College Algebra Lial et al. Eleventh Edition ISBN 978-1-2922-38-9 9 781292 2389 College Algebra Lial Hornsb Schneider Daniels Eleventh Edition
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(Modeling) Cost, Revenue, and Profit Analsis A firm will break even (no profit and no loss) as long as revenue just equals cost. The value of (the number of items produced and sold) where C12 = R12 is called the break-even point. Assume that each of the following can be epressed as a linear function. Find (a) the cost function, (b) the revenue function, and (c) the profit function. (d) Find the break-even point and decide whether the product should be produced, given the restrictions on sales. See Eample 9. Fied Cost Variable Cost Price of Item 87. $ 5 $ 1 $ 35 No more than 18 units can be sold. 88. $27 $15 $28 No more than 25 units can be sold. 89. $165 $4 $35 All units produced can be sold. 9. $ 18 $ 11 $ 2 No more than 3 units can be sold. (Modeling) Break-Even Point The manager of a small compan that produces roof tile has determined that the total cost in dollars, C12, of producing units of tile is given b C12 = 2 + 1, while the revenue in dollars, R12, from the sale of units of tile is given b R12 = 24. 91. Find the break-even point and the cost and revenue at the break-even point. 92. Suppose the variable cost is actuall $22 per unit, instead of $2. How does this affect the break-even point? Is the manager better off or not? Quiz (Sections 1 4) 1. For A1-4, 22 and B1-8, -32, find d1a, B2, the distance between A and B. 2. Two-Year College Enrollment Enrollments in two-ear colleges for selected ears are shown in the table. Use the midpoint formula to estimate the enrollments for 22 and 26. Year Enrollment (in millions) 2 5.95 24 6.55 28 6.97 Source: U.S. Center for Education Statistics. 3. Sketch the graph of =- 2 + 4 b plotting points. 4. Sketch the graph of 2 + 2 = 16. 5. Determine the radius and the coordinates of the center of the circle with equation 2 + 2-4 + 8 + 3 =. 23
For Eercises 6 8, refer to the graph of ƒ12 = + 3. 6. Find ƒ1-12. f() = + 3 7. Give the domain and the range of ƒ. 8. Give the largest interval over which the function ƒ is (a) decreasing, (b) increasing, (c) constant. ( 1, 2) ( 5, 2) ( 3, ) 9. Find the slope of the line through the given points. (a) 11, 52 and 15, 112 (b) 1-7, 42 and 1-1, 42 (c) 16, 122 and 16, -42 1. Motor Vehicle Sales The graph shows a straight line segment that approimates new motor vehicle sales in the United States from 25 to 29. Determine the average rate of change from 25 to 29, and interpret the results. New Vehicle Sales Number of Vehicles (in thousands) 2, 15, 1, 5, 17,445 1,61 25 27 29 Year Source: U.S. Bureau of Economic Analsis. 5 Equations of Lines and Linear Models Point-Slope Form Slope-Intercept Form Vertical and Horizontal Lines Parallel and Perpendicular Lines Modeling Data Solving Linear Equations in One Variable b Graphing Point-Slope Form The graph of a linear function is a straight line. We now develop various forms for the equation of a line. Figure 43 shows the line passing through the fied point 1 1, 1 2 having slope m. (Assuming that the line has a slope guarantees that it is not vertical.) Let 1, 2 be an other point on the line. Since the line is not vertical, - 1. Now use the definition of slope. m = - 1-1 Slope formula (Section 4) m1-1 2 = - 1 Multipl each side b - 1. or - 1 = m1-1 2 Interchange sides. This result is the point-slope form of the equation of a line. An other point on (, ) Slope = m the line Point-Slope Form Figure 43 ( 1, 1 ) Fied point The point-slope form of the equation of the line with slope m passing through the point 1 1, 1 2 is 1 m1 1 2. 231
LOOKING AHEAD TO CALCULUS A standard problem in calculus is to find the equation of the line tangent to a curve at a given point. The derivative (see Looking Ahead to Calculus in Section 4 ) is used to find the slope of the desired line, and then the slope and the given point are used in the pointslope form to solve the problem. EXAMPLE 1 Using the Point-Slope Form (Given a Point and the Slope) Write an equation of the line through 1-4, 12 having slope -3. SOLUTION Here 1 =-4, 1 = 1, and m =-3. - 1 = m1-1 2 Point-slope form - 1 = -33-1-424 1 =-4, 1 = 1, m =-3-1 =-31 + 42 Be careful with signs. - 1 =-3-12 Distributive propert =-3-11 Add 1. Now Tr Eercise 23. EXAMPLE 2 Using the Point-Slope Form (Given Two Points) Write an equation of the line through 1-3, 22 and 12, -42. Write the result in standard form A + B = C. SOLUTION Find the slope first. m = -4-2 2-1-32 = - 6 Definition of slope 5 The slope m is - 6 5. Either 1-3, 22 o r 12, -42 can be used for 1 1, 1 2. We choose 1-3, 22. - 1 = m1-1 2 Point-slope form - 2 = - 6 5 3-1-324 1 =-3, 1 = 2, m =- 6 5 51-22 =-61 + 32 Multipl b 5. 5-1 =-6-18 Distributive propert 6 + 5 =-8 Standard form (Section 4) Verif that we obtain the same equation if we use 12, -42 instead of 1-3, 22 in the point-slope form. Now Tr Eercise 13. NOTE The lines in Eamples 1 and 2 both have negative slopes. Keep in mind that a slope of the form - A - A B ma be interpreted as either B or - A B. Slope-Intercept Form As a special case of the point-slope form of the equation of a line, suppose that a line passes through the point 1, b2, so the line has -intercept b. If the line has slope m, then using the point-slope form with 1 = and 1 = b gives the following. - 1 = m1-1 2 Point-slope form - b = m1-2 1 =, 1 = b 232 Slope = m + b Solve for. -intercept Since this result shows the slope of the line and the -intercept, it is called the slope-intercept form of the equation of the line.
Slope-Intercept Form The slope-intercept form of the equation of the line with slope m and -intercept b is m b. EXAMPLE 3 Finding the Slope and -Intercept from an Equation of a Line Find the slope and -intercept of the line with equation 4 + 5 =-1. SOLUTION Write the equation in slope-intercept form. 4 + 5 =-1 5 =-4-1 Subtract 4. = - 4-2 Divide b 5. 5 m b The slope is - 4 5 and the -intercept is -2. Now Tr Eercise 35. NOTE Generalizing from Eample 3, the slope m of the graph of A + B = C is - A B, and the -intercept b is C B. EXAMPLE 4 Using the Slope-Intercept Form (Given Two Points) Write an equation of the line through 11, 12 and 12, 42. Then graph the line using the slope-intercept form. SOLUTION In Eample 2, we used the point-slope form in a similar problem. Here we show an alternative method using the slope-intercept form. First, find the slope. m = 4-1 2-1 = 3 = 3 Definition of slope 1 Now substitute 3 for m in = m + b and choose one of the given points, sa 11, 12, to find the value of b. = m + b Slope-intercept form (, 2) (2, 4) (1, 1) Figure 44 = 3 2 changes 3 units changes 1 unit The slope-intercept form is 1 = 3112 + b m = 3, = 1, = 1 -intercept b = -2 Solve for b. = 3-2. The graph is shown in Figure 44. We can plot 1, -22 and then use the definition of slope to arrive at 11, 12. Verif that 12, 42 also lies on the line. Now Tr Eercise 13. 233
EXAMPLE 5 Finding an Equation from a Graph Use the graph of the linear function ƒ shown in Figure 45 to complete the following. (a) Find the slope, -intercept, and -intercept. (b) Write the equation that defines ƒ. SOLUTION (a) The line falls 1 unit each time the -value increases b 3 units. Therefore, the slope is - 1 3 =- 1 3. The graph intersects the -ais at the point 1, -12 and intersects the -ais at the point 1-3, 2. Therefore, the -intercept is -1 and the -intercept is -3. (b) The slope is m =- 1 3, and the -intercept is b =-1. = ƒ12 = m + b Slope-intercept form 3 1 1 3 = f() Figure 45 ƒ12 =- 1 3-1 m =- 1 3, b =-1 Now Tr Eercise 39. Vertical line = 3 ( 3, 1) (a) Vertical and Horizontal Lines We first saw graphs of vertical and horizontal lines in Section 4. The vertical line through the point 1a, b2 passes through all points of the form 1a, 2, for an value of. Consequentl, the equation of a vertical line through 1a, b2 is = a. For eample, the vertical line through 1-3, 12 has equation =-3. See Figure 46 (a). Since each point on the -ais has -coordinate, the equation of the -ais is. The horizontal line through the point 1a, b2 passes through all points of the form 1, b2, for an value of. Therefore, the equation of a horizontal line through 1a, b2 i s = b. For eample, the horizontal line through 11, -32 has equation =-3. S e e Figure 46 (b). Since each point on the -ais has -coordinate, the equation of the -ais is. 1 Horizontal = 3 line (1, 3) (b) Figure 46 Equations of Vertical and Horizontal Lines An equation of the vertical line through the point 1a, b2 is a. An equation of the horizontal line through the point 1a, b2 is b. Parallel and Perpendicular Lines Since two parallel lines are equall steep, the should have the same slope. Also, two distinct lines with the same steepness are parallel. The following result summarizes this discussion. (The statement p if and onl if q means if p then q and if q then p. ) Parallel Lines Two distinct nonvertical lines are parallel if and onl if the have the same slope. 234