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1 SlopeIntercept Form Determining the Rate of Change and yintercept Learning Goals In this lesson, you will: Graph lines using the slope and yintercept. Calculate the yintercept of a line when given the slope and one point that lies on the line. Write equations of lines in slopeintercept form if given two points that lie on the line or the slope and one point that lies on the line. Write equations in pointslope form if given the slope and one point that lies on the line. Graph lines in standard form by using the intercepts. Convert equations from pointslope form and standard form to slopeintercept form. Discuss the advantages and disadvantages of pointslope and standard form. Key Terms pointslope form standard form A synonym is a word that has the same or almost the same definition of another word. An example of synonyms is "prefer" and "like." In many cases, journalists use synonyms if their writing has many words that repeat within an article or a blog. Sometimes, synonyms can also make an awkwardly written article read more smoothly. Of course, literary critics may sometimes criticize a writer for using too complex synonyms when more common words could easily be used. Can you think of other advantages and disadvantages for using synonyms? 3.6 Determining the Rate of Change and yintercept Determining the Rate of Change and yintercept 195
2 Problem 1 In a previous lesson students were introduced to rise run to determine rate of change from a graph and the last lesson dealt with the yintercept. In this lesson, those two concepts are connected as students use slopeintercept form to graph the equations of lines. Ask a student to read the introduction aloud. Complete the example using steps 1 through 5 as a class. Problem 1 Using SlopeIntercept Form to Graph a Line As you learned previously, the slopeintercept form of a linear equation is where m is the slope of the line. However, you did not learn what b represented. In the slopeintercept form, b is the yintercept of the line. Remember that the slope of the line is the "steepness" of that line. Douglas is giving away tickets to a concert that he won from a radio station contest. Currently, he has 10 tickets remaining. He gives a pair of tickets to each person who asks for them. An equation to represent this context is: y 5 number of tickets available x 5 number of people who request tickets y 5 22x 1 10 y Notice the equation is written in slopeintercept form. 12 Discuss Phase, Example Could the slope be written another way? When the slope is positive, how could you rewrite it using negative signs? x Follow these steps to graph the equation: Step 1: Write the coordinates for the yintercept. b 5 10; (0, 10) Step 2: Plot the yintercept on the coordinate plane shown. Step 3: Write the slope as a ratio. m or 2 21 Step 4: Use the slope and count from the yintercept. To identify another point on the graph, start at the yintercept and count either down (negative) or up (positive) for the rise. Then, count either left (negative) or right (positive) for the run. (1, 8) Continue the counting process to plot the next points. (2, 6), (3, 4), (4, 2), (5, 0) Step 5: Connect the points to make a straight line. 196 Chapter 3 Analyzing Linear Equations 196 Chapter 3 Analyzing Linear Equations
3 Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Share Phase, Questions 1 through 3 How does the placement of negative signs in both the numerator and denominator affect where you place the points on the graph? How do you get the points on the left side of the yaxis? After graphing the line, how can you use the equation to verify that your graph is correct? Graph each line. Be careful to take into account the scales on the axes. 1. y x y x y y x86 42 x86 First think about the y intercept and then interpret the slope y 5 10x y How will you know by the equation if your graph will go up to the right or down to the right? x Determining the Rate of Change and yintercept Determining the Rate of Change and yintercept 197
4 Problem 2 Students write an equation when the slope and a point on the line other than the yintercept are provided. They will calculate the yintercept using the slopeintercept form of a line along with the given two pieces of information and their algebra skills. Ask a student to read the introduction aloud. Discuss the worked example as a class. Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Problem 2 Using SlopeIntercept Form to Calculate the yintercept So far, you have been able to determine the yintercept of a line when given the linear equation in the slopeintercept form. However, you can determine the yintercept of a line when given the slope of that line and one point that lies on the line. Given: m and the point (4, 5) that lies on the line. 2 Step 1: Substitute the values of m, x, and y into the equation for a line. The x and yvalues are obtained from the point that is given (4) 1 b Step 2: Solve the equation for b. The yintercept is b b b Discuss Phase, Worked Example Graph the line given the slope and point provided. Determine the yintercept from the graph. Why might graphing not always be the best method for determining the yintercept? How does the algebraic method demonstrated in the example compare to the graphing method? Write the equation of the line in slopeintercept form. 198 Chapter 3 Analyzing Linear Equations Share Phase, Questions 1 through 3 Write the equation of the line in slopeintercept form. What would the graph look like? 7719_CL_C3_CH03_pp indd 198 Calculate the yintercept of each line when given the slope and one point that lies on the line. 1. m 5 9; (2, 11) 2. m ; (16, 252) (2) 1 b (2.25)(16) 1 b b b b b b b 3. m 5 23 ; (50, 7) (50) 1 b b b b Can you estimate where the line will cross the axis, y based on the slope and one point? Is knowing the slope and yintercept any more helpful than knowing the slope and any other point that lies on the line? Explain. 13/03/14 11:26 AM 198 Chapter 3 Analyzing Linear Equations
5 Problem 3 Students write equations when two points that lie on the line are provided, rather than the slope and the yintercept. Students will calculate the slope using y 2 y 2 1 x 2 2 x. They then 1 calculate the yintercept using slopeintercept form and the process practiced in Problem 2. Ask a student to read the introduction aloud. Discuss the worked example as a class. Problem 3 Using SlopeIntercept Form to Write Equations of Lines So far, you have determined the yintercept from the slopeintercept form of a linear equation, and the yintercept from the slope and a point on that lies on the line given. Now, you will write the equation of a line when given two points that lie on the line. Given: Points (15, 213) and (5, 27) that lie on a line. Step 1: Calculate the slope using y 2 y 2 1 x 2 2 x (213) So, this time you have to calculate the slope first. Discuss Phase, Worked Example Do the two points provide enough information to graph the line? Explain. Do the two points provide enough information to visualize the graph of the line without graphing it? Can you tell whether it is increasing or decreasing? Can you tell where it is located on the coordinate plane? How do you know what point to use when calculating the yintercept? Is it easier to visualize the graph from the two points or it s equation in slopeintercept form? Explain. What two pieces of information are most helpful when graphing the equation of a line? Step 2: Calculate the yintercept by using the slope and one of the points (5) 1 b b b b Step 3: Substitute m and b into the equation. y 5 24x 1 47 The equation for a line in which points (15, 213) and (5, 27) lie on that line is y 5 24x Determining the Rate of Change and yintercept Determining the Rate of Change and yintercept 199
6 Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. Share Phase, Questions 1 through 4 How did you decide what point to use when calculating the yintercept? Does it matter whether you calculate the slope or yintercept first? Explain. What would the graph of this equation look like? For Questions 3 and 4: Was there a more efficient way to solve this problem using the information provided? Write an equation of a line using the given information. Show your work. 1. (7, 15) and (239, 28) m (7) 1 b b b b y x (429, 956) and (249, 836) m (249) 1 b b b y x (6, 19) and (0, 235) m The yintercept is given: (0, 235) y 5 9x Chapter 3 Analyzing Linear Equations 200 Chapter 3 Analyzing Linear Equations
7 4. The slope is 28. The point (3, 12) lies on the line (3) 1 b b 36 5 b y 5 28x 1 36 Problem 4 The pointslope form of a linear equation is derived from the slope formula. Students use the pointslope form to write equations of lines when given a point and the slope; however, they still convert the equation to slopeintercept form in order to determine the yintercept and visualize its graph. Students will evaluate the usefulness of the pointslope form of a linear equation. Problem 4 Another Form of a Linear Equation Let s develop a second form of a linear equation. Step 1: Begin with the formula for slope. m 5 y 2 y 2 1 x 2 2 x 1 Step 2: Rewrite the equation to remove the fraction by multiplying both sides of the equation by (x 2 2 x 1 ). Step 3: After simplifying, the result is: m(x 2 2 x 1 ) 5 ( y 2 y 2 1 x 2 2 x 1 ) (x 2 x ) 2 1 Ask a student to read the introduction aloud. Discuss the worked example and complete Questions 1 through 3 as a class. m(x 2 2 x 1 ) 5 (y 2 2 y 1 ) Step 4: Remove the subscripts for the second point. m(x 2 x 1 ) 5 (y 2 y 1 ) The formula m( x 2 x 1 ) 5 ( y 2 y 1 ) is the pointslope form of a linear equation that passes through the point (x 1, y 1 ) and has slope m. Step 5: Finally, substitute the values for m, x, and y into the pointslope form of the equation. The x and yvalues should be substituted in for x 1 and y Determining the Rate of Change and yintercept Determining the Rate of Change and yintercept 201
8 Discuss Phase, Questions 1 through 3 What would the graph look like using the pointslope form of the equation of a line? What two pieces of information are most helpful when visualizing the graph of a line? What other method could you use to write the equation of a line given a point and a slope? 1. Write the equation of a line in pointslope form with a slope of 28 and the point (3, 12) that lies on the line. m(x 2 x 1 ) 5 (y 2 y 1 ) 28(x 2 3) 5 y While this equation took little time to write, it is difficult to visualize its graph or even its yintercept. To determine the yintercept, manipulate the equation using algebra to write the equation in form. Show all work. 28(x 2 3) 5 y x y x y y 5 28x What is the yintercept of this line? The yintercept is (0, 36). Have students complete Questions 4 and 5 with a partner. Then share the responses as a class. 4. Write the equation of each line in pointslope form. Then, state the yintercept of the line. Show all work. a. slope 5 25; (16, 32) lies on the line m(x 2 x 1 ) 5 ( y 2 y 1 ) 25(x 2 16) 5 y x y x y The yintercept is (0, 112). Share Phase, Questions 4 and 5 Is the yintercept obvious when an equation is written in pointslope form? Explain. What information is obvious when an equation is written in pointslope form? Write the equation of the line and determine the yintercept using another method. What b. m 5 2 ; (9, 218) lies on the line 3 m(x 2 x 1 ) 5 ( y 2 y 1 ) 2 (x 2 9) 5 y x y x y The yintercept is (0, 224). method do you prefer? Why? 202 Chapter 3 Analyzing Linear Equations 202 Chapter 3 Analyzing Linear Equations
9 c. rate of change is 24.5; (280, 55) lies on the line m(x 2 x 1 ) 5 ( y 2 y 1 ) 24.5(x 1 80) 5 y x y x y x y The yintercept is (0, 2305). 5. What are the advantages and disadvantages of using pointslope form? The advantage of pointslope form is that it is easy to write the equation. The disadvantage is that I cannot determine the yintercept from this form. I still have to convert it to slopeintercept form to know the yintercept and graph it. Problem 5 The standard form of a linear equation is defined. Students interpret an equation written in standard form. Students will graph a line in standard form using the intercepts rather than converting it to slopeintercept form. Problem 5 Exploring Standard Form of a Linear Equation Tickets for the school play cost $5.00 for students and $8.00 for adults. On opening night, $1600 was collected in ticket sales. This situation can be modeled by the equation 5x 1 8y You can define the variables as shown. x 5 number of student tickets sold y 5 number of adult tickets sold This equation was not written in slopeintercept form. It was written in standard form. Have students complete Questions 1 through 8 with a partner. Then share the responses as a class. The standard form of a linear equation is Ax 1 By 5 C, where A, B, and C are constants and A and B are not both zero. 1. Explain what each term of the equation represents in the problem situation. 5x is the cost of student tickets multiplied by the number of student tickets sold. 8y is the cost of adult tickets multiplied by the number of adult tickets sold is the total collected in ticket sales. Share Phase, Question 1 What is different about this question that makes it easier to write in standard form? 3.6 Determining the Rate of Change and yintercept 203 How can you tell if an equation is written in standard form or slopeintercept form? 3.6 Determining the Rate of Change and yintercept 203
10 2. What is the independent variable? What is the dependent variable? Explain your reasoning. In this context, either variable could be the independent variable or dependent variable. The number of student tickets sold could depend upon the number of adult tickets sold to get to the $1600 collected in ticket sales, or vice versa. Remember, the x intercept crosses the x axis so the value of y is 0. The y intercept crosses the y axis so the value of x is Calculate the xintercept and yintercept for this equation. Show your work. 5x 1 8y x 1 8y x 1 8(0) (0) 1 8y x y x y The xintercept is (320, 0). The yintercept is (0, 200). 4. What are the meanings of the xintercept and yintercept? The xintercept means that if 320 student tickets are sold, then no adult tickets were sold to collect the $1600. The yintercept means that if 200 adult tickets are sold, then no student tickets were sold to collect the $ Chapter 3 Analyzing Linear Equations 204 Chapter 3 Analyzing Linear Equations
11 Share Phase, Questions 5 through 7 Explain why this line could have been graphed with the axes reversed. Does the coefficient of x represent the slope in all equations of lines? Explain. 5. Use the xintercept and yintercept to graph the equation of the line. Number of Adult Tickets Sold y Number of Student Tickets Sold x 6. What is the slope of this line? Show your work. y 2 2 y 1 x 2 2 x x 1 8y y 5 25x y x What does the slope mean in this problem situation? The slope is It represents the fact that the number of adult tickets sold decreases by 5 for every 8 student tickets sold. 3.6 Determining the Rate of Change and yintercept Determining the Rate of Change and yintercept 205
12 Share Phase, Question 8 How could the graph have been used to solve this question? 8. If 100 student tickets were sold, how many adult tickets were sold? Show your work. If 100 student tickets were sold, then about 138 adult tickets must have been sold to equal the $1600 in sales. I used the equation 5x 1 8y (100) 1 8y y y y y Talk the Talk Students match graphs of lines to their standard form equations. Students evaluate the usefulness of the standard form of a linear equation. Have students complete Questions 1 and 2 with a partner. Then share the responses as a class. Talk the Talk Notice that there are no values on the x and yaxis. What strategies can you use to determine which graph goes with which equation? 1. Match each graph with the correct equation written in standard form. Show your work and explain your reasoning. y Line 1 Line 2 Line 3 a. 3x 2 12y in slopeintercept form is y x 1 5. b. 6x 2 2y in slopeintercept form is y 5 3x 1 5. x a. 3x 2 12y b. 6x 2 2y c. 9x 2 9y Chapter 3 Analyzing Linear Equations 206 Chapter 3 Analyzing Linear Equations
13 Share Phase, Talk the Talk What form are the equations written in? What form should the equations be rewritten in? Which form would be most helpful? Explain. What do the graphs of all three lines have in common? What is different about all three graphs? How can you determine which line matches each equation without having a scale on either axes? How is knowing the slopes of each equation helpful? c. 9x 2 9y in slopeintercept form is y 5 x 1 5. Because all three lines had the same yintercept, I needed to use the slopes of the lines to distinguish among them. I converted the equations in standard form to slopeintercept form. The one with the greatest coefficient of x matches the steepest line, so equation (b) matches line 1. Following this reasoning, equation (c) matches line 2. Equation (a) matches line What are the advantages and disadvantages of using standard form? One advantage of standard form is that it is easy to write the equation for some types of contexts. Another advantage of standard form is that it is easy to calculate both the xintercept and yintercept, and then use them to graph the equation. The disadvantage is that you still have to convert it to slopeintercept form to know the slope. Be prepared to share your solutions and methods. 3.6 Determining the Rate of Change and yintercept Determining the Rate of Change and yintercept 207
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