.5 Graphing Linear Equations in Slope-Intercept Form COMMON CORE Learning Standards HSA-CED.A. HSF-IF.B. HSF-IF.C.7a HSF-LE.B.5 Essential Question How can ou describe the graph of the equation = m + b? 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 6 slope = 6 Finding Slopes and -Intercepts Work with a partner. Find the slope and -intercept of each line. a. 6 b. MAKING CONJECTURES To be proficient in math, ou first need to collect and organize data. Then make conjectures about the patterns ou observe in the data. = + Writing a Conjecture = Work with a partner. Graph each equation. Then cop and complete the table. Use the completed table to write a conjecture about the relationship between the graph of = m + b and the values of m and b. Equation Description of graph Slope of graph -Intercept a. = + Line b. = c. = + d. = Communicate Your Answer. 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) b choosing one equation from Eploration and () varing onl m and () varing onl b. Section.5 Graphing Linear Equations in Slope-Intercept Form 5
.5 Lesson What You Will Learn Core Vocabular slope, p. 6 rise, p. 6 run, p. 6 slope-intercept form, p. 8 constant function, p. 8 Previous dependent variable independent variable Find the slope of a line. Use the slope-intercept form of a linear equation. Use slopes and -intercepts to solve real-life problems. The Slope of a Line Core Concept Slope The slope m of a nonvertical line passing through two points (, ) and (, ) is the ratio of ( the rise (change in ) to the run (change in )., ) slope = m = rise run change in = change in = (, ) run = O rise = When the line rises from left to right, the slope is positive. When the line falls from left to right, the slope is negative. Finding the Slope of a Line Describe the slope of each line. Then find the slope. a. (, ) b. (0, ) STUDY TIP When finding slope, ou can label either point as (, ) and the other point as (, ). The result is the same. (, ) a. The line rises from left to right. So, the slope is positive. Let (, ) = (, ) and (, ) = (, ). 6 (, ) b. The line falls from left to right. So, the slope is negative. Let (, ) = (0, ) and (, ) = (, ). READING In the slope formula, is read as sub one and is read as sub two. The numbers and in and are called subscripts. m = = ( ) ( ) = 6 = Monitoring Progress Describe the slope of the line. Then find the slope... (, ) (, ) m = = 0 = = Help in English and Spanish at BigIdeasMath.com (, ) (, ). (5, ) 8 (, ) 6 Chapter Graphing Linear Functions
Finding Slope from a Table STUDY TIP As a check, ou can plot the points represented b the table to verif that the line through them has a slope of. The points represented b each table lie on a line. How can ou find the slope of each line from the table? What is the slope of each line? a. b. c. 0 7 0 0 8 6 5 9 a. Choose an two points from the table and use the slope formula. Use the points (, ) = (, 0) and (, ) = (7, ). m = 0 = 7 = 6, or The slope is. b. Note that there is no change in. Choose an two points from the table and use the slope formula. Use the points (, ) = (, ) and (, ) = (5, ). m = = 5 ( ) = 0, or 0 6 The change in is 0. The slope is 0. c. Note that there is no change in. Choose an two points from the table and use the slope formula. Use the points (, ) = (, 0) and (, ) = (, 6). m = 6 = 0 ( ) = 6 0 The change in is 0. Because division b zero is undefined, the slope of the line is undefined. Monitoring Progress Help in English and Spanish at BigIdeasMath.com The points represented b the table lie on a line. How can ou find the slope of the line from the table? What is the slope of the line?. 6 8 5. 5 5 5 5 0 5 0 5 9 6 Concept Summar Slope Positive slope Negative slope Slope of 0 Undefined slope O O O O The line rises from left to right. The line falls from left to right. The line is horizontal. The line is vertical. Section.5 Graphing Linear Equations in Slope-Intercept Form 7
Using the Slope-Intercept Form of a Linear Equation Core Concept 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 (0, b) = m + b slope -intercept A linear equation written in the form = 0 + b, or = b, is a constant function. The graph of a constant function is a horizontal line. Identifing Slopes and -Intercepts Find the slope and the -intercept of the graph of each linear equation. a. = b. = 6.5 c. 5 = STUDY TIP For a constant function, ever input has the same output. For instance, in Eample b, ever input has an output of 6.5. STUDY TIP When ou rewrite a linear equation in slope-intercept form, ou are epressing as a function of. a. = m + b Write the slope-intercept form. slope -intercept = + ( ) Rewrite the original equation in slope-intercept form. The slope is, and the -intercept is. b. The equation represents a constant function. The equation can also be written as = 0 + 6.5. The slope is 0, and the -intercept is 6.5. c. Rewrite the equation in slope-intercept form b solving for. 5 = Write the original equation. + 5 + 5 Add 5 to each side. = 5 = 5 = 5 + Simplif. Divide each side b. Simplif. The slope is 5, and the -intercept is. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the slope and the -intercept of the graph of the linear equation. 6. = 6 + 7. = 8 8. + = 0 8 Chapter Graphing Linear Functions
STUDY TIP You can use the slope to find points on a line in either direction. In Eample, note that the slope can be written as. So, ou could move unit left and units up from (0, ) to find the point (, ). REMEMBER You can also find the -intercept b substituting 0 for in the equation + = and solving for. Graph + =. Identif the -intercept. Using Slope-Intercept Form to Graph Step Rewrite the equation in slope-intercept form. = + Step Find the slope and the -intercept. m = and b = Step The -intercept is. So, plot (0, ). Step Use the slope to find another point on the line. slope = rise run = Plot the point that is unit right and units down from (0, ). Draw a line through the two points. The line crosses the -ais at (, 0). So, the -intercept is. (0, ) Graphing from a Verbal Description A linear function g models a relationship in which the dependent variable increases units for ever unit the independent variable increases. Graph g when g(0) =. Identif the slope, -intercept, and -intercept of the graph. Because the function g is linear, it has a constant rate of change. Let represent the independent variable and represent the dependent variable. 5 (0, ) (, 0) Step Find the slope. When the dependent variable increases b, the change in is +. When the independent variable increases b, the change in is +. So, the slope is, or. Step Find the -intercept. The statement g(0) = indicates that when = 0, =. So, the -intercept is. Plot (0, ). Step Use the slope to find another point on the line. A slope of can be written as. Plot the point that is unit left and units down from (0, ). Draw a line through the two points. The line crosses the -ais at (, 0). So, the -intercept is. The slope is, the -intercept is, and the -intercept is. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the linear equation. Identif the -intercept. 9. = 0. + =. + = 6. A linear function h models a relationship in which the dependent variable decreases units for ever 5 units the independent variable increases. Graph h when h(0) =. Identif the slope, -intercept, and -intercept of the graph. Section.5 Graphing Linear Equations in Slope-Intercept Form 9
Solving Real-Life Problems In most real-life problems, slope is interpreted as a rate, such as miles per hour, dollars per hour, or people per ear. Modeling with Mathematics A submersible that is eploring the ocean floor begins to ascend to the surface. The elevation h (in feet) of the submersible is modeled b the function h(t) = 650t,000, where t is the time (in minutes) since the submersible began to ascend. a. Graph the function and identif its domain and range. b. Interpret the slope and the intercepts of the graph. STUDY TIP Because t is the independent variable, the horizontal ais is the t-ais and the graph will have a t-intercept. Similarl, the vertical ais is the h-ais and the graph will have an h-intercept.. Understand the Problem You know the function that models the elevation. You are asked to graph the function and identif its domain and range. Then ou are asked to interpret the slope and intercepts of the graph.. Make a Plan Use the slope-intercept form of a linear equation to graph the function. Onl graph values that make sense in the contet of the problem. Eamine the graph to interpret the slope and the intercepts.. Solve the Problem a. The time t must be greater than or equal to 0. The elevation h is below sea level and must be less than or equal to 0. Use the slope of 650 and the h-intercept of,000 to graph the function in Quadrant IV. The domain is 0 t 0, and the range is,000 h 0. b. The slope is 650. So, the submersible ascends at a rate of 650 feet per minute. The h-intercept is,000. So, the elevation of the submersible after 0 minutes, or when the ascent begins, is,000 feet. The t-intercept is 0. So, the submersible takes 0 minutes to reach an elevation of 0 feet, or sea level.. Look Back You can check that our graph is correct b substituting the t-intercept for t in the function. If h = 0 when t = 0, the graph is correct. h = 650(0),000 Substitute 0 for t in the original equation. h = 0 Simplif. Elevation (feet) Elevation of a Submersible Time (minutes) 0 8 6 0 t 0 (0, 0),000 8,000,000 h (0,,000) Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? The elevation of the submersible is modeled b h(t) = 500t 0,000. (a) Graph the function and identif its domain and range. (b) Interpret the slope and the intercepts of the graph. 0 Chapter Graphing Linear Functions
.5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE The of a nonvertical line passing through two points is the ratio of the rise to the run.. VOCABULARY What is a constant function? What is the slope of a constant function?. WRITING What is the slope-intercept form of a linear equation? Eplain wh this form is called the slope-intercept form.. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. = 5 = 8 = + = + Monitoring Progress and Modeling with Mathematics In Eercises 5 8, describe the slope of the line. Then find the slope. (See Eample.) 5. 7. (, ) (, ) (, ) (, ) 5 6. 8. (, ) (, ) (0, ) (5, ) In Eercises 9, the points represented b the table lie on a line. Find the slope of the line. (See Eample.) 9. 9 5 0. 5 9. ANALYZING A GRAPH The graph shows the distance (in miles) that a bus travels in hours. Find and interpret the slope of the line. Distance (miles) 50 00 50 Bus Trip (, 0) (, 60) 0 0 Time (hours). ANALYZING A TABLE The table shows the amount (in hours) of time ou spend at a theme park and the admission fee (in dollars) to the park. The points represented b the table lie on a line. Find and interpret the slope of the line. 0.. 5 8 6 6 6 6 0 0 0 0 0 8 Time (hours), Admission (dollars), 6 5.99 7 5.99 8 5.99 Section.5 Graphing Linear Equations in Slope-Intercept Form
In Eercises 5, find the slope and the -intercept of the graph of the linear equation. (See Eample.) 5. = + 6. = 7 7. = 6 8. = 9. + = 0. + = 6. 5 = 8. 0 = + ERROR ANALYSIS In Eercises and, describe and correct the error in finding the slope and the -intercept of the graph of the equation.. = The slope is, and the -intercept is 0. 5. GRAPHING FROM A VERBAL DESCRIPTION A linear function r models the growth of our right inde fingernail. The length of the fingernail increases 0.7 millimeter ever week. Graph r when r (0) =. Identif the slope and interpret the -intercept of the graph. 6. GRAPHING FROM A VERBAL DESCRIPTION A linear function m models the amount of milk sold b a farm per month. The amount decreases 500 gallons for ever $ increase in price. Graph m when m(0) = 000. Identif the slope and interpret the - and -intercepts of the graph. 7. MODELING WITH MATHEMATICS The function shown models the depth d (in inches) of snow on the ground during the first 9 hours of a snowstorm, where t is the time (in hours) after the snowstorm begins. (See Eample 6.). = 6 The slope is, and the -intercept is 6. d(t) = t + 6 In Eercises 5, graph the linear equation. Identif the -intercept. (See Eample.) 5. = + 7 6. = + 7. = 8. = 9. + = 0. + = 8. + 5 = 0. + 6 = 0 In Eercises and, graph the function with the given description. Identif the slope, -intercept, and -intercept of the graph. (See Eample 5.). A linear function f models a relationship in which the dependent variable decreases units for ever units the independent variable increases. The value of the function at 0 is.. A linear function h models a relationship in which the dependent variable increases unit for ever 5 units the independent variable decreases. The value of the function at 0 is. a. Graph the function and identif its domain and range. b. Interpret the slope and the d-intercept of the graph. 8. MODELING WITH MATHEMATICS The function c() = 0.5 + 70 represents the cost c (in dollars) of renting a truck from a moving compan, where is the number of miles ou drive the truck. a. Graph the function and identif its domain and range. b. Interpret the slope and the c-intercept of the graph. 9. COMPARING FUNCTIONS A linear function models the cost of renting a truck from a moving compan. The table shows the cost (in dollars) when ou drive the truck miles. Graph the function and compare the slope and the -intercept of the graph with the slope and the c-intercept of the graph in Eercise 8. Miles, Cost (dollars), 0 0 50 80 00 0 Chapter Graphing Linear Functions
ERROR ANALYSIS In Eercises 0 and, describe and correct the error in graphing the function. 0.. MATHEMATICAL CONNECTIONS The graph shows the relationship between the base length and the side length (of the two equal sides) of an isosceles triangle in meters. The perimeter of a second isosceles triangle is 8 meters more than the perimeter of the first triangle. + = (0, ) 8 = 6. + = (0, ) 0 0 8 a. Graph the relationship between the base length and the side length of the second triangle. b. How does the graph in part (a) compare to the graph shown? 5. ANALYZING EQUATIONS Determine which of the equations could be represented b each graph.. MATHEMATICAL CONNECTIONS Graph the four equations in the same coordinate plane. = = + 8 = = 7 = = = 0 = = 7 = 9 = 6 = + 5 a. What enclosed shape do ou think the lines form? Eplain. a. b. b. Write a conjecture about the equations of parallel lines.. MATHEMATICAL CONNECTIONS The graph shows the relationship between the width and the length of a rectangle in inches. The perimeter of a second rectangle is 0 inches less than the perimeter of the first rectangle. a. Graph the relationship between the width and length of the second rectangle. b. How does the graph in part (a) compare to the the graph shown? 6 8 = 0 0 0 8 6 c. d. 6. MAKING AN ARGUMENT Your friend sas that ou can write the equation of an line in slope-intercept form. Is our friend correct? Eplain our reasoning. Section.5 Graphing Linear Equations in Slope-Intercept Form
7. WRITING Write the definition of the slope of a line in two different was. 8. THOUGHT PROVOKING Your famil goes on vacation to a beach 00 miles from our house. You reach our destination 6 hours after departing. Draw a graph that describes our trip. Eplain what each part of our graph represents. 9. ANALYZING A GRAPH The graphs of the functions g() = 6 + a and h() = + b, where a and b are constants, are shown. The intersect at the point (p, q). (p, q) 50. HOW DO YOU SEE IT? You commute to school b walking and b riding a bus. The graph represents our commute. Distance (miles) d Commute to School 0 0 8 6 t Time (minutes) a. Describe our commute in words. b. Calculate and interpret the slopes of the different parts of the graph. PROBLEM SOLVING In Eercises 5 and 5, find the value of k so that the graph of the equation has the given slope or -intercept. a. Label the graphs of g and h. b. What do a and b represent? c. Starting at the point (p, q), trace the graph of g until ou get to the point with the -coordinate p +. Mark this point C. Do the same with the graph of h. Mark this point D. How much greater is the -coordinate of point C than the -coordinate of point D? Maintaining Mathematical Proficienc Find the coordinates of the figure after the transformation. (Skills Review Handbook) 5. Translate the rectangle units left. A D B C 55. Dilate the triangle with respect to the origin using a scale factor of. X Y Z 5. = k 5; m = 5. = + 5 k; b = 0 6 5. ABSTRACT REASONING To show that the slope of a line is constant, let (, ) and (, ) be an two points on the line = m + b. Use the equation of the line to epress in terms of and in terms of. Then use the slope formula to show that the slope between the points is m. Reviewing what ou learned in previous grades and lessons 56. Reflect the trapezoid in the -ais. Q T R S Determine whether the equation represents a linear or nonlinear function. Eplain. (Section.) 57. 9 = 58. = + 5 59. + = 60. = 6 Chapter Graphing Linear Functions