Finite Mathematics Chapter 1

Transcription

1 Finite Mathematics Chapter 1 Section 1.2 Straight Lines The equation of a horizontal line is of the form y # (namely b ), since m 0. 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 (x 1, y 1 ) and (x 2, y 2 ). If x 1 = x 2, 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 (x 1, y 1 ) and (x 2, y 2 ). If y 1 = y 2, then L is a horizontal line, and the slope is zero. Slope of a Nonvertical Line If (x 1, y 1 ) and (x 2, y 2 ) are two distinct points on a nonvertical line L, then the slope m of L is given by y y y m x x x P a g e

2 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. 2 P a g e

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) 1 st coordinate y: is the y-coordinate (move on the y-axis) 2 nd coordinate Example: Plot the points A: (0,4) B: (3,2) C: (-2,5) D: (-4, -5) E: (2, -4) 3 P a g e

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 x 1, y 1 and x 2, y 2, then the slope of the line is given by m the change in y rise 2 1. the change in x run x2 x1 The ratio of the vertical change to the horizontal change for any two points on the line. vertical change Slope = horizontal change y y Types of Slope y y m x x 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 the slope of the line, b is the y -intercept, and Notice that the slope-intercept form is solved for y. y mx b, where m is x, y represents any point on the line. 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) 4 P a g e

5 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 L 1 be a line that passes through the points ( 2, 9) and (1, 3), and let L 2 be the line that passes through the points ( 4, 10) and (3, 4). Determine whether L 1 and L 2 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 y 1 = m(x x 1 ) where x 1 and y 1 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 y 1 = m(x x 1 ) 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) 5 P a g e

6 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 ( 1, 3) and ( 2, 2). Find an equation of the line that passes through the points ( 3, 2) and (4, 1). 6 P a g e

7 Perpendicular Lines If L 1 and L 2 are two distinct nonvertical lines that have slopes m 1 and m 2, respectively, then L 1 is perpendicular to L 2 (written L 1 L 2 ) if and only if 1 m 1 m Example - The line passes through ( 3, 2) 2 and is perpendicular to the line 4 y 8 x. Example - Find the equation of the line that passes through the point (3, 1) and is perpendicular to the line described by y 3 2( x 1) 7 P a g e

8 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. 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, #) Example: Graph 3x + 2y = 6 Find the x-intercept. Find the y-intercept. 8 P a g e

9 Slope-Intercept Form The slope-intercept form of the equation of a nonvertical line is given by the slope of the line, b is the y -intercept, and Notice that the slope-intercept form is solved for y. y mx b x, y represents any point on the line., where m is 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 rise to locate another point on the line. run 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. 9 P a g e

10 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 10 P a g e

11 Equations of Straight Lines Vertical line: x = a Horizontal line: y = b Point-slope form: y y 1 = m(x x 1 ) Slope-intercept form: y = mx + b General Form: Ax + By + C = 0 11 P a g e

12 Section 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: 12 P a g e

13 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? ( 2, 8),(3, 0),( 1, ( 2, 5),(3, 5),( 1, (2, 5) 5) 5),(3, 0),(2, 0),, 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? 13 P a g e

14 Function notation and evaluating functions (finding range values)... EXAMPLE: ( x) 2x 3 f is read f of x is equal to 2 3 f is the name of the function. x is representative of an element in the domain of f. x. f (x) is representative of an element of the range of f, and means the same as y. 2x 3 is the function rule. EVALUATE: f ( 5) f ($) f (x 1) 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. 14 P a g e

15 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 = x 2 This equation defines y as a function of x, since for each admissible value of x there corresponds precisely one number y = πx 2 giving the area of the circle. The area function may be written as f(x) = πx 2 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(5 2 )= 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 P a g e

16 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 Expenditure A mathematical model giving the approximate U.S. health-care expenditures over the period in question is given by S ( t ) t where t is measured in years, with t = 0 corresponding to 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? 16 P a g e

17 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. 17 P a g e

18 Section Intersections of Straight Lines Finding the Point of Intersection Suppose we are given two straight lines L 1 and L 2 with equations y = m 1 x + b 1 and y = m 2 x + b 2 (where m 1, b 1, m 2, and b 2 are constants) that intersect at the point P(x 0, y 0 ). The point P(x 0, y 0 ) lies on the line L 1 and so satisfies the equation y = m 1 x + b 1. The point P(x 0, y 0 ) also lies on the line L 2 and so satisfies y = m 2 x + b 2 as well. Therefore, to find the point of intersection P(x 0, y 0 ) of the lines L 1 and L 2, we solve for x and y the system composed of the two equations y = m 1 x + b 1 and y = m 2 x + b 2 Technology Help: Page 40 Example - Find the point of intersection of the straight lines that have equations y = x + 1 and y = 2x P a g e

19 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. 19 P a g e

20 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 5 x 3 p 30 0 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 52 x 30 p 45 0 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. 20 P a g e

21 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? 21 P a g e

22 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, x (x is in thousand dollars). Write a profit function for producing and selling x thousand notebook computers, and indicate the domain of this function. 22 P a g e

23 Section 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 P 1 (x 1, y 1 ), P 2 (x 2, y 2 ), P 3 (x 3, y 3 ), P 4 (x 4, y 4 ), and P 5 (x 5, y 5 ) 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. 23 P a g e

24 Suppose we try to fit a straight line, best fit, L to the data points P 1, P 2, P 3, P 4, and P 5. The line will miss these points by the amounts d 1, d 2, d 3, d 4, and d 5 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 d 1, d 2, d 3, d 4, and d 5, that is d be made as small as possible. d d d d Suppose we are given n data points: P 1 (x 1, y 1 ), P 2 (x 2, y 2 ), P 3 (x 3, y 3 ),..., P n (x n, y n ) 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 ( x 1 x 2 x 3 x n) m nb y 1 y 2 y 3 y n Use second and Use first ( x x x x ) m ( x x x x ) b n y x y x y x y x simultaneously. n n n 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. 24 P a g e

25 Example - Find the equation of the least-squares line for the data P 1 (1, 1), P 2 (2, 3), P 3 (3, 4), P 4 (4, 3), and P 5 (5, 6) 25 P a g e

26 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: 2 nd zero (Catalog) scroll down to Diagnostic On then press Enter, then Enter Stat Edit then enter your data into L 1, and L 2 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). 26 P a g e

27 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 Expenditure, y 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, $2, $2, $3, $3, P a g e

28 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 (x 1, y 1 ) and (x 2, y 2 ) are two distinct points on this line, then the slope of the line is y2 y1 y m x x x 2 1 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? 28 P a g e