6.1 Slope-Intercept Form

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1 Name Class Date 6.1 Slope-Intercept Form Essential Question: How can ou represent a linear function in a wa that reveals its slope and -intercept? Resource Locker Eplore Graphing Lines Given Slope and -intercept Graphs of linear equations can be used to model man real-life situations. Given the slope and -intercept, ou can graph the line, and use the graph to answer questions. Andrew wants to bu a smart phone that costs $5. His parents will pa for the phone, and Andrew will pa them $5 each month until the entire amount is repaid. The loan repament represents a linear situation in which the amount that Andrew owes his parents is dependent on the number of paments he has made. When =, =. The -intercept of the graph of the equation that represents the situation is Houghton Mifflin Harcourt Publishing Compan Image Credits: Echo/ Cultura/Gett Images B The rate of change in the amount Andrew owes over time is The slope is. per month. Use the -intercept to plot a point on the graph of the equation. The -intercept is, so plot the point. Using the definition of slope, plot a second point. Change in Slope = Change in =_ =. 1 Start at the point ou plotted. Count and unit right and plot another point. Draw a line through the points ou plotted. units down Amount ($) Amount Andrew Owes Time (Months) Module Lesson 1

2 Reflect 1. Discussion How can ou use the same method to find two more points on that same line? 2. How man months will it take Andrew to pa off his loan? Eplain our answer. Eplain 1 Creating Linear Equations in Slope-Intercept Form You can use the slope formula to derive the slope-intercept form of a linear equation. Consider a line with slope m and -intercept b. 2-1 The slope formula is m = _ Substitute (, b) for ( 1, 1 ) and (, ) for ( 2, 2 ). - b m = _ - - b m = _ m = - b Multipl both sides b ( ). m + b = Add b to both sides. Slope-Intercept Form of an Equation = m + b If a line has slope m and -intercept (, b), then the line is described b the equation = m + b. Eample 1 Write the equation of each line in slope-intercept form. Slope is 3, and (2, 5) is on the line. Step 1: Find the -intercept. = m + b Write the slope intercept form. 5 = 3 (2) + b Substitute 3 for m, 2 for, and 5 for. 5 = 6 + b Multipl. 5-6 = 6 + b - 6 Subtract 6 from both sides. -1 = b Simplif. Step 2: Write the equation. = m + b Write the slope intercept form. = 3 + (-1) Substitute 3 for m and -1 for b. = 3-1 Houghton Mifflin Harcourt Publishing Compan Module 6 2 Lesson 1

3 B The line passes through (, 5) and (2, 13). Step 1: Use the points to find the slope. m = _ Substitute (, 5) for ( 1, 1 ) and (, ) for ( 2, 2 ). m = _ = _ = Step 2: Substitute the slope and - and -coordinates of either of the points in the equation = m + b. Step 3: Substitute for m and for b in the equation = m + b. The equation of the line is. = m + b = ( ) + b = + b - = + b - = b Your Turn Write the equation of each line in slope-intercept form. 3. Slope is 1, and (3, 2) is on the line.. The line passes through (1, ) and (3, 1). Eplain 2 Graphing from Slope-Intercept Form Writing an equation in slope-intercept form can make it easier to graph the equation. Eample 2 Write each equation in slope-intercept form. Then graph the line. Houghton Mifflin Harcourt Publishing Compan = 5 - The equation = 5 - is alread in slope-intercept form. Slope: m = 5 = 5_ 1 -intercept: b = Step 1: Plot (, ) Step 2: Count 5 units up and 1 unit to the right and plot another point. Step 3: Draw a line through the points. 2 2 Module 6 21 Lesson 1

4 B = 6 Step 1: Write the equation in slope-intercept form b solving for = 6 - Slope: 2 6 = + 6 -intercept: 2 = + Step 2: Graph the line. Plot (, ). Move unit down and units to the right to plot a second point. Draw a line through the points. Your Turn Write each equation in slope-intercept form. Then graph the line = = Eplain 3 Determining Solutions of Equations in Two Variables Given a real-world linear situation described b a table, a graph, or a verbal description, ou can write an equation in slope-intercept form. You can use that equation to solve problems. Eample 3 Identif the slope and -intercept of the graph that represents each linear situation and interpret what the mean. Then write an equation in slope-intercept form and use it to solve the problem. For one tai compan, the cost in dollars of a tai ride is a linear function of the distance in miles traveled. The initial charge is $2.5, and the charge per mile is $.35. Find the cost of riding a distance of 1 miles. The rate of change is $.35 per mile, so the slope, m, is.35. The initial cost is the cost to travel miles, $2.5, so the -intercept, b, is 2.5. Houghton Mifflin Harcourt Publishing Compan Then an equation is = = To find the cost of riding 1 miles, substitute 1 for. =.35 (1) = 6 (6, 1) is a solution of the equation, and the cost of riding a distance of 1 miles is $6. Module 6 22 Lesson 1

5 B A chairlift descends from a mountain top to pick up skiers at the bottom. The height in feet of the chairlift is a linear function of the time in minutes since it begins descending as shown in the graph. Find the height of the chairlift 2 minutes after it begins descending. Height (ft) Height of a Chairlift (, 5) (, 2) Time (min) The graph contains the points (, ) and (, 2). - 5 The slope is =. - It represents the rate at which the chairlift. The graph passes through the point (, ), so the -intercept is. It represents the height Houghton Mifflin Harcourt Publishing Compan Image Credits: Moodboard/Corbis of the chairlift minutes after it begins descending. Let be the time in minutes after the chairlift begins to descend. Let be the height of the chairlift in feet. The equation is. To find the height after 2 minutes, substitute 2 for and simplif. = ( ) + 5 = + 5 = is a solution of the equation, and the height of the chairlift 2 minutes after it begins descending is feet. Reflect 7. In the eample involving the tai, how would the equation change if the cost per mile increased or decreased? How would this affect the graph? Module 6 23 Lesson 1

6 Your Turn Identif the slope and -intercept of the graph that represents the linear situation and interpret what the mean. Then write an equation in slope-intercept form and use it to solve the problem.. A local club charges an initial membership fee as well as a monthl cost. The cost C in dollars is a linear function of the number of months of membership. Find the cost of the membership after months. Membership Cost Time (months) Cost ($) Elaborate 9. What are some advantages to using slope-intercept form? 1. What are some disadvantages of slope-intercept form? 11. Essential Question Check-In When given a real-world situation that can be described b a linear equation, how can ou identif the slope and -intercept of the graph of the equation? Evaluate: Homework and Practice For each situation, determine the slope and -intercept of the graph of the equation that describes the situation. 1. John gets a new job and receives a $5 signing bonus. After that, he makes $2 a da. Online Homework Hints and Help Etra Practice Houghton Mifflin Harcourt Publishing Compan Module 6 2 Lesson 1

7 2. Jennifer is 2 miles north of her house, and she is driving north on the highwa at a rate of 55 miles per hour. Sketch a graph that represents the situation. 3. Morwenna rents a truck. She pas $2 plus $.25 per mile. 2 Cost of a Rental Truck. An investor invests $5 in a certain stock. After the first si months, the value of the stock has increased at a rate of $2 per month. 62 Value of Investment Cost ($) Amount ($) Time (months) Distance (mi) Write the equation of each line in slope-intercept form. 5. Slope is 3, and (1, 5) is on the line. 6. Slope is, and (5, 3) is on the line. Houghton Mifflin Harcourt Publishing Compan 7. Slope is _ 1, and (, 2) is on the line.. Slope is 5, and (2, 6) is on the line. 9. Slope is - 2_ 3, and (-6, -5) is on the line. 1. Slope is - 1_, and (-3, 2) is on the line. 2 Module 6 25 Lesson 1

8 11. Passes through (5, 7) and (3, 1) 12. Passes through (-6, 1) and (-3, ) 13. Passes through (6, 6) and (, 2) 1. Passes through (-1, -5) and (2, 6) Write each equation in slope-intercept form. Identif the slope and -intercept. Then graph the line described b the equation. 15. = = = 2_ 3-1. = - 1_ = = -12 Houghton Mifflin Harcourt Publishing Compan Module 6 26 Lesson 1

9 = = Sports A figure skating school offers introductor lessons at $25 per session. There is also a registration fee of $3. Write a linear equation in slope-intercept form that represents the situation. You want to take at least 6 lessons. Can ou pa for those lessons using a $2 gift certificate? If so, how much mone, if an, will be left on the gift certificate? If not, eplain wh not. Houghton Mifflin Harcourt Publishing Compan Image Credits: YinYang/ Gett Images 2. Represent Real World Problems Lorena and Benita are saving mone. The began on the same da. Lorena started with $. Each week she adds $. The graph describes Benita s savings plan. Which girl will have more mone in 6 weeks? How much more will she have? Eplain our reasoning. Amount saved ($) Time (weeks) Module 6 27 Lesson 1

10 H.O.T. Focus on Higher Order Thinking 25. Analze Relationships Julio and Jake start their reading assignments the same da. Jake is reading a 16-page book at a rate of 2 pages per da. Julio s book is 1 pages long and his reading rate is 1 1 times Jake s rate. After 5 das, who will have more pages left to read? How man more? Eplain our reasoning. 26. Eplain the Error John has $2 in his bank account when he gets a job. He begins making $17 dollars a da. A student found that the equation that represents this situation is = What is wrong with the student s equation? Describe and correct the student s error. 27. Justif Reasoning Is it possible to write the equation of ever line in slopeintercept form? Eplain our reasoning. Lesson Performance Task The graph shows the cost of a gm membership in each of two ears. What are the values that represent the sign-up fee and the membership monthl fee? How did the values change between the ears? a. Write an equation in slope-intercept form for each of the two lines in the graph. b. What are the values that represent the sign-up fee and the membership cost? How did the values change between the ears? Cost (Dollars) Gm Memberships Year 2 Year Time (Months) Houghton Mifflin Harcourt Publishing Compan Module 6 2 Lesson 1