diamonds taken from an on-line trader are given in the following table. Find

the linear model that best fits this data.

Weight (carats) Price

0.5 $1,677

0.6 $2,353

0.7 $2,718

0.8 $3,218

0.9 $3,982


Slope as a Rate of Change

If x and y are related by the equation y = mx + b, where m and b are constants with m not equal to zero,

then x and y are linearly related. If (x1, y1) and (x2, y2) are two distinct points on this line, then the slope

of the line is

This ratio is called the rate of change of y with respect to x. Since the slope of a line is unique, the rate

of change of two linearly related variables is constant. Some examples of familiar rates of change are

miles per hour, price per pound, and revolutions per minute.

Example of Rate of Change: Rate of Descent

Parachutes are used to deliver cargo to areas that cannot be reached by other means of conveyance.

The rate of descent of the cargo is the rate of change of altitude with respect to time. The absolute

value of the rate of descent is called the speed of the cargo. At low altitudes, the altitude of the cargo

and the time in the air are linearly related. If a linear model relating altitude a (in feet) and time in the

air t (in seconds) is given by a = –14.1t +2,880, how fast is the cargo moving when it lands?

2 1

2 1

y y ym

x x x

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