Graphing and Writing Linear Equations. Graphing Linear Equations. Slope of a Line. Graphing Proportional Relationships. Graphing Linear Equations in Slope-Intercept Form. Graphing Linear Equations in Standard Form. Writing Equations in Slope-Intercept Form.7 Writing Equations in Point-Slope Form Oka Descartes, stand on the - ais and tr to intercept the pass when I throw.
What You Learned Before I estimate that we are on a slope of about -.. What do ou think? Eample Evaluate + ( + ) when = and = 7. + ( + ) = ()(7) + ( + 7) Substitute for and 7 for. = 8(7) + ( + 7) Use order of operations. = + () Simplif. = + Multipl. = 89 Add. Evaluate the epression when a = and b =.. 8ab. a b. b. a + (b a ) a Eample Write the ordered pair that corresponds to point U. Point U is units to the left of the origin and units down. So, the -coordinate is, and the -coordinate is. The ordered pair (, ) corresponds to point U. Eample Which point is located at (, )? Start at the origin. Move units right and units down. N R U Q V S P T Point T is located at (, ). Use the graph to answer the question.. Write the ordered pair that corresponds to point Q.. Write the ordered pair that corresponds to point P. 7. Which point is located at (, )? 8. Which point is located in Quadrant II?
. Graphing Linear Equations How can ou draw its graph? How can ou recognize a linear equation? ACTIVITY: Graphing a Linear Equation Work with a partner. a. Use the equation = + to complete the table. (Choose an two -values and find the -values.) b. Write the two ordered pairs given b the table. These are called solution points of the equation. = + Solution Points c. PRECISION Plot the two solution points. Draw a line eactl through the two points. d. Find a different point on the line. Check that this point is a solution O point of the equation = +. e. LOGIC Do ou think it is true that an point on the line is a solution point of the equation = +? Eplain. f. Choose five additional -values for the table. (Choose positive and negative -values.) Plot the five corresponding solution points. Does each point lie on the line? Graphing Equations In this lesson, ou will understand that lines represent solutions of linear equations. graph linear equations. = + Solution Points g. LOGIC Do ou think it is true that an solution point of the equation = + is a point on the line? Eplain. h. Wh do ou think = a + b is called a linear equation? Chapter Graphing and Writing Linear Equations
ACTIVITY: Using a Graphing Calculator Use a graphing calculator to graph = +. a. Enter the equation = + into our calculator. Math Practice Recognize Usefulness of Tools What are some advantages and disadvantages of using a graphing calculator to graph a linear equation? b. Check the settings of the viewing window. The boundaries of the graph are set b the minimum and the maimum - and -values. The numbers of units between the tick marks are set b the - and -scales. c. Graph = + on our calculator. This is the standard viewing window. d. Change the settings of the viewing window to match those shown. Compare the two graphs. 8. IN YOUR OWN WORDS How can ou recognize a linear equation? How can ou draw its graph? Write an equation that is linear. Write an equation that is not linear.. Use a graphing calculator to graph = in the standard viewing window. a. Can ou tell where the line crosses the -ais? Can ou tell where the line crosses the -ais? b. How can ou adjust the viewing window so that ou can determine where the line crosses the - and -aes?. CHOOSE TOOLS You want to graph =..8. Would ou graph it b hand or b using a graphing calculator? Wh? Use what ou learned about graphing linear equations to complete Eercises and on page. Section. Graphing Linear Equations
. Lesson Lesson Tutorials Ke Vocabular linear equation, p. solution of a linear equation, p. Remember An ordered pair (, ) is used to locate a point in a coordinate plane. Linear Equations A linear equation is an equation whose graph is a line. The points on the line are solutions of the equation. You can use a graph to show the solutions of a linear equation. The graph below represents the equation = +. (, ) (, ) (, ) (, ) (, ) O (, ) (, ) EXAMPLE Graphing a Linear Equation Graph = +. Step : Make a table of values. Check = + (, ) = ( ) + (, ) = () + (, ) = () + (, ) (, ) (, ) Step : Plot the ordered pairs. Step : Draw a line through the points. O (, ) Graphing Horizontal and Vertical Lines The graph of = b is a horizontal line passing through (, b). The graph of = a is a vertical line passing through (a, ). b (, b) O a (a, ) O Chapter Graphing and Writing Linear Equations
EXAMPLE Graphing a Horizontal Line and a Vertical Line a. Graph =. b. Graph =. The graph of = is a horizontal line passing through (, ). Draw a horizontal line through this point. The graph of = is a vertical line passing through (, ). Draw a vertical line through this point. O (, ) (, ) O Eercises Graph the linear equation. Use a graphing calculator to check our graph, if possible.. =. = +. =. =. EXAMPLE V I D E O Real-Life Application The wind speed (in miles per hour) of a tropical storm is = +, where is the number of hours after the storm enters the Gulf of Meico. a. Graph the equation. b. When does the storm become a hurricane? a. Make a table of values. = + (, ) = () + (, ) 7 = () + 8 (, 8) 7 A tropical storm becomes a hurricane when wind speeds are at least 7 miles per hour. = () + 7 (, 7) = () + 7 (, 7) 8 Plot the ordered pairs and draw a line through the points. b. From the graph, ou can see that = 7 when =. So, the storm becomes a hurricane hours after it enters the Gulf of Meico.. WHAT IF? The wind speed of the storm is =. +. When does the storm become a hurricane? Section. Graphing Linear Equations
. Eercises Help with Homework. VOCABULARY What tpe of graph represents the solutions of the equation = +?. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. =.. + = = + + = 9+(-)= +(-)= +(-9)= 9+(-)= PRECISION Cop and complete the table. Plot the two solution points and draw a line eactl through the two points. Find a different solution point on the line... = = + Graph the linear equation. Use a graphing calculator to check our graph, if possible.. =. = 7. = 8. = 9. =. = 7. = +. =. =. =.7. =.. = 7. ERROR ANALYSIS Describe and correct the error in graphing the equation. 8. MESSAGING You sign up for an unlimited tet-messaging plan for our cell phone. The equation = represents the cost (in dollars) for sending tet messages. Graph the equation. What does the graph tell ou? O (, ) 9. MAIL The equation = + represents the cost (in dollars) of mailing a package that weighs pounds. a. Graph the equation. b. Use the graph to estimate how much it costs to mail the package. c. Use the equation to find eactl how much it costs to mail the package. Chapter Graphing and Writing Linear Equations
Solve for. Then graph the equation. Use a graphing calculator to check our graph.. =. + =. + =. +. =.. SAVINGS You have $ in our savings account and plan to deposit $. each month. a. Graph a linear equation that represents the balance in our account. b. How man months will it take ou to save enough mone to bu acres of land on Mars?. GEOMETRY The sum S of the interior angle measures of a polgon with n sides is S = (n ) 8. a. Plot four points (n, S ) that satisf the equation. Is the equation a linear equation? Eplain our reasoning. b. Does the value n =. make sense in the contet of the problem? Eplain our reasoning.. SEA LEVEL Along the U.S. Atlantic coast, the sea level is rising about millimeters per ear. How man millimeters has sea level risen since ou were born? How do ou know? Use a linear equation and a graph to justif our answer. Video time: min. sec. 7. Problem Solving One second of video on our digital camera uses the same amount of memor as two pictures. Your camera can store pictures. a. Write and graph a linear equation that represents the number of pictures our camera can store when ou take seconds of video. b. How man pictures can our camera store in addition to the video shown? Write the ordered pair corresponding to the point. (Skills Review Handbook) B 8. point A 9. point B A. point C. point D. MULTIPLE CHOICE A debate team has female members. The ratio of females to males is :. How man males are on the debate team? (Skills Review Handbook) O A B C D D C Section. Graphing Linear Equations 7
. Slope of a Line describe the line? How can ou use the slope of a line to Slope is the rate of change between an two points on a line. It is the measure of the steepness of the line. To find the slope of a line, find the ratio of the change in (vertical change) to the change in (horizontal change). change in slope = change in 7 Slope 7 ACTIVITY: Finding the Slope of a Line Work with a partner. Find the slope of each line using two methods. Method : Use the two black points. Method : Use the two pink points. Do ou get the same slope using each method? Wh do ou think this happens? a. b. Graphing Equations In this lesson, ou will find slopes of lines b using two points. find slopes of lines from tables. c. d. 8 Chapter Graphing and Writing Linear Equations
ACTIVITY: Using Similar Triangles Work with a partner. Use the figure shown. a. ABC is a right triangle formed b drawing a horizontal line segment from point A and a vertical line segment from point B. Use this method to draw another right triangle, DEF. b. What can ou conclude about ABC and DEF? Justif our conclusion. c. For each triangle, find the ratio of the length of the vertical side to the length of the horizontal side. What do these ratios represent? d. What can ou conclude about the slope between an two points on the line? 9 8 7 A(, ) B(, ) D(9, ) C(, ) E(, ) 7 8 9 ACTIVITY: Drawing Lines with Given Slopes Math Practice Interpret a Solution What does the slope tell ou about the graph of the line? Eplain. Work with a partner. a. Draw two lines with slope. One line passes through (, ), and the other line passes through (, ). What do ou notice about the two lines? b. Draw two lines with slope. One line passes through (, ), and the other line passes through (, ). What do ou notice about the two lines? c. CONJECTURE Make a conjecture about two different nonvertical lines in the same plane that have the same slope. d. Graph one line from part (a) and one line from part (b) in the same coordinate plane. Describe the angle formed b the two lines. What do ou notice about the product of the slopes of the two lines? e. REPEATED REASONING Repeat part (d) for the two lines ou did not choose. Based on our results, make a conjecture about two lines in the same plane whose slopes have a product of.. IN YOUR OWN WORDS How can ou use the slope of a line to describe the line? Use what ou learned about the slope of a line to complete Eercises on page. Section. Slope of a Line 9
. Lesson Lesson Tutorials Ke Vocabular slope, p. rise, p. run, p. Reading In the slope formula, is read as sub one, and is read as sub two. The numbers and in and are called subscripts. Slope The slope m of a line is a ratio of the change in (the rise) to the change in (the run) between an two points, (, ) and (, ), on the line. m = rise run change in = change in = Positive Slope O The line rises from left to right. (, ) (, ) Rise Run O Negative Slope O The line falls from left to right. EXAMPLE Finding the Slope of a Line Describe the slope of the line. Then find the slope. a. (, ) (, ) b. (, ) (, ) Stud Tip When finding slope, ou can label either point as (, ) and the other point as (, ). The line rises from left to The line falls from left to right. So, the slope is positive. right. So, the slope is negative. Let (, ) = (, ) and Let (, ) = (, ) and (, ) = (, ). (, ) = (, ). m = m = ( ) = ( ) = = ( ) =, or Chapter Graphing and Writing Linear Equations
Eercises 7 9 Find the slope of the line.. (, ) (, ). (, ) (, ). (, ) (, ) EXAMPLE Finding the Slope of a Horizontal Line Find the slope of the line. m = = ( ) (, ) (, ) = 7, or The slope is. Stud Tip EXAMPLE The slope of ever horizontal line is. The slope of ever vertical line is undefined. Finding the Slope of a Vertical Line Find the slope of the line. m = = = (, ) (, ) Because division b zero is undefined, the slope of the line is undefined. Eercises Find the slope of the line through the given points.. (, ), (7, ). (, ), (, ). (, ), (, ) 7. (, 8), (, ) 8. How do ou know that the slope of ever horizontal line is? How do ou know that the slope of ever vertical line is undefined? Section. Slope of a Line
EXAMPLE Finding Slope from a Table The points in the table lie on a line. How can ou find the slope of the line from the table? What is the slope? 7 8 Choose an two points from the table and use the slope formula. Use the points (, ) = (, 8) and (, ) = (, ). m = = 8 =, or The slope is. Check 8 7 (, 8) (, ) (7, ) (, ) 7 8 9 Eercises The points in the table lie on a line. Find the slope of the line. 9.. 7 8 Slope Positive Slope Negative Slope Slope of Undefined Slope O O O O The line rises The line falls The line is The line is from left to right. from left to right. horizontal. vertical. Chapter Graphing and Writing Linear Equations
. Eercises Help with Homework. CRITICAL THINKING Refer to the graph. a. Which lines have positive slopes? b. Which line has the steepest slope? c. Do an lines have an undefined slope? Eplain.. OPEN-ENDED Describe a real-life situation in which ou need to know the slope. A B C. REASONING The slope of a line is. What do ou know about the line? 9+(-)= +(-)= +(-9)= 9+(-)= Draw a line through each point using the given slope. What do ou notice about the two lines?. slope =. slope =. slope = Find the slope of the line. 7. (, ) (, ) 8. (, ) (, ) 9. (, ) (, ). (, ) (, ). (, ) (, ). (, ) (, ) Section. Slope of a Line
Find the slope of the line through the given points.. (, ), (, ). (, ), (, 8). ( 7, ), ( 7, ). (, ), (, ) 7. (, ), (, ) 8. (, ), (, ) 9. ERROR ANALYSIS Describe and correct the error in finding the slope of the line.. CRITICAL THINKING Is it more difficult to walk up the ramp or the hill? Eplain. m = = = (, ) (, ) ft hill ramp 8 ft 8 ft ft The points in the table lie on a line. Find the slope of the line.. 7. 7 8.. 8 8 8 ft ft. PITCH Carpenters refer to the slope of a roof as the pitch of the roof. Find the pitch of the roof.. PROJECT The guidelines for a wheelchair ramp suggest that the ratio of the rise to the run be no greater than :. a. CHOOSE TOOLS Find a wheelchair ramp in our school or neighborhood. Measure its slope. Does the ramp follow the guidelines? b. Design a wheelchair ramp that provides access to a building with a front door that is. feet above the sidewalk. Illustrate our design. Use an equation to find the value of k so that the line that passes through the given points has the given slope. 7. (, ), (, k); m = 8. (, k), (, ); m = 9. (, k), (, 7); m =. (, ), (k, ); m = Chapter Graphing and Writing Linear Equations
. TURNPIKE TRAVEL The graph shows the cost of traveling b car on a turnpike. a. Find the slope of the line. b. Eplain the meaning of the slope as a rate of change. Cost (dollars).. Turnpike Travel. BOAT RAMP Which is steeper: the boat ramp or a road with a % grade? Eplain. (Note: Road grade is the vertical increase divided b the horizontal distance.) 8 Miles driven ft ft. REASONING Do the points A(, ), B(, ), and C(, ) lie on the same line? Without using a graph, how do ou know?. BUSINESS A small business earns a profit of $ in Januar and $7, in Ma. What is the rate of change in profit for this time period?. STRUCTURE Choose two points in the coordinate plane. Use the slope formula to find the slope of the line that passes through the two points. Then find the slope using the formula. Eplain wh our results are the same.. The top and the bottom of the slide are level with the ground, which has a slope of. ft a. What is the slope of the main portion of the slide? b. How does the slope change when the bottom of the slide is onl inches above the ground? Is the slide steeper? Eplain. ft 8 in. ft 8 ft Solve the proportion. 7. b = (Skills Review Handbook) 8. 7 = n 9. 8 =. MULTIPLE CHOICE What is the prime factorization of 8? (Skills Review Handbook) A 7 B 7 C 7 D Section. Slope of a Line
Etension. Slopes of Parallel and Perpendicular Lines Lesson Tutorials Graphing Equations In this etension, ou will identif parallel and perpendicular lines. Parallel Lines and Slopes Lines in the same plane that do not intersect are parallel lines. Nonvertical parallel lines have the same slope. All vertical lines are parallel. EXAMPLE Identifing Parallel Lines Which two lines are parallel? How do ou know? Find the slope of each line. Blue Line Red Line Green Line m = m = m = = ( ) = = =, or =, or =, or (, ) (, ) (, ) (, ) (, ) (, ) The slopes of the blue and green lines are. The slope of the red line is. The blue and green lines have the same slope, so the are parallel. Which lines are parallel? How do ou know?. (, ) (, ) (, ) (, ) (, ) (, ). (, ) (, ) (, ) (, ) (, ) (, ) Are the given lines parallel? Eplain our reasoning.. =, =. =, =. =, =. GEOMETRY The vertices of a quadrilateral are A(, ), B(, ), C(, ), and D(, ). How can ou use slope to determine whether the quadrilateral is a parallelogram? Is it a parallelogram? Justif our answer. Chapter Graphing and Writing Linear Equations
Perpendicular Lines and Slope Lines in the same plane that intersect at right angles are perpendicular lines. Two nonvertical lines are perpendicular when the product of their slopes is. Vertical lines are perpendicular to horizontal lines. EXAMPLE Identifing Perpendicular Lines Which two lines are perpendicular? How do ou know? Find the slope of each line. Blue Line Red Line Green Line (, ) (, ) (, ) m = m = m = = ( ) =, or = ( ) = 7 ( ) = ( ) = 7 (, ) 7 (, ) (, ) The slope of the red line is 7. The slope of the green line is 7. Because 7 7 =, the red and green lines are perpendicular. Which lines are perpendicular? How do ou know? 7. 8. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Are the given lines perpendicular? Eplain our reasoning. 9. =, = 8. = 8, = 7. =, =. GEOMETRY The vertices of a parallelogram are J (, ), K(, ), L(, ), and M(, ). How can ou use slope to determine whether the parallelogram is a rectangle? Is it a rectangle? Justif our answer. Etension. Slopes of Parallel and Perpendicular Lines 7
.. Graphing Proportional Relationships equation = m? How can ou describe the graph of the ACTIVITY: Identifing Proportional Relationships Work with a partner. Tell whether and are in a proportional relationship. Eplain our reasoning. a. Earnings (dollars) 7 Mone 7 Hours worked b. Height (meters) Helicopter 7 Time (seconds) Graphing Equations In this lesson, ou will write and graph proportional relationships. c. e. Cost (dollars) 8 Tickets 7 Number of tickets Laps, Time (seconds), 9 8 d. Cost (dollars) f. Cups of Sugar, Cups of Flour, ACTIVITY: Analzing Proportional Relationships Pizzas 7 Number of pizzas Work with a partner. Use onl the proportional relationships in Activit to do the following. Find the slope of the line. Find the value of for the ordered pair (, ). What do ou notice? What does the value of represent? 8 Chapter Graphing and Writing Linear Equations
ACTIVITY: Deriving an Equation Work with a partner. Let (, ) represent an point on the graph of a proportional relationship. (, ) (, m) (, ) Math Practice View as Components What part of the graph can ou use to find the side lengths? a. Eplain wh the two triangles are similar. b. Because the triangles are similar, the corresponding side lengths are proportional. Use the vertical and horizontal side lengths to complete the steps below. = m = m Simplif. Ratios of side lengths = m Multiplication Propert of Equalit What does the final equation represent? c. Use our result in part (b) to write an equation that represents each proportional relationship in Activit.. IN YOUR OWN WORDS How can ou describe the graph of the equation = m? How does the value of m affect the graph of the equation?. Give a real-life eample of two quantities that are in a proportional relationship. Write an equation that represents the relationship and sketch its graph. Use what ou learned about proportional relationships to complete Eercises on page. Section. Graphing Proportional Relationships 9
. Lesson Lesson Tutorials Direct Variation Stud Tip In the direct variation equation = m, m represents the constant of proportionalit, the slope, and the unit rate. Words When two quantities and are proportional, the relationship can be represented b the direct variation equation = m, where m is the constant of proportionalit. Graph The graph of = m is a line with a slope of m that passes through the origin. (, ) (, m) EXAMPLE Graphing a Proportional Relationship Cost (dollars) Internet Plan (, ) (, ) Data used (gigabtes) The cost (in dollars) for gigabtes of data on an Internet plan is represented b =. Graph the equation and interpret the slope. The equation shows that the slope m is. So, the graph passes through (, ) and (, ). Plot the points and draw a line through the points. Because negative values of do not make sense in this contet, graph in the first quadrant onl. The slope indicates that the unit cost is $ per gigabte. Stud Tip EXAMPLE In Eample, the slope indicates that the weight of an object on Titan is one-seventh its weight on Earth. Writing and Using a Direct Variation Equation The weight of an object on Titan, one of Saturn s moons, is proportional to the weight of the object on Earth. An object that weighs pounds on Earth would weigh pounds on Titan. a. Write an equation that represents the situation. Use the point (, ) to find the slope of the line. = m Direct variation equation = m() Substitute for and for. 7 = m Simplif. So, an equation that represents the situation is = 7. b. How much would a chunk of ice that weighs. pounds on Titan weigh on Earth?. = 7 Substitute. for.. = Multipl each side b 7. So, the chunk of ice would weigh. pounds on Earth. Chapter Graphing and Writing Linear Equations
Eercises 7 8. WHAT IF? In Eample, the cost is represented b =. Graph the equation and interpret the slope.. In Eample, how much would a spacecraft that weighs kilograms on Earth weigh on Titan? Distance (meters) 8 (, ) EXAMPLE Two-Person Lift (, ) 7 Time (seconds) Comparing Proportional Relationships The distance (in meters) that a four-person ski lift travels in seconds is represented b the equation =.. The graph shows the distance that a two-person ski lift travels. a. Which ski lift is faster? Interpret each slope as a unit rate. Four-Person Lift =. The slope is.. The four-person lift travels. meters per second. Two-Person Lift change in slope = change in = 8 = The two-person lift travels meters per second. So, the four-person lift is faster than the two-person lift. b. Graph the equation that represents the four-person lift in the same coordinate plane as the two-person lift. Compare the steepness of the graphs. What does this mean in the contet of the problem? The graph that represents the four-person lift is steeper than the graph that represents the two-person lift. So, the four-person lift is faster. Distance (meters) 8 Ski Lift four-person two-person 7 Time (seconds) Eercise 9. The table shows the distance (in meters) that a T-bar ski lift travels in seconds. Compare its speed to the ski lifts in Eample. (seconds) (meters) 9 Section. Graphing Proportional Relationships
. Eercises Help with Homework. VOCABULARY What point is on the graph of ever direct variation equation?. REASONING Does the equation = + represent a proportional relationship? Eplain. 9+(-)= +(-)= +(-9)= 9+(-)= Tell whether and are in a proportional relationship. Eplain our reasoning. If so, write an equation that represents the relationship... 7 8 8 7. 9. 8 8 7. TICKETS The amount (in dollars) that ou raise b selling fundraiser tickets is represented b the equation =. Graph the equation and interpret the slope. 8. KAYAK The cost (in dollars) to rent a kaak is proportional to the number of hours that ou rent the kaak. It costs $7 to rent the kaak for hours. a. Write an equation that represents the situation. b. Interpret the slope. c. How much does it cost to rent the kaak for hours? Distance (miles) 7 Car Gasoline (gallons) 9. MILEAGE The distance (in miles) that a truck travels on gallons of gasoline is represented b the equation = 8. The graph shows the distance that a car travels. a. Which vehicle gets better gas mileage? Eplain how ou found our answer. b. How much farther can the vehicle ou chose in part (a) travel than the other vehicle on 8 gallons of gasoline? Chapter Graphing and Writing Linear Equations
. BIOLOGY Toenails grow about millimeters per ear. The table shows fingernail growth. a. Do fingernails or toenails grow faster? Eplain. Weeks Fingernail Growth (millimeters).7...8 b. In the same coordinate plane, graph equations that represent the growth rates of toenails and fingernails. Compare the steepness of the graphs. What does this mean in the contet of the problem?. REASONING The quantities and are in a proportional relationship. What do ou know about the ratio of to for an point (, ) on the line?. PROBLEM SOLVING The graph relates the temperature change (in degrees Fahrenheit) to the altitude change (in thousands of feet). a. Is the relationship proportional? Eplain. b. Write an equation of the line. Interpret the slope. c. You are at the bottom of a mountain where the temperature is 7 F. The top of the mountain is feet above ou. What is the temperature at the top of the mountain? Temperature ( F) Altitude Change 7 8 9 Altitude (thousands of feet). Consider the distance equation d = rt, where d is the distance (in feet), r is the rate (in feet per second), and t is the time (in seconds). a. You run feet per second. Are distance and time proportional? Eplain. Graph the equation. b. You run for seconds. Are distance and rate proportional? Eplain. Graph the equation. c. You run feet. Are rate and time proportional? Eplain. Graph the equation. d. One of these situations represents inverse variation. Which one is it? Wh do ou think it is called inverse variation? Graph the linear equation. (Section.). =. =. = 7. MULTIPLE CHOICE What is the value of? (Section.) A B C D 9 Section. Graphing Proportional Relationships
Stud Help Graphic Organizer You can use a process diagram to show the steps involved in a procedure. Here is an eample of a process diagram for graphing a linear equation. Graphing a linear equation Make a table of values. Eample Graph =. Plot the ordered pairs. (, ) (, ) (, ) Draw a line through the points. (, ) (, ) (, ) = Make process diagrams with eamples to help ou stud these topics.. finding the slope of a line. graphing a proportional relationship After ou complete this chapter, make process diagrams for the following topics.. graphing a linear equation using a. slope and -intercept b. - and -intercepts. writing equations in slope-intercept form. writing equations in point-slope form Here is a process diagram with suggestions for what to do if a hena knocks on our door. Chapter Graphing and Writing Linear Equations
.. Quiz Progress Check Graph the linear equation. (Section.). = + 8. =. =. =. Find the slope of the line. (Section.). (, ) (, ). (, ) (, ) 7. (, ) (, ) 8. (, ) (, ) 9. What is the slope of a line that is parallel to the line in Eercise? What is the slope of a line that is perpendicular to the line in Eercise? (Section.). Are the lines = and = parallel? Are the perpendicular? Justif our answer. (Section.). BANKING A bank charges $ each time ou use an out-of-network ATM. At the beginning of the month, ou have $ in our bank account. You withdraw $ from our bank account each time ou use an out-of-network ATM. Graph a linear equation that represents the balance in our account after ou use an out-of-network ATM times. (Section.). MUSIC The number of hours of cello lessons that ou take after weeks is represented b the equation =. Graph the equation and interpret the slope. (Section.). DINNER PARTY The cost (in dollars) to provide food for guests at a dinner part is proportional to the number of guests attending the part. It costs $ to provide food for guests. (Section.) a. Write an equation that represents the situation. b. Interpret the slope. c. How much does it cost to provide food for guests? Sections.. Quiz
.. Graphing Linear Equations in Slope-Intercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope of each line. Find the point where each line crosses the -ais. Complete the table. Equation a. = + b. = + Slope of Graph Point of Intersection with -ais c. = d. = + e. = + f. = g. = Graphing Equations In this lesson, ou will find slopes and -intercepts of graphs of linear equations. graph linear equations written in slope-intercept form. h. = i. = + j. = k. Do ou notice an relationship between the slope of the graph and its equation? between the point of intersection with the -ais and its equation? Compare the results with those of other students in our class. Chapter Graphing and Writing Linear Equations
ACTIVITY: Deriving an Equation Work with a partner. a. Look at the graph of each equation in Activit. Do an of the graphs represent a proportional relationship? Eplain. b. For a nonproportional linear relationship, the graph crosses the -ais at some point (, b), where b does not equal. Let (, ) represent an other point on the graph. You can use the formula for slope to write the equation for a nonproportional linear relationship. Use the graph to complete the steps. (, b) (, ) = m Slope formula = m Substitute values. Math Practice Use Prior Results How can ou use the results of Activit to help support our answer? = m Simplif. = m Multiplication Propert of Equalit = m Simplif. = m + Addition Propert of Equalit c. What do m and b represent in the equation?. IN YOUR OWN WORDS How can ou describe the graph of the equation = m + b? a. How does the value of m affect the graph of the equation? b. How does the value of b affect the graph of the equation? c. Check our answers to parts (a) and (b) with three equations that are not in Activit.. LOGIC Wh do ou think = m + b is called the slope-intercept form of the equation of a line? Use drawings or diagrams to support our answer. Use what ou learned about graphing linear equations in slope-intercept form to complete Eercises on page 7. Section. Graphing Linear Equations in Slope-Intercept Form 7
. Lesson Lesson Tutorials Ke Vocabular -intercept, p. 8 -intercept, p. 8 slope-intercept form, p. 8 Intercepts The -intercept of a line is the -coordinate of the point where the line crosses the -ais. It occurs when =. (, b) -intercept b -intercept a The -intercept of a line is the -coordinate of the point where the line crosses the -ais. It occurs when =. O (a, ) Stud Tip Linear equations can, but do not alwas, pass through the origin. So, proportional relationships are a special tpe of linear equation in which b =. Slope-Intercept Form Words A linear equation written in the form = m + b is in slope-intercept form. The slope of the line is m, and the -intercept of the line is b. Algebra = m + b slope -intercept EXAMPLE Identifing Slopes and -Intercepts Find the slope and the -intercept of the graph of each linear equation. a. = = + ( ) Write in slope-intercept form. The slope is, and the -intercept is. b. = = + Add to each side. The slope is, and the -intercept is. Find the slope and the -intercept of the graph of the linear equation. Eercises 7. = 7. = 8 Chapter Graphing and Writing Linear Equations
EXAMPLE Graphing a Linear Equation in Slope-Intercept Form Graph = +. Identif the -intercept. Step : Find the slope and the -intercept. = + slope -intercept Check Step : The -intercept is. So, plot (, ). Step : Use the slope to find another point and draw the line. m = rise run = Plot the point that is unit right and units down from (, ). Draw a line through the two points. (, ) The line crosses the -ais at (, ). So, the -intercept is. EXAMPLE Real-Life Application The cost (in dollars) of taking a tai miles is =. +. (a) Graph the equation. (b) Interpret the -intercept and the slope. a. The slope of the line is. =. Use the slope and the -intercept to graph the equation. The -intercept is. So, plot (, ). 7 (, ) Use the slope to plot another point, (, 7). Draw a line through the points. b. The slope is.. So, the cost per mile is $.. The -intercept is. So, there is an initial fee of $ to take the tai. Eercises 8 Graph the linear equation. Identif the -intercept. Use a graphing calculator to check our answer.. =. = +. In Eample, the cost (in dollars) of taking a different tai miles is = +.. Interpret the -intercept and the slope. Section. Graphing Linear Equations in Slope-Intercept Form 9
. Eercises Help with Homework. VOCABULARY How can ou find the -intercept of the graph of + =?. CRITICAL THINKING Is the equation = in slope-intercept form? Eplain.. OPEN-ENDED Describe a real-life situation that ou can model with a linear equation. Write the equation. Interpret the -intercept and the slope. 9+(-)= +(-)= +(-9)= 9+(-)= Match the equation with its graph. Identif the slope and the -intercept.. = +. =. = + A. B. C. Find the slope and the -intercept of the graph of the linear equation. 7. = 8. = 7 + 9. =. =. +. + =. = 8.. =. =. = +.. ERROR ANALYSIS Describe and correct the error in finding the slope and the -intercept of the graph of the linear equation. = The slope is, and the -intercept is. 7. SKYDIVING A skdiver parachutes to the ground. The height (in feet) of the skdiver after seconds is = +. a. Graph the equation. b. Interpret the -intercept and the slope. 7 Chapter Graphing and Writing Linear Equations
Graph the linear equation. Identif the -intercept. Use a graphing calculator to check our answer. 8. = + 9. = 7. = 8 + 9. =.. + 9 =. =. APPLES You go to a harvest festival and pick apples. a. Which equation represents the cost (in dollars) of going to the festival and picking pounds of apples? Eplain. = +.7 =.7 + b. Graph the equation ou chose in part (a).. REASONING Without graphing, identif the equations of the lines that are (a) parallel and (b) perpendicular. Eplain our reasoning. = + = = = + = + = + = + =. Si friends create a website. The website earns mone b selling banner ads. The site has banner ads. It costs $ a month to operate the website. a. A banner ad earns $. per click. Write a linear equation that represents the monthl income (in dollars) for clicks. b. Graph the equation in part (a). On the graph, label the number of clicks needed for the friends to start making a profit. Solve the equation for. (Section.) 7. = 8. + = 9. =. 7 + = 8. MULTIPLE CHOICE Which point is a solution of the equation 8 =? (Section.) A (, ) B (, ) C (, ) D (, ) Section. Graphing Linear Equations in Slope-Intercept Form 7
. Graphing Linear Equations in Standard Form equation a + b = c? How can ou describe the graph of the ACTIVITY: Using a Table to Plot Points Work with a partner. You sold a total of $ worth of tickets to a school concert. You lost track of how man of each tpe of ticket ou sold. adult Number of adult tickets + student Number of student tickets = a. Let represent the number of adult tickets. Let represent the number of student tickets. Write an equation that relates and. b. Cop and complete the table showing the different combinations of tickets ou might have sold. Number of Adult Tickets, Number of Student Tickets, Graphing Equations In this lesson, ou will graph linear equations written in standard form. c. Plot the points from the table. Describe the pattern formed b the points. d. If ou remember how man adult tickets ou sold, can ou determine how man student tickets ou sold? Eplain our reasoning. 9 8 7 7 8 9 7 Chapter Graphing and Writing Linear Equations
ACTIVITY: Rewriting an Equation Work with a partner. You sold a total of $ worth of cheese. You forgot how man pounds of each tpe of cheese ou sold. Pounds pound of swiss + Pounds of pound cheddar = Math Practice Understand Quantities What do the equation and the graph represent? How can ou use this information to solve the problem? a. Let represent the number of pounds of swiss cheese. Let represent the number of pounds of cheddar cheese. Write an equation that relates and. b. Rewrite the equation in slope-intercept form. Then graph the equation. c. You sold pounds of cheddar cheese. How man pounds of swiss cheese did ou sell? d. Does the value =. make sense in the contet of the problem? Eplain. 9 8 7 7 8 9. IN YOUR OWN WORDS How can ou describe the graph of the equation a + b = c?. Activities and show two different methods for graphing a + b = c. Describe the two methods. Which method do ou prefer? Eplain.. Write a real-life problem that is similar to those shown in Activities and.. Wh do ou think it might be easier to graph + = without rewriting it in slope-intercept form and then graphing? Use what ou learned about graphing linear equations in standard form to complete Eercises and on page 7. Section. Graphing Linear Equations in Standard Form 7
. Lesson Lesson Tutorials Ke Vocabular standard form, p. 7 Stud Tip An linear equation can be written in standard form. Standard Form of a Linear Equation The standard form of a linear equation is a + b = c where a and b are not both zero. EXAMPLE Graphing a Linear Equation in Standard Form Graph + =. Step : Write the equation in slope-intercept form. + = = Write the equation. Add to each side. = Divide each side b. Step : Use the slope and the -intercept to graph the equation. slope = + ( ) -intercept Check (, ) Use the slope to plot another point, (, ). The -intercept is. So, plot (, ). Draw a line through the points. Eercises Graph the linear equation. Use a graphing calculator to check our graph.. + =. + =. + =. + = 7 Chapter Graphing and Writing Linear Equations
EXAMPLE Graphing a Linear Equation in Standard Form Graph + = using intercepts. Step : To find the -intercept, To find the -intercept, substitute for. substitute for. + = + = + () = + = = = Step : Graph the equation. Check The -intercept is. So, plot (, ). (, ) (, ) The -intercept is. So, plot (, ). Draw a line through the points. EXAMPLE Real-Life Application You have $ to spend on apples and bananas. (a) Graph the equation. +. =, where is the number of pounds of apples and is the number of pounds of bananas. (b) Interpret the intercepts. a. Find the intercepts and graph the equation. -intercept -intercept. +. =. +. =. +.() =.() +. = = = (, ) b. The -intercept shows that ou can bu pounds of apples when ou do not bu an bananas. The -intercept shows that ou can bu pounds of bananas when ou do not bu an apples. 8.. (, ) Eercises 8 Graph the linear equation using intercepts. Use a graphing calculator to check our graph.. = 8. + = 7. WHAT IF? In Eample, ou bu pounds of oranges instead of bananas. Oranges cost $. per pound. Graph the equation. +. =. Interpret the intercepts. Section. Graphing Linear Equations in Standard Form 7
. Eercises Help with Homework. VOCABULARY Is the equation = + in standard form? Eplain.. WRITING Describe two was to graph the equation + =. 9+(-)= +(-)= +(-9)= 9+(-)= Define two variables for the verbal model. Write an equation in slope-intercept form that relates the variables. Graph the equation.. $. pound Pounds of peaches + $. pound Pounds of apples = $. miles Hours hour biked + miles Hours hour walked = miles Write the linear equation in slope-intercept form.. + = 7. = 7. + = Graph the linear equation. Use a graphing calculator to check our graph. 8. 8 + 9 = 7 9. =. + = Match the equation with its graph.. =. + =. + 8 = A. B. C.. ERROR ANALYSIS Describe and correct the error in finding the -intercept.. BRACELET A charm bracelet costs $, plus $ for each charm. The equation + = represents the cost of the bracelet, where is the number of charms. a. Graph the equation. b. How much does the bracelet shown cost? + = () + = = = 7 Chapter Graphing and Writing Linear Equations
Graph the linear equation using intercepts. Use a graphing calculator to check our graph.. = 7. + = 8 8. = 9. SHOPPING The amount of mone ou spend on CDs and DVDs is given b the equation + 8 =. Find the intercepts and graph the equation. Boat: $ /da Gear: $ /da. SCUBA Five friends go scuba diving. The rent a boat for das and scuba gear for das. The total spent is $. a. Write an equation in standard form that represents the situation. b. Graph the equation and interpret the intercepts.. MODELING You work at a restaurant as a host and a server. You earn $9. for each hour ou work as a host and $7. for each hour ou work as a server. a. Write an equation in standard form that models our earnings. b. Graph the equation. Basic Information Pa to the Order of:... John Doe # of hours worked as... host: # of hours worked as... server: Earnings for this pa... period: $.. LOGIC Does the graph of ever linear equation have an -intercept? Eplain our reasoning. Include an eample.. For a house call, a veterinarian charges $7, plus $ an hour. a. Write an equation that represents the total fee (in dollars) the veterinarian charges for a visit lasting hours. b. Find the -intercept. Does this value make sense in this contet? Eplain our reasoning. c. Graph the equation. The points in the table lie on a line. Find the slope of the line. (Section.).. 8. MULTIPLE CHOICE Which value of makes the equation = 9 true? (Section.) A B C D Section. Graphing Linear Equations in Standard Form 77
. Writing Equations in Slope-Intercept Form How can ou write an equation of a line when ou are given the slope and the -intercept of the line? ACTIVITY: Writing Equations of Lines Work with a partner. Find the slope of each line. Find the -intercept of each line. Write an equation for each line. What do the three lines have in common? a. 7 b. Writing Equations In this lesson, ou will write equations of lines in slope-intercept form. c. 8 7 d. 78 Chapter Graphing and Writing Linear Equations
ACTIVITY: Describing a Parallelogram Math Practice Analze Givens What do ou need to know to write an equation? Work with a partner. Find the area of each parallelogram. Write an equation that represents each side of each parallelogram. a. b. 7 ACTIVITY: Interpreting the Slope and the -Intercept Work with a partner. The graph shows a trip taken b a car, where t is the time (in hours) and is the distance (in miles) from Phoeni. a. Find the -intercept of the graph. What does it represent? b. Find the slope of the graph. What does it represent? c. How long did the trip last? d. How far from Phoeni was the car at the end of the trip? e. Write an equation that represents the graph. Distance (miles) Car Trip 8 t Time (hours). IN YOUR OWN WORDS How can ou write an equation of a line when ou are given the slope and the -intercept of the line? Give an eample that is different from those in Activities,, and.. Two sides of a parallelogram are represented b the equations = + and = +. Give two equations that can represent the other two sides. Use what ou learned about writing equations in slope-intercept form to complete Eercises and on page 8. Section. Writing Equations in Slope-Intercept Form 79
. Lesson Lesson Tutorials Stud Tip EXAMPLE After writing an equation, check that the given points are solutions of the equation. Writing Equations in Slope-Intercept Form Write an equation of the line in slope-intercept form. a. b. 7 (, ) (, ) Because the line crosses the -ais at (, ), the -intercept is. So, the equation is = +. (, ) (, ) Find the slope and the -intercept. m = = =, or slope -intercept Find the slope and the -intercept. m = = =, or Because the line crosses the -ais at (, ), the -intercept is. slope -intercept So, the equation is = + ( ), or =. Eercises Write an equation of the line in slope-intercept form... (, ) (, ) (, ) (, ) 8 Chapter Graphing and Writing Linear Equations
Remember EXAMPLE The graph of = a is a horizontal line that passes through (, a). Writing an Equation Which equation is shown in the graph? A = B = C = D = Find the slope and the -intercept. The line is horizontal, so the change in is. change in m = change in = = (, ) Because the line crosses the -ais at (, ), the -intercept is. So, the equation is = + ( ), or =. The correct answer is A. (, ) EXAMPLE Engineers used tunnel boring machines like the ones shown above to dig an etension of the Metro Gold Line in Los Angeles. The new tunnels are.7 miles long and feet wide. Real-Life Application The graph shows the distance remaining to complete a tunnel. (a) Write an equation that represents the distance (in feet) remaining after months. (b) How much time does it take to complete the tunnel? a. Find the slope and the -intercept. change in m = change in = = Because the line crosses the -ais at (, ), the -intercept is. So, the equation is = +. b. The tunnel is complete when the distance remaining is feet. So, find the value of when =. = + Write the equation. = + Substitute for. = Distance remaining (feet) Subtract from each side. 7 = Divide each side b. It takes 7 months to complete the tunnel. (, ) Tunnel Digging (, ) 7 Time (months) Eercises. Write an equation of the line that passes through (, ) and (, ).. WHAT IF? In Eample, the points are (, ) and (, ). How long does it take to complete the tunnel? Section. Writing Equations in Slope-Intercept Form 8
. Eercises Help with Homework. PRECISION Eplain how to find the slope of a line given the intercepts of the line.. WRITING Eplain how to write an equation of a line using its graph. 9+(-)= +(-)= +(-9)= 9+(-)= Write an equation that represents each side of the figure... Write an equation of the line in slope-intercept form.. (, ) (, ). (, ) (, ) 7. (, ) (, ) 8. (, ) (, ) 9. (, ) (, ). (, ) (, ). ERROR ANALYSIS Describe and correct the error in writing an equation of the line.. BOA A boa constrictor is 8 inches long at birth and grows 8 inches per ear. Write an equation that represents the length (in feet) of a boa constrictor that is ears old. = + (, ) (, ) 8 Chapter Graphing and Writing Linear Equations
Write an equation of the line that passes through the points.. (, ), (, ). (, ), (, ). (, ), (, ). WALKATHON One of our friends gives ou $ for a charit walkathon. Another friend gives ou an amount per mile. After miles, ou have raised $. total. Write an equation that represents the amount of mone ou have raised after miles. 7. BRAKING TIME During each second of braking, an automobile slows b about miles per hour. a. Plot the points (, ) and (, ). What do the points represent? b. Draw a line through the points. What does the line represent? c. Write an equation of the line. 8. PAPER You have sheets of notebook paper. After week, ou have 7% of the sheets left. You use the same number of sheets each week. Write an equation that represents the number of pages remaining after weeks. 9. The palm tree on the left is ears old. The palm tree on the right is 8 ears old. The trees grow at the same rate. a. Estimate the height (in feet) of each tree. b. Plot the two points (, ), where is the age of each tree and is the height of each tree. c. What is the rate of growth of the trees? d. Write an equation that represents the height of a palm tree in terms of its age. ft Plot the ordered pair in a coordinate plane. (Skills Review Handbook). (, ). (, ). (, ). (, 7). MULTIPLE CHOICE Which of the following statements is true? (Section.) A The -intercept is. B The -intercept is. C The -intercept is. D The -intercept is. Section. Writing Equations in Slope-Intercept Form 8
.7 Writing Equations in Point-Slope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the line that has the given slope and passes through the given point. Find the -intercept of the line. Write an equation of the line. a. m = b. m = 8 7 7 Writing Equations In this lesson, ou will write equations of lines using a slope and a point. write equations of lines using two points. c. m = 7 d. m = 7 8 Chapter Graphing and Writing Linear Equations
ACTIVITY: Deriving an Equation Math Practice Construct Arguments How does a graph help ou derive an equation? Work with a partner. a. Draw a nonvertical line that passes through the point (, ). b. Plot another point on our line. Label this point as (, ). This point represents an other point on O the line. c. Label the rise and the run of the line through the points (, ) and (, ). d. The rise can be written as. The run can be written as. Eplain wh this is true. e. Write an equation for the slope m of the line using the epressions from part (d). f. Multipl each side of the equation b the epression in the denominator. Write our result. What does this result represent? (, ) ACTIVITY: Writing an Equation Work with a partner. For months, ou saved $ a month. You now have $7 in our savings account. Draw a graph that shows the balance in our account after t months. Use our result from Activit to write an equation that represents the balance A after t months. Balance (dollars) Savings Account A 7 7 7 8 9 t Time (months). Redo Activit using the equation ou found in Activit. Compare the results. What do ou notice?. Wh do ou think = m( ) is called the point-slope form of the equation of a line? Wh do ou think it is important?. IN YOUR OWN WORDS How can ou write an equation of a line when ou are given the slope and a point on the line? Give an eample that is different from those in Activit. Use what ou learned about writing equations using a slope and a point to complete Eercises on page 88. Section.7 Writing Equations in Point-Slope Form 8
.7 Lesson Lesson Tutorials Ke Vocabular point-slope form, p. 8 Point-Slope Form Words A linear equation written in the form = m( ) is in point-slope form. The line passes through the point (, ), and the slope of the line is m. slope (, ) Algebra = m( ) (, ) passes through (, ) O EXAMPLE Writing an Equation Using a Slope and a Point Write in point-slope form an equation of the line that passes through the point (, ) with slope. = m( ) Write the point-slope form. = [ ( )] Substitute for m, for, and for. = ( + ) Simplif. So, the equation is = ( + ). Check Check that (, ) is a solution of the equation. = ( + ) Write the equation. =? ( + ) Substitute. = Simplif. Eercises Write in point-slope form an equation of the line that passes through the given point and has the given slope.. (, ); m =. (7, ); m =. ( 8, ); m = 8 Chapter Graphing and Writing Linear Equations
Stud Tip EXAMPLE You can use either of the given points to write the equation of the line. Use m = and (, ). ( ) = ( ) + = + = + 8 Writing an Equation Using Two Points Write in slope-intercept form an equation of the line that passes through the points (, ) and (, ). Find the slope: m = = = = Then use the slope m = and the point (, ) to write an equation of the line. = m( ) Write the point-slope form. = ( ) Substitute for m, for, and for. = + = + 8 Distributive Propert Write in slope-intercept form. EXAMPLE Real-Life Application You finish parasailing and are being pulled back to the boat. After seconds, ou are feet above the boat. (a) Write and graph an equation that represents our height (in feet) above the boat after seconds. (b) At what height were ou parasailing? a. You are being pulled down at the rate of feet per second. So, the slope is. You are feet above the boat after seconds. So, the line passes through (, ). Use the point-slope form. = ( ) Substitute for m,, and. feet per second = + = + Distributive Propert Write in slope-intercept form. So, the equation is = +. b. You start descending when =. The -intercept is. So, ou were parasailing at a height of feet. (, ) 7 Eercises 7 Write in slope-intercept form an equation of the line that passes through the given points.. (, ), (, ). (, ), (, ). ( 8, ), (, 9) 7. WHAT IF? In Eample, ou are feet above the boat after seconds. Write and graph an equation that represents our height (in feet) above the boat after seconds. Section.7 Writing Equations in Point-Slope Form 87
.7 Eercises Help with Homework. VOCABULARY From the equation = ( + ), identif the slope and a point on the line.. WRITING Describe how to write an equation of a line using (a) its slope and a point on the line and (b) two points on the line. 9+(-)= +(-)= +(-9)= 9+(-)= Use the point-slope form to write an equation of the line with the given slope that passes through the given point.. m =. m =. m = Write in point-slope form an equation of the line that passes through the given point and has the given slope.. (, ); m = 7. (, 8); m = 8. (, ); m = 9. (7, ); m = 7. (, ); m =. (, ); m = Write in slope-intercept form an equation of the line that passes through the given points.. (, ), (, ). (, ), (, ). (, ), (, 7). (, ), (8, ). ( 9, ), (, ) 7. (, ), (, ) 8. CHEMISTRY At C, the volume of a gas is liters. For each degree the temperature T (in degrees Celsius) increases, the volume V (in liters) of the gas increases b. Write an equation that represents the volume of the gas in terms of the temperature. 88 Chapter Graphing and Writing Linear Equations
9. CARS After it is purchased, the value of a new car decreases $ each ear. After ears, the car is worth $8,. a. Write an equation that represents the value V (in dollars) of the car ears after it is purchased. b. What was the original value of the car?. REASONING Write an equation of a line that passes through the point (8, ) that is (a) parallel and (b) perpendicular to the graph of the equation =.. CRICKETS According to Dolbear s law, ou can predict the temperature T (in degrees Fahrenheit) b counting the number of chirps made b a snow tree cricket in minute. For each rise in temperature of. F, the cricket makes an additional chirp each minute. a. A cricket chirps times in minute when the temperature is F. Write an equation that represents the temperature in terms of the number of chirps in minute. b. You count chirps in minute. What is the temperature? c. The temperature is 9 F. How man chirps would ou epect the cricket to make? Leaning Tower of Pisa (.7, ). WATERING CAN You water the plants in our classroom at a constant rate. After seconds, our watering can contains 8 ounces of water. Fifteen seconds later, the can contains 8 ounces of water. a. Write an equation that represents the amount (in ounces) of water in the can after seconds. b. How much water was in the can when ou started watering the plants? c. When is the watering can empt? 7.7 m. Problem Solving The Leaning Tower of Pisa in Ital was built between 7 and. a. Write an equation for the ellow line. b. The tower is meters tall. How far off center is the top of the tower? Graph the linear equation. (Section.). =. = +. = 7. MULTIPLE CHOICE What is the -intercept of the equation + =? (Section.) A B C D Section.7 Writing Equations in Point-Slope Form 89
..7 Quiz Find the slope and the -intercept of the graph of the linear equation. (Section.) ). = 8. = + Find the - and -intercepts of the graph of the equation. (Section.). =. + = Write an equation of the line in slope-intercept form. (Section.). (, ) (, ). (, ) (, ) 7. Progress Check (, ) (, ) Write in point-slope form an equation of the line that passes through the given point and has the given slope. (Section.7) 8. (, ); m = 9. (, ); m =. (, ); m =. (8, ); m = 8 Write in slope-intercept form an equation of the line that passes through the given points. (Section.7). (, ) (, ). (, ), (, ). STATE FAIR The cost (in dollars) of one person buing admission to a fair and going on rides is = +. (Section.) a. Graph the equation. b. Interpret the -intercept and the slope.. PAINTING You used $9 worth of paint for a school float. (Section.) a. Graph the equation 8 + = 9, where is the number of gallons of blue paint and is the number of gallons of white paint. b. Interpret the intercepts.. CONSTRUCTION A construction crew is etending a highwa sound barrier that is miles long. The crew builds of a mile per week. Write an equation that represents the length (in miles) of the barrier after weeks. (Section.) 9 Chapter Graphing and Writing Linear Equations
Chapter Review Review Ke Vocabular linear equation p. solution of a linear equation, p. slope, p. rise, p. run, p. Vocabular Help -intercept, p. 8 -intercept, p. 8 slope-intercept form, p. 8 standard form, p. 7 point-slope form, p. 8 Review Eamples and Eercises. Graphing Linear Equations (pp. 7) Graph =. Step : Make a table of values. = (, ) = ( ) 7 (, 7) = ( ) (, ) = () (, ) = () (, ) Step : Plot the ordered pairs. Step : Draw a line through the points. (, ) (, ) (, ) (, ) (, ) (, ) (, 7) 7 (, 7) 7 Graph the linear equation.. =. =. = 9. =. = +. = Chapter Review 9
. Slope of a Line (pp. 8 7) Find the slope of each line in the graph. Red Line: m = ( ) = = 8 The slope of the red line is undefined. Blue Line: m = = ( ) = 7, or 7 Green Line: m = = =, or (, ) (, ) (, ) (, ) (, ) (, ) The points in the table lie on a line. Find the slope of the line. 7. 8. 9. Are the lines = and = parallel? Are the perpendicular? Eplain.. Graphing Proportional Relationships (pp. 8 ) The cost (in dollars) for tickets to a movie is represented b the equation = 7. Graph the equation and interpret the slope. The equation shows that the slope m is 7. So, the graph passes through (, ) and (, 7). Plot the points and draw a line through the points. Because negative values of do not make sense in this contet, graph in the first quadrant onl. The slope indicates that the unit cost is $7 per ticket. Cost (dollars) Movie Tickets (, 7) (, ) Number of tickets. RUNNING The number of miles ou run after weeks is represented b the equation = 8. Graph the equation and interpret the slope.. STUDYING The number of hours that ou stud after das is represented b the equation =.. Graph the equation and interpret the slope. 9 Chapter Graphing and Writing Linear Equations
. Graphing Linear Equations in Slope-Intercept Form (pp. 7) Graph =.. Identif the -intercept. Step : Find the slope and the -intercept. slope =. + ( ) Step : The -intercept is. So, plot (, ). Step : Use the slope to find another point and draw the line. m = rise run = -intercept Plot the point that is units right and unit up from (, ). Draw a line through the two points. The line crosses the -ais at (, ). So, the -intercept is. (, ) (, ). Graph the linear equation. Identif the -intercept. Use a graphing calculator to check our answer.. =. = + 8. = 8. Graphing Linear Equations in Standard Form (pp. 7 77) Graph 8 + =. Step : Write the equation in slope-intercept form. 8 + = = 8 + Write the equation. Subtract 8 from each side. = + Divide each side b. Step : Use the slope and the -intercept to graph the equation. slope = + -intercept The -intercept is. So, plot (, ). Draw a line through the points. (, ) 8 (, ) Use the slope to plot another point, (, ). Chapter Review 9
Graph the linear equation.. + =. + = 8 7. + = 8. + 8 = 9. A dog kennel charges $ per night to board our dog and $ for each hour of platime. The amount of mone ou spend is given b + = 8, where is the number of nights and is the number of hours of platime. Graph the equation and interpret the intercepts.. Writing Equations in Slope-Intercept Form (pp. 78 8) Write an equation of the line in slope-intercept form. a. Find the slope and the -intercept. (, ) (, ) m = = =, or Because the line crosses the -ais at (, ), the -intercept is. slope -intercept So, the equation is = +, or = +. b. (, ) (, ) Find the slope and the -intercept. ( ) = m = =, or Because the line crosses the -ais at (, ), the -intercept is. slope -intercept So, the equation is = + ( ), or =. 9 Chapter Graphing and Writing Linear Equations
Write an equation of the line in slope-intercept form.. (, ) (, ). (, ) (, ). (, ) (, ). (, ) (, ). Write an equation of the line that passes through (, 8) and (, 8).. Write an equation of the line that passes through (, ) and (, )..7 Writing Equations in Point-Slope Form (pp. 8 89) Write in slope-intercept form an equation of the line that passes through the points (, ) and (, ). Find the slope. m = = =, or Then use the slope and one of the given points to write an equation of the line. Use m = and (, ). = m( ) Write the point-slope form. = ( ) Substitute for m, for, and for. = 8 = 7 Distributive Propert Write in slope-intercept form. (, ) (, ) So, the equation is = 7.. Write in point-slope form an equation of the line that passes through the point (, ) with slope. 7. Write in slope-intercept form an equation of the line that passes through the points (, ) and (, ). Chapter Review 9
Chapter Test Test Practice Find the slope and the -intercept of the graph of the linear equation.. =. = +. =. = + 8.. +. =.. + = 7 Graph the linear equation. 7. = + 8. = 9. + =. Which lines are parallel? Which lines. The points in the table lie on a line. are perpendicular? Eplain. Find the slope of the line. (, ) (, ) (, ) (, ) (, ) (,.) (, ) (, ) Write an equation of the line in slope-intercept form.. (, ) (, ). (, ) (, ) Write in slope-intercept form an equation of the line that passes through the given points.. (, ), (, ). (, ), (, ). (, ), (, ) 7. VOCABULARY The number of new vocabular words that ou learn after weeks is represented b the equation =. a. Graph the equation and interpret the slope. b. How man new vocabular words do ou learn after weeks? c. How man more vocabular words do ou learn after weeks than after weeks? 9 Chapter Graphing and Writing Linear Equations
Cumulative Assessment. Which equation matches the line shown in the graph? Test-Taking Strateg Estimate the Answer A. = B. = + C. = D. = + O Using estimation, ou can see that there are about hairs. So, it has to be C.. The equation = is written in standard form. Which point lies on the graph of this equation? F. (, ) H. (, ) G. (, ) I. (, ). Which line has a slope of? A. C. O B. D. O O Cumulative Assessment 97