Find the Slope of a Line from its Graph. Find the Slope of a Line given Two Points

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1 Math "Determining the Equation of a Line" Objectives: * Find the slope of a line determined b two points. * Write the equation of a line, given the slope and a point contained in the line. * Write the equation of a line, given two points contained in the line. * Write the equation of a line parallel or perpendicular to another line and containing a speci ed point. * Solve application problems involving the slope of a line. Find the Slope of a Line from its Graph De nition: "Slope of a Line" The slope of a line (m) is a ratio that compares the vertical change to the corresponding horizontal change as we move along the line from one point to another: Eample 1: (Finding the slope of a line from its graph) Find the slope of the line graphed below Find the Slope of a Line given Two Points De nition: "Slope of a Line" If P 1 and P 2 are an two di erent points on a line, P 1 with coordinates ( 1 ; 1 ) and P 2 with coordinates ( 2 ; 2 ) ; then the slope of the line (denoted b m) is where 1 6= 2 : Eample 2: (Finding the slope of a line given two points) Find the slope of the line that passes through the given points. a) (3; 1) ; ( 6; 2) b) ( 3; 6) ; (; 8) Page: 1 Notes b Bibiana Lopez

2 College Algebra b Kaufmann and Schwitters 2.3 Find the Slope of Horizontal and Vertical Lines Eample 3: (Finding the slope of a horizontal line) Find the slope of the following lines. a) = 3 b) = Slopes of Horizontal and Vertical Lines: Horizontal lines have a slope of Vertical lines have Tpes of Slopes: Positive Slope Negative Slope Zero Slope Unde ned Slope m > 0 m < 0 m H = 0 m V = undefined Application of Slope The concept of slope has man real-world applications even though the word "slope" is often not used. For eample, hospital beds are constructed so that both the head end and the foot end can be raised or lowered; that is, the slope of either end of the bed can be change. A roofer, when making an estimate to replace a roof, is concerned not onl about the total area to be covered but also about the pinch of the roof. The concept of slope is also used in the construction of stairwas. In highwa construction, the word grade is used to describe the slope. Page: 2 Notes b Bibiana Lopez

3 College Algebra b Kaufmann and Schwitters 2.3 Eample : (Applied problem) A certain highwa has a 3% grade. How man feet does it rise in a horizontal distance of 1 mile? Use Slope-Intercept Form to Write the Equation of a Line Slope-Intercept Form: The equation of the line with slope m and intercept (0; b) is Eample 5: (Using the slope-intercept form) Use the slope-intercept form to write an equation of the line with slope 5 and intercept 0; 1 : Write the answer in standard form. Eample 6: (Using the slope-intercept form) Find the slope and -intercept of the line graphed below. Write the answer in standard form. Then use the slope-intercept form to write the equation of the line Page: 3 Notes b Bibiana Lopez

4 College Algebra b Kaufmann and Schwitters 2.3 Eample 7: (Using the slope-intercept form) Use the slope-intercept form to write an equation of the line that has the given slope and passes through the given point. a) slope 3; and passes through ( 2; 5) b) slope 2; and passes through ( 2; 8) Eample 8: (Using the slope-intercept form) Find the slope and the intercept of each line. a) 2 + = 9 b) 5 = 0 Use the Point-Slope Form to Write the Equation of a Line Point-Slope Form: The equation of the line passing through ( 1 ; 1 ) and with slope m is Eample 9: (Using the point-slope form) Find an equation of the line that has slope 5 and passes trough (2; 6). Write the equation in standard form. Page: Notes b Bibiana Lopez

5 College Algebra b Kaufmann and Schwitters 2.3 Eample 10: (Using the point-slope form) Find an equation of the line passing through ( 2; 5) and (; 3). Write the equation in slope-intercept form. Eample 11: (Application problems) After purchasing a new drill press, a machine shop owner had his accountant prepare a depreciation worksheet for ta purposes. Depreciation Worksheet Drill press... $1,970 (new) Salvage value... $270 (in 10 ears) a) Assuming straight-line depreciation, write an equation that gives the value (v) of the drill press after ears of use. b) Find the value of the drill press after ears of use. Use Slope as an Aid When Graphing If we know the slope and the intercept of a line, we can graph the line without constructing a table of solutions. Eample 12: (Use slope as an aid when graphing) Find the slope and -intercept of the line with the equation 3 2 = and graph the line. Page: 5 Notes b Bibiana Lopez

6 College Algebra b Kaufmann and Schwitters 2.3 Parallel and Perpendicular Lines Properties of Parallel and Perpendicular Lines: If two nonvertical lines have slopes of m 1 and m 2; then 1: The two lines are parallel if and onl if 2: The two lines are perpendicular if and onl if Eample 13: (Determining whether lines are parallel or perpendicular) Determine whether the following lines are parallel, perpendicular or neither. 2 = 5 3 = 12 a) b) 2 = = 27 Eample 1: (Finding equations of parallel and perpendicular lines) Write the equation of each line that satis es the given conditions. Write the answer in standard form. a) Contains the point (1; 2) and is parallel to the line 3 = 5 b) Contains the point ( 3; 5) and is perpendicular = Page: 6 Notes b Bibiana Lopez