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4-7 Point-Slope Form Connection: Relating Slope-Intercept, Point-Slope, and Standard Forms Essential question: What properties of linear functions does each linear function form illustrate? TEACH Standards for Mathematical Content 1 F-IF.3.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F-LE.1.2 Construct linear … functions … given … two input-output pairs (include reading these from a table).* EXAMPLE Questioning Strategies • Would you use standard form or the slopeintercept form to find the x-intercept? Explain. Standard form; to find the x-intercept, substitute 0 for y, and solve the equation for x. • Could you use standard form to graph the equation in part A? Explain. Yes; you can use Prerequisites Identifying Linear Functions Using Intercepts The Slope Formula Slope-Intercept Form standard form to find the x- and y-intercepts and use those to graph the equation. EXTRA EXAMPLE Write an equation for 9x - 6y = 18 in slopeintercept form. Then use that form to identify the slope and y-intercept. y = _ 32 x - 3; slope: _32 , Math Background Students have learned three different forms for writing linear equations. In standard form, Ax + By = C where A, B, and C are real numbers and A and B are not both 0. In slope-intercept form, y = mx + b where m is the slope of the line and b is the y-intercept. In point-slope form, y - y1 = m(x - x1 ) where m is the slope of the line and (x1 , y1 ) is a point on the line. y-intercept: -3 2 EXAMPLE No; however, using standard form will take fewer steps than using point-slope form directly to find the x- and y-intercepts. INTR O D U C E • Would you use the standard form or the pointslope form to graph the equation? Explain. Point- Tell students they will be given a linear equation to graph. Ask them what information they need to know about the line in order to graph it. Make a list of the students’ responses on the board. (The list should include slope, y-intercept, x-intercept, and points on the line.) Then tell students they can pick the form of the equation (standard, slope-intercept, or point-slope). Ask students which form they would pick and why. slope form; the slope and a point on the line are easily seen in the point-slope form, and the slope and a point on the line are used to graph the line. EXTRA EXAMPLE Write an equation for y - 6 = -2(x - 1) in standard form. Then use that form to identify the x- and y-intercepts. 2x + y = 8; x-intercept: 4, y-intercept: 8 Avoid Common Errors When multiplying to eliminate a fraction in an equation, remind students to multiply both sides of the equation by the number, not just the side of the equation with the fraction. Similarly, when dividing a variable term by a number to isolate a variable, remind students to divide every term on both sides of the equation by that number. Chapter 4 225 Lesson 7 © Houghton Mifflin Harcourt Publishing Company Questioning Strategies • Do you have to use the standard form of the equation to find the x- and y-intercepts? Explain. Name Class Notes 4-7 Date Point-Slope Form Connection: Relating Slope-Intercept, Point-Slope, and Standard Forms Essential question: What properties of linear functions does each linear function form illustrate? F-IF.3.8 1 EXAMPLE Writing Equations in Slope-Intercept Form Write an equation for -4x + 5y = 10 in slope-intercept form. Then use that form to identify the slope and y-intercept. Rewrite the equation in slope-intercept form by solving for y. A -4x + 5y = 10 Standard form of the equation 5y = 4x + 10 y= __4 x 5 10 + __ 5 4x + 2 y = __ 5 B Add 4x to each side. Divide each side by 5. Simplify. Identify the slope and y-intercept of the line. The equation is in the form y = mx + b, where m is the slope and b is the y-intercept. So, the slope of the line is __4 5 and the y-intercept is 2 . © Houghton Mifflin Harcourt Publishing Company REFLECT 1a. In the Example, which equation form would you use to graph the equation? Explain. Slope-intercept form; the slope and y-intercept are most helpful when graphing a line, and the slope and y-intercept are easily determined when the equation is in slope-intercept form. 1b. Rewrite Ax + By = C in slope-intercept form. Explain how you can use this form to identify the slope and y-intercept of a line when its equation is given in standard form. A A y = - __ x + _C ; in the rewritten equation, - __ represents the slope of the line and _C B B B B represents the y-intercept of the line. For a linear equation given in standard form, C A calculate - __ for the slope and _ for the y-intercept. B B Chapter 4 225 Lesson 7 F-IF.3.8 2 EXAMPLE Writing Equations in Standard Form 2 (x - 9) in standard form. Then use that form to Write an equation for y - 4 = __ 3 identify the x- and y-intercepts. Rewrite the equation in standard form by collecting the x- and y-terms on one side of the equation. y - 4 = __23 (x - 9) 3(y - 4) = 3 r__23 (x - 9) 3y - 12 = 2x - 18 3y = 2x - 6 -2x + 3y = -6 B Multiply each side by 3. Simplify. Add 12 to each side. Subtract 2x from each side. Find the x-intercept by substituting 0 for y and solving for x. ( -2x + 3 0 ) = -6 -2x = -6 x= The x-intercept is C Point-slope form of the equation 3 Substitute 0 for y. Simplify. Solve for x. 3 . Find the y-intercept by substituting 0 for x and solving for y. ( -2 0 ) + 3y = -6 3y = -6 y = -2 Substitute 0 for x. Simplify. Solve for y. The y-intercept is -2 . REFLECT 2a. Find the x-intercept of the line in the Example by substituting 0 for y and solving for x in the point-slope form of the equation. Is it easier to find the x-intercept using the standard form or the point-slope form? Explain your answer. Sample answer: The x-intercept is 3; standard form; there are fewer mathematical calculations needed to solve for x when using standard form. © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company A 2b. Explain why you substitute 0 for x in the equation to find the y-intercept. The y-intercept of the graph of a line is where the graph crosses the y-axis. The point that corresponds to the y-intercept b is (0, b). To find the value of b, you have to substitute 0 for x in the equation and solve for y. Chapter 4 Chapter 4 226 Lesson 7 226 Lesson 7 3 Summarize Have students complete the table by stating which form of a linear equation they would use to write an equation for the line with the given information. EXAMPLE Questioning Strategies • In Part C, what is the point-slope form if you substitute (-4, 9)? How can you show that this equation is equivalent to the point-slope equation using (8, 6)? y - 9 = -__14(x + 4); Change both point-slope equations to either standard form or slope-intercept form. • Suppose the slope and a point on the line were given. Which equation form would you use to write an equation for the line? Explain. If the point were the y-intercept, I would use slope-intercept form because the slope and the y-intercept can be substituted directly into this form. If the point were not the y-intercept, I would use point-slope form because the slope and the point can be substituted directly into this form. Given Information Equation Form slope and y-intercept slope-inter. two points on the line point-slope a horizontal line and one point on the line standard slope and point (x, y) on the line where x ≠ 0 point-slope a vertical line and one point on the line standard slope and point (x, y) on the line where x = 0 slope-inter. Highlighting the Standards EXTRA EXAMPLE Determine which form of a linear equation to use to write an equation for the line with a slope of __45 that passes through (0, -5). Then write an equation for the line. slope-intercept form; y = _45 x - 5 Teaching Strategy Before beginning 3 EXAMPLE , have students read the summary table at the bottom of the page. Discuss how this information is helpful in choosing an appropriate equation form when writing a linear equation. To reinforce students’ understanding, ask general questions such as “Which form will you use to write a linear equation given the slope and the y-intercept of the line?” CLOSE Essential Question What properties of linear functions does each linear function form illustrate? PR ACTICE Slope-intercept form shows the slope and the y-intercept of the line. Point-slope form shows the slope and a point that lies on the line. Standard form shows whether a line is vertical or horizontal and can be used to find the x- and y-intercepts of lines that are not horizontal or vertical. Where skills are taught Where skills are practiced 1 EXAMPLE EXS. 2, 3 2 EXAMPLE EXS. 1, 4 3 EXAMPLE EXS. 5-8 Exercise 9: Students determine whether it is easier to transform an equation given in standard form to slope-intercept form or to point-slope form. Exercise 10: Students write formulas for finding the x- and y-intercepts of any line given in standard form. Chapter 4 227 Lesson 7 © Houghton Mifflin Harcourt Publishing Company Exercises 1-4 provide opportunities to address Mathematical Practices Standard 2 (Reason abstractly and quantitatively) and Standard 7 (Look for and make use of structure). Using properties of equality, students manipulate a linear equation so that it is written in a specific form. Then students analyze the terms in the rewritten equation. These terms illustrate characteristics of the line, such as its slope or y-intercept. Students should understand that transforming a linear equation does not change the characteristics of the line—it changes the form of the equation so that the characteristics of the line are easily determined. Notes F-LE.1.2 3 EXAMPLE Choosing an Appropriate Form of a Linear Equation Determine which form of a linear equation to use to write an equation for the line that passes through (-4, 9) and (8, 6). Then write an equation for the line. A To determine which form to use, identify the given information. Two points on the line are given, and neither of them is the y-intercept. So, write the point-slope equation using B 9 6- y -y 2 1 ____________ = m = ______ x -x = 2 C form. Find the slope of the line using the two points. 8 1 - (-4) -3 ___ 12 y - y1 = m(x - x1) 6 __1 =4 4 point-slope Write the equation in y- 1 = - __ form using the point (8, 6). General form of the equation (x - 8 ) Substitute values for x1, y1, and m. REFLECT 3a. Could (-4, 9) have been used to write an equation of the line in Part C? Explain. Yes; both given points are on the line so either point can be used to write an equation for the line. 3b. Suppose you are given the y-intercept and the slope of a line. Which linear equation form would you use to write an equation for the line? Explain your reasoning. © Houghton Mifflin Harcourt Publishing Company Slope-intercept form; the given information can be substituted in the general slope-intercept form and no calculations need to be made. The following table provides a summary of the information presented in the Examples above. Form Equation Information When to Use slope-intercept y = mx + b · m is the slope. · b is the y-intercept. · When given the slope and the y-intercept point-slope y - y1 = m(x - x1) · m is the slope. · (x1, y1) lies on the line. · When given the slope and one point on the line · When given two points on the line standard Ax + By = C · A, B, and C are real numbers. · A and B are not both 0. · When given a horizontal or vertical line and one point on the line Chapter 4 227 Lesson 7 PRACTICE Rewrite the equation to find the characteristics of the line. 1. Rewrite y = -__32 x + 6 in standard form. Identify the x-intercept of the line. 7 7 x + 2; the slope is _ . The y-intercept is 2. y=_ 9 9 3. Rewrite y + 1 = 4(x + 3) in slope-intercept form. Identify the y-intercept of the line. y = 4x + 11; the y-intercept is 11. 4. Rewrite y + 25 = -__53(x - 12) in standard form. Identify the x- and y-intercepts of the line. 5x + 3y = -15; the x-intercept is -3. The y-intercept is -5. Determine which form of a linear equation to use to write an equation for the line with the given characteristics. Then write an equation for the line in that form. 5. passes through (-5, -4), (7, 5) 3 (x + 5) point-slope form; y + 4 = _ 6. slope = -__27 ; y-intercept = 9 2 slope-intercept form; y = -_ x+9 7 4 11 ; passes through (-3, 6) 8. slope = __ 13 7. vertical line through (8, 1) 11 (x + 3) point-slope form; y - 6 = __ standard form; x = 8 13 9. You are asked to rewrite -9x + 4y = 14 to find the slope of the line. Would you rewrite the equation in slope-intercept form or point-slope form? Explain. Possible answer: Slope-intercept form; writing the equation in slope-intercept form will take fewer steps than writing the equation in point-slope form. 10. Find the x- and y-intercepts of the graph of Ax + By = C. Explain how you can use these intercepts to find the x- and y-intercepts of any line in standard form. C C The x-intercept is __ , and the y-intercept is _ ; for an equation written in standard A B C C form, substitute values in __ to find the x-intercept of the line and in _ to find the A B © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company 3x + 2y = 12; the x-intercept is 4. 2. Rewrite -7x + 9y = 18 in slope-intercept form. Identify the slope and y-intercept of the line. y-intercept of the line. Chapter 4 Chapter 4 228 Lesson 7 228 Lesson 7 Problem Solving ADD I T I O NA L P R AC TI C E AND PRO BL E M S O LV I N G 1. Possible answer: y - 130 = 1.2(x - 10); y = 1.2x + 118; 136 Assign these pages to help your students practice and apply important lesson concepts. For additional exercises, see the Student Edition. 2. y = 3x + 32 Answers 1 x + 5; $7.50 3. y = _ 10 Additional Practice 4. B 1. y - 2 = 3(x + 4) 2. y + 1 = -(x - 6) 5. F 6. 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