Finite Mathematics – Chapter 1
Section 1.2 – Straight Lines
The equation of a horizontal line is of the form (namely ), since .
The equation of a vertical line is of the form (namely the -intercept of the line).
Slope of a Vertical Line
u Let L denote the unique straight line that passes through the two distinct points (x1, y1) and (x2, y2).
u If x1 = x2, then L is a vertical line, and the slope is undefined.
Slope of a Nonvertical Line
u If (x1, y1) and (x2, y2) are two distinct points on a nonvertical line L, then the slope m of L is given by
If the graph of a function rises from left to right, it is said to be increasing.
u 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
n The horizontal line is called the x-axis.
n The vertical line is called the y-axis.
n The point of intersection is the origin.
Plotting Points
n Each point in the xy-plane corresponds to a unique ordered pair (a, b).
n Plot the point (2, 4). Starting from the origin:
Move 2 units right
Move 4 units up
Slope is a measure of the steepness of a line and is denoted by the letter .
If a nonvertical line passes through two distinct points and , then the slope of the line is given by .
Note: If a line rises from left to right, the slope is positive.
If a line falls from left to right, the slope is negative.
If a line is horizontal, the slope is zero.
If a line is vertical, the slope of the line is undefined. (i.e. A vertical line has no slope.)
The slope-intercept form of the equation of a nonvertical line is given by , where is the slope of the line, is the -intercept, and represents any point on the line.
Notice that the slope-intercept form is solved for .
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)
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 and .
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 and is perpendicular to the line .
Example - Find the equation of the line that passes through the point (3, 1) and is perpendicular to the line described by
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.
Slope-Intercept Form
The slope-intercept form of the equation of a nonvertical line is given by , where is the slope of the line, is the
-intercept, and represents any point on the line.
Notice that the slope-intercept form is solved for .
To graph a linear equation in slope-intercept form . . .
a) Identify the slope of the line and the y-intercept.
b) Plot the y-intercept.
c) Beginning at the point that you plotted, use the slope of the line to locate another point on the line.
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
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.
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.
Example - Let x and y denote the radius and area of a circle, respectively.
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
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.
Linear Function
The function f defined by y=mx+b
where m and b are constants, is called a linear function.
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 / 2010Expenditure / 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?
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
P(x) = R(x) - C(x)
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.
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
Example - Find the point of intersection of the straight lines that have equations y = x + 1 and y = – 2x + 4
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.
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.
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 we try to fit a straight line 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.
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)
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 / 5Expenditure, 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.
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