3.5 Equations of Lines

Transcription

1 Section. Equations of Lines 6. Professional plumbers suggest that a sewer pipe should be sloped 0. inch for ever foot. Find the recommended slope for a sewer pipe. (Source: Rules of Thumb b Tom Parker, Houghton Mifflin Compan). a. On a single screen, graph = +, = +, and = +. Notice the change in slope for each graph. b. On a single screen, graph = - +, = - +, and = - +. Notice the change in slope for each graph. c. Determine whether the following statement is true or false for slope m of a given line. As 0 m 0 becomes greater, the line becomes steeper.. Support the result of Eercise 67 b graphing the pair of equations on a graphing calculator.. Support the result of Eercise 70 b graphing the pair of equations on a graphing calculator. (Hint: Use the window showing [-, ] on the -ais and [-0, 0] on the -ais.). Equations of Lines S Graph a Line Using Its Slope and -Intercept. Use the Slope Intercept Form to Write the Equation of a Line. Use the Point Slope Form to Write the Equation of a Line. Write Equations of Vertical and Horizontal Lines. Find Equations of Parallel and Perpendicular Lines. Graphing a Line Using Its Slope and -Intercept In the last section, we learned that the slope intercept form of a linear equation is = m + b. Recall that when an equation is written in this form, the slope of the line is the same as the coefficient m of. Also, the -intercept of the line is 0, b. For eample, the slope of the line defined b = + is and its -intercept is 0,. We ma also use the slope intercept form to graph a linear equation. EXAMPLE Graph = -. Solution Recall that the slope of the graph of = - is and the -intercept is 0, -. To graph the line, we first plot the -intercept 0, -. To find another point on the line, we recall that slope is rise run =. Another point ma then be plotted b starting at 0, -, rising unit up, and then running units to the right. We are now at the point, -. The graph is the line through these two points. m ~ 6 Run Rise (, ) (0, ) 6 Notice that the line does have a -intercept of 0, - and a slope of. Graph = +. EXAMPLE Graph + =. Solution First, we solve the equation for to write it in slope intercept form. In slope intercept form, the equation is = - +. Net we plot the -intercept 0,. To find another point on the line, we use the slope -, which can be written as rise run = -. We start at 0, and move down units since the numerator of the slope

2 66 CHAPTER Graphs and Functions is -; then we move units to the right since the denominator of the slope is. We arrive at the point,. The line through these points is the graph, shown below to the left. Down (0, ) Right (, ) m s Left (, 6) 6 Up (0, ) m s The slope - can also be written as, so to find another point, we could start - at 0, and move up units and then units to the left. We would arrive at the point -, 6. The line through -, 6 and 0, is the same line as shown previousl through, and 0,. See the graph above to the right. Graph + = 6. Using Slope Intercept Form to Write Equations of Lines We ma also use the slope intercept form to write the equation of a line given its slope and -intercept. The equation of a line is a linear equation in variables that, if graphed, would produce the line described. EXAMPLE Write an equation of the line with -intercept 0, - and slope of. Solution We want to write a linear equation in variables that describes the line with -intercept 0, - and has a slope of. We are given the slope and the -intercept. Let m = and b = - and write the equation in slope intercept form, = m + b. = m + b = + - Let m = and b = -. = - Simplif. Write an equation of the line with -intercept (0, ) and slope of -. CONCEPT CHECK What is wrong with the following equation of a line with -intercept 0, and slope? = + Answer to Concept Check: -intercept and slope were switched, should be = +

3 Section. Equations of Lines 67 Using Point Slope Form to Write Equations of Lines When the slope of a line and a point on the line are known, the equation of the line can also be found. To do this, use the slope formula to write the slope of a line that passes through points, and,. We have m = - - Multipl both sides of this equation b - to obtain - = m - This form is called the point slope form of the equation of a line. Point Slope Form of the Equation of a Line The point slope form of the equation of a line is slope T - = m - æ point æ where m is the slope of the line and, is a point on the line. EXAMPLE Find an equation of the line with slope - containing the point, -. Write the equation in slope intercept form = m + b. Solution Because we know the slope and a point of the line, we use the point slope form with m = - and, =, -. - = m = = - + Point slope form Let m = - and, =, -. Appl the distributive propert. = - - Write in slope intercept form. In slope intercept form, the equation is = - -. Find an equation of the line with slope - containing the point -,. Write the equation in slope intercept form = m + b. Helpful Hint Remember, slope intercept form means the equation is solved for. EXAMPLE Find an equation of the line through points (, 0) and -, -. Write the equation using function notation. Solution First, find the slope of the line. m = = - -8 = 8 Net, use the point slope form. Replace, b either (, 0) or -, - in the point slope equation. We will choose the point (, 0). The line through (, 0) with slope 8 is - = m = = - 8 = - 0 Point slope form. Let m = 8 and, =, 0. Multipl both sides b 8. Appl the distributive propert.

4 68 CHAPTER Graphs and Functions To write the equation using function notation, we solve for, then replace with f. 8 = - 0 = Divide both sides b 8. f = 8 - Write using function notation. Find an equation of the line through points -, and (, 0). Write the equation using function notation. Helpful Hint If two points of a line are given, either one ma be used with the point slope form to write an equation of the line. Find an equation of the line graphed. Write the equation in stan- EXAMPLE 6 dard form. Solution First, find the slope of the line b identifing the coordinates of the noted points on the graph. The points have coordinates -, and (, ). - m = - - = Net, use the point slope form. We will choose (, ) for,, although it makes no difference which Point slope form Let m = and, =,. point we choose. The line through (, ) with slope is - = m - - = - - = = - 9 Multipl both sides b. Appl the distributive propert. To write the equation in standard form, move - and -terms to one side of the equation and an numbers (constants) to the other side. - 0 = = Subtract from both sides and add 0 to both sides. The equation of the graphed line is - + =. 6 Find an equation of the line graphed in the margin. Write the equation in standard form. The point slope form of an equation is ver useful for solving real-world problems. EXAMPLE 7 Predicting Sales Southern Star Realt is an established real estate compan that has enjoed constant growth in sales since 000. In 00, the compan sold 00 houses, and in 007, the compan sold 7 houses. Use these figures to predict the number of houses this compan will sell in the ear 06.

5 Section. Equations of Lines 69 Solution. UNDERSTAND. Read and reread the problem. Then let = the number of ears after 000 and = the number of houses sold in the ear corresponding to. The information provided then gives the ordered pairs (, 00) and (7, 7). To better visualize the sales of Southern Star Realt, we graph the linear equation that passes through the points (, 00) and (7, 7). 80 Number of houses sold (, 00) (7, 7) Years after 000. TRANSLATE. We write a linear equation that passes through the points (, 00) and (7, 7). To do so, we first find the slope of the line m = = = Then, using the point slope form and the point, 00 to write the equation, we have - = m = = - 0 = + 70 Let m = and, =, 00. Multipl. Add 00 to both sides.. SOLVE. To predict the number of houses sold in the ear 06, we use = + 70 and complete the ordered pair (6, ), since = 6. = Let = 6. = 0. INTERPRET. Check: Verif that the point (6, 0) is a point on the line graphed in Step. State: Southern Star Realt should epect to sell 0 houses in the ear Southwest Florida, including Fort Mers and Cape Coral, has been a growing real estate market in past ears. In 00, there were 7 house sales in the area, and in 006, there were 998 house sales. Use these figures to predict the number of house sales there will be in 0. Writing Equations of Vertical and Horizontal Lines Two special tpes of linear equations are linear equations whose graphs are vertical and horizontal lines.

6 70 CHAPTER Graphs and Functions EXAMPLE 8 Find an equation of the horizontal line containing the point (, ). Solution Recall that a horizontal line has an equation of the form = b. Since the line contains the point (, ), the equation is =, as shown to the right. (, ) 8 Find the equation of the horizontal line containing the point 6, -. EXAMPLE 9 Find an equation of the line containing the point (, ) with undefined slope. Solution Since the line has undefined slope, the line must be vertical. A vertical line has an equation of the form = c. Since the line contains the point (, ), the equation is =, as shown to the right. (, ) 9 Find an equation of the line containing the point 6, - with undefined slope. Finding Equations of Parallel and Perpendicular Lines Net, we find equations of parallel and perpendicular lines. EXAMPLE 0 Find an equation of the line containing the point (, ) and parallel to the line + = -6. Write the equation in standard form. Solution Because the line we want to find is parallel to the line + = -6, the two lines must have equal slopes. Find the slope of + = -6 b writing it in the form = m + b. In other words, solve the equation for. + = -6 = Subtract from both sides. = Divide b. = - - Write in slope intercept form. The slope of this line is -. Thus, a line parallel to this line will also have a slope of -. The equation we are asked to find describes a line containing the point (, ) with a slope of -. We use the point slope form.

7 Section. Equations of Lines 7 Helpful Hint Multipl both sides of the equation + = 0 b - and it becomes - - = -0. Both equations are in standard form, and their graphs are the same line. - = m - - = = = = 0 Let m = -, =, and =. Multipl both sides b. Appl the distributive propert. Write in standard form. 0 Find an equation of the line containing the point 8, - and parallel to the line + =. Write the equation in standard form. EXAMPLE Write a function that describes the line containing the point (, ) that is perpendicular to the line + = -6. Solution In the previous eample, we found that the slope of the line + = -6 is -. A line perpendicular to this line will have a slope that is the negative reciprocal of -, or. From the point slope equation, we have - = m - - = - - = = - = - = - f = - Let =, =, and m =. Multipl both sides b. Appl the distributive propert. Add 8 to both sides. Divide both sides b. Write using function notation. Write a function that describes the line containing the point 8, - that is perpendicular to the line + =. Forms of Linear Equations A + B = C = m + b - = m - = c = c Standard form of a linear equation A and B are not both 0. Slope intercept form of a linear equation The slope is m, and the -intercept is (0, b). Point slope form of a linear equation The slope is m, and, is a point on the line. Horizontal line The slope is 0, and the -intercept is (0, c). Vertical line The slope is undefined, and the -intercept is (c, 0). Parallel and Perpendicular Lines Nonvertical parallel lines have the same slope. The product of the slopes of two nonvertical perpendicular lines is -.

8 7 CHAPTER Graphs and Functions Vocabular, Readiness & Video Check State the slope and the -intercept of each line with the given equation.. = - +. = - 7. =. = -. = = - + Decide whether the lines are parallel, perpendicular, or neither. 7. = + 6 = - 8. = = = -9 + = = - = - 6 Martin-Ga Interactive Videos See Video. Watch the section lecture video and answer the following questions.. Complete these statements based on Eample. To graph a line using its slope and -intercept, first write the equation in form. Graph the one point ou now know, the. Use the to find a second point.. From Eample, given a -intercept point, how do ou know which value to use for b in the slope intercept form?. Eample discusses how to find an equation of a line given two points. Under what circumstances might the slope intercept form be chosen over the point slope form to find an equation?. Solve Eamples and 6 again, this time using the point -, in each eercise.. Solve Eample 7 again, this time write the equation of the line in function notation, parallel to the given line through the given point.. Eercise Set Graph each linear equation. See Eamples and.. = -. = +. + = 7. + = = = -6 Use the slope intercept form of the linear equation to write the equation of each line with the given slope and -intercept. See Eample. 7. Slope -; -intercept (0, ) 8. Slope ; -intercept 0, -6. Slope - ; -intercept (0, 0) Find an equation of the line with the given slope and containing the given point. Write the equation in slope intercept form. See Eample.. Slope ; through (, ). Slope ; through (, ). Slope -; through, - 6. Slope -; through, - 7. Slope ; through -6, 9. Slope ; -intercept a0, b 0. Slope -; -intercept a0, - b. Slope ; -intercept (0, 0) 7 8. Slope ; through -9, 9. Slope - 9 ; through -, Slope - ; through, -6

9 Section. Equations of Lines 7 Find an equation of the line passing through the given points. Use function notation to write the equation. See Eample.. (, 0), (, 6). (, 0), (7, 8). -,, -6,. 7, -, (, 6). -, -, -, , -, -, , -8, -6, , -,, a, 0 b and a -, 7 0 b 0. a, - b and a, b Find an equation of each line graphed. Write the equation in standard form. See Eample Use the graph of the following function f() to find each value.. f(0) 6. f - 7. f() 8. f() 9. Find such that f = Find such that f =. Write an equation of each line. See Eamples 8 and 9.. Slope 0; through -, -. Horizontal; through -,. Vertical; through (, 7). Vertical; through (, 6). Horizontal; through (0, ) 6. Undefined slope; through (0, ) Find an equation of each line. Write the equation using function notation. See Eamples 0 and. 7. Through (, 8); parallel to f = - 8. Through (, ); parallel to f = - 9. Through, -; perpendicular to = Through -, 8; perpendicular to - =. Through -, -; parallel to + =. Through -, -; perpendicular to + = MIXED Find the equation of each line. Write the equation in standard form unless indicated otherwise. See Eamples through, and 8 through.. Slope ; through -,. Slope ; through -,. Through (, 6) and (, ); use function notation. 6. Through (, 9) and (8, 6) 7. With slope - ; -intercept 8. With slope -; -intercept ; use function notation Through -7, - and 0, Through, -8 and -, - 6. Slope - ; through -, 0 6. Slope - ; through, - 6. Vertical line; through -, Horizontal line; through (, 0) 6. Through 6, -; parallel to the line + = Through 8, -; parallel to the line 6 + = 67. Slope 0; through -9, 68. Undefined slope; through 0, Through (6, ); parallel to the line 8 - = Through (, ); perpendicular to the line - = 8 7. Through, -6; perpendicular to = 9 7. Through -, -; parallel to = 9 7. Through, -8 and -6, -; use function notation. 7. Through -, - and -6, ; use function notation.

10 7 CHAPTER Graphs and Functions Solve. See Eample A rock is dropped from the top of a 00-foot building. After second, the rock is traveling feet per second. After seconds, the rock is traveling 96 feet per second. Let be the rate of descent and be the number of seconds since the rock was dropped. a. Write a linear equation that relates time to rate. [Hint: Use the ordered pairs (, ) and (, 96).] b. Use this equation to determine the rate of travel of the rock seconds after it was dropped. 76. A fruit compan recentl released a new applesauce. B the end of its first ear, profits on this product amounted to $0,000. The anticipated profit for the end of the fourth ear is $66,000. The ratio of change in time to change in profit is constant. Let be ears and be profit. a. Write a linear equation that relates profit and time. [Hint: Use the ordered pairs (, 0,000) and (, 66,000).] b. Use this equation to predict the compan s profit at the end of the seventh ear. c. Predict when the profit should reach $6, The Whammo Compan has learned that b pricing a newl released Frisbee at $6, sales will reach 000 per da. Raising the price to $8 will cause the sales to fall to 00 per da. Assume that the ratio of change in price to change in dail sales is constant and let be the price of the Frisbee and be number of sales. a. Find the linear equation that models the price sales relationship for this Frisbee. [Hint: The line must pass through (6, 000) and (8, 00).] b. Use this equation to predict the dail sales of Frisbees if the price is set at $ The Pool Fun Compan has learned that, b pricing a newl released Fun Noodle at $, sales will reach 0,000 Fun Noodles per da during the summer. Raising the price to $ will cause the sales to fall to 8000 Fun Noodles per da. Let be price and be the number sold. a. Assume that the relationship between sales price and number of Fun Noodles sold is linear and write an equation describing this relationship. [Hint: The line must pass through (, 0,000) and (, 8000).] b. Use this equation to predict the dail sales of Fun Noodles if the price is $ The number of people emploed in the United States as registered nurses was 69 thousand in 008. B 08, this number is epected to rise to 00 thousand. Let be the number of registered nurses (in thousands) emploed in the United States in the ear, where = 0 represents 008. (Source: U.S. Bureau of Labor Statistics) a. Write a linear equation that models the number of people (in thousands) emploed as registered nurses in ear. b. Use this equation to estimate the number of people emploed as registered nurses in In 008, IBM had 98,00 emploees worldwide. B 00, this number had increased to 6,7. Let be the number of IBM emploees worldwide in the ear, where = 0 represents 008. (Source: IBM Corporation) a. Write a linear equation that models the growth in the number of IBM emploees worldwide, in terms of the ear. b. Use this equation to predict the number of IBM emploees worldwide in In 00, the average price of a new home sold in the United States was $7,900. In 00, the average price of a new home in the United States was $97,000. Let be the average price of a new home in the ear, where = 0 represents the ear 00. (Source: Based on data from U.S. census) a. Write a linear equation that models the average price of a new home in terms of the ear. [Hint: The line must pass through the points (0, 97,000) and (, 7,900).] b. Use this equation to predict the average price of a new home in The number of McDonald s restaurants worldwide in 00 was,77. In 00, there were,06 McDonald s restaurants worldwide. Let be the number of McDonald s restaurants in the ear, where = 0 represents the ear 00. (Source: McDonald s Corporation) a. Write a linear equation that models the growth in the number of McDonald s restaurants worldwide in terms of the ear. [Hint: The line must pass through the points (0,,06) and (,,77).] b. Use this equation to predict the number of McDonald s restaurants worldwide in 0. REVIEW AND PREVIEW Solve. Write the solution in interval notation. See Section Ú CONCEPT EXTENSIONS Ú - Answer true or false. 89. A vertical line is alwas perpendicular to a horizontal line. 90. A vertical line is alwas parallel to a vertical line. Eample: Find an equation of the perpendicular bisector of the line segment whose endpoints are (, 6) and (0, - ). Perpendicular bisector 7 6 (, 6) Line segment (0, ) 9 Solution: A perpendicular bisector is a line that contains the midpoint of the given segment and is perpendicular to the segment.

11 Integrated Review 7 Step. The midpoint of the segment with endpoints (, 6) and 0, - is (, ). Step. The slope of the segment containing points (, 6) and 0, - is. Step. A line perpendicular to this line segment will have Step. slope of -. The equation of the line through the midpoint (, ) with a slope of - will be the equation of the perpendicular bisector. This equation in standard form is + = 9. Find an equation of the perpendicular bisector of the line segment whose endpoints are given. See the previous eample. 9., -; -, 9. -6, -; -8, , 6; -, - 9. (, 8); (7, ) 9. (, ); -, , 8; -, Describe how to see if the graph of - = 7 passes through the points., -.0 and 0, -.7. Then follow our directions and check these points. Use a graphing calculator with a TRACE feature to see the results of each eercise. 98. Eercise 6; graph the equation and verif that it passes through (, 9) and (8, 6). 99. Eercise ; graph the function and verif that it passes through (, 6) and (, ). 00. Eercise 6; graph the equation. See that it has a negative slope and passes through, Eercise 6; graph the equation. See that it has a negative slope and passes through -, Eercise 8: Graph the equation and verif that it passes through (, ) and is parallel to = Eercise 7: Graph the equation and verif that it passes through (, 8) and is parallel to = -. Integrated Review LINEAR EQUATIONS IN TWO VARIABLES Sections.. Below is a review of equations of lines. Forms of Linear Equations A + B = C Standard form of a linear equation. A and B are not both 0. = m + b Slope intercept form of a linear equation. The slope is m, and the -intercept is (0, b). - = m - Point slope form of a linear equation. The slope is m, and, is a point on the line. = c Horizontal line The slope is 0, and the -intercept is (0, c). = c Vertical line The slope is undefined and the -intercept is (c, 0). Parallel and Perpendicular Lines Nonvertical parallel lines have the same slope. The product of the slopes of two nonvertical perpendicular lines is -. Graph each linear equation.. = -. - = 6. = -. =. Find the slope of the line containing each pair of points.. -, -,, - 6. (, ), (0, ) Find the slope and -intercept of each line. 7. = = 7 Determine whether each pair of lines is parallel, perpendicular, or neither. 9. = 8-6 = = + + =

12 76 CHAPTER Graphs and Functions Find the equation of each line. Write the equation in the form = a, = b, or = m + b. For Eercises through 7, write the equation in the form f = m + b.. Through (, 6) and (, ). Vertical line; through -, -0. Horizontal line; through (, 0). Through, -9 and -6, -. Through -, with slope - 6. Slope -; -intercept a0, b 7. Slope ; -intercept 0, - 8. Through a, 0b with slope 9. Through -, -; parallel to - = 0. Through (0, ); perpendicular to - = 0. Through, -; perpendicular to + =. Through -, 0; parallel to + =. Undefined slope; through -,. m = 0; through -,.6 Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions S Graph Piecewise-Defined Functions. Vertical and Horizontal Shifts. Reflect Graphs. Graphing Piecewise-Defined Functions Throughout Chapter, we have graphed functions. There are man special functions. In this objective, we stud functions defined b two or more epressions. The epression used to complete the function varies with and depends upon the value of. Before we actuall graph these piecewise-defined functions, let s practice finding function values. EXAMPLE Evaluate f, f -6, and f0 for the function f = e Then write our results in ordered pair form. if if Solution Take a moment and stud this function. It is a single function defined b two epressions depending on the value of. From above, if 0, use f = +. If 7 0, use f = - -. Thus f = - - = - since 7 0 f = - Ordered pairs:, - f -6 = -6 + = -9 since -6 0 f -6 = -9-6, -9 f0 = 0 + = since 0 0 f0 = 0, Evaluate f, f -, and f0 for the function f = e if if Now, let s graph a piecewise-defined function.